Daily Papers

We show that deep belief networks with binary hidden units can approximate any multivariate probability density under very mild integrability requirements on the parental density of the visible nodes. The approximation is measured in the L^q-norm for qin[1,infty] (q=infty corresponding to the supremum norm) and in Kullback-Leibler divergence. Furthermore, we establish sharp quantitative bounds on the approximation error in terms of the number of hidden units.

Zero-shot and in-context learning enable solving tasks without model fine-tuning, making them essential for developing generative model solutions. Therefore, it is crucial to understand whether a pretrained model can be prompted to approximate any function, i.e., whether it is a universal in-context approximator. While it was recently shown that transformer models do possess this property, these results rely on their attention mechanism. Hence, these findings do not apply to fully recurrent architectures like RNNs, LSTMs, and the increasingly popular SSMs. We demonstrate that RNNs, LSTMs, GRUs, Linear RNNs, and linear gated architectures such as Mamba and Hawk/Griffin can also serve as universal in-context approximators. To streamline our argument, we introduce a programming language called LSRL that compiles to these fully recurrent architectures. LSRL may be of independent interest for further studies of fully recurrent models, such as constructing interpretability benchmarks. We also study the role of multiplicative gating and observe that architectures incorporating such gating (e.g., LSTMs, GRUs, Hawk/Griffin) can implement certain operations more stably, making them more viable candidates for practical in-context universal approximation.

It has been shown that deep neural networks of a large enough width are universal approximators but they are not if the width is too small. There were several attempts to characterize the minimum width w_{min} enabling the universal approximation property; however, only a few of them found the exact values. In this work, we show that the minimum width for L^p approximation of L^p functions from [0,1]^{d_x} to mathbb R^{d_y} is exactly max{d_x,d_y,2} if an activation function is ReLU-Like (e.g., ReLU, GELU, Softplus). Compared to the known result for ReLU networks, w_{min}=max{d_x+1,d_y} when the domain is mathbb R^{d_x}, our result first shows that approximation on a compact domain requires smaller width than on mathbb R^{d_x}. We next prove a lower bound on w_{min} for uniform approximation using general activation functions including ReLU: w_{min}ge d_y+1 if d_x<d_yle2d_x. Together with our first result, this shows a dichotomy between L^p and uniform approximations for general activation functions and input/output dimensions.

The study of universal approximation properties (UAP) for neural networks (NN) has a long history. When the network width is unlimited, only a single hidden layer is sufficient for UAP. In contrast, when the depth is unlimited, the width for UAP needs to be not less than the critical width w^*_{min}=max(d_x,d_y), where d_x and d_y are the dimensions of the input and output, respectively. Recently, cai2022achieve shows that a leaky-ReLU NN with this critical width can achieve UAP for L^p functions on a compact domain K, i.e., the UAP for L^p(K,R^{d_y}). This paper examines a uniform UAP for the function class C(K,R^{d_y}) and gives the exact minimum width of the leaky-ReLU NN as w_{min}=max(d_x+1,d_y)+1_{d_y=d_x+1}, which involves the effects of the output dimensions. To obtain this result, we propose a novel lift-flow-discretization approach that shows that the uniform UAP has a deep connection with topological theory.

A recurrent neural network (RNN) is a widely used deep-learning network for dealing with sequential data. Imitating a dynamical system, an infinite-width RNN can approximate any open dynamical system in a compact domain. In general, deep networks with bounded widths are more effective than wide networks in practice; however, the universal approximation theorem for deep narrow structures has yet to be extensively studied. In this study, we prove the universality of deep narrow RNNs and show that the upper bound of the minimum width for universality can be independent of the length of the data. Specifically, we show that a deep RNN with ReLU activation can approximate any continuous function or L^p function with the widths d_x+d_y+2 and max{d_x+1,d_y}, respectively, where the target function maps a finite sequence of vectors in R^{d_x} to a finite sequence of vectors in R^{d_y}. We also compute the additional width required if the activation function is tanh or more. In addition, we prove the universality of other recurrent networks, such as bidirectional RNNs. Bridging a multi-layer perceptron and an RNN, our theory and proof technique can be an initial step toward further research on deep RNNs.

In this work, we propose a concise neural operator architecture for operator learning. Drawing an analogy with a conventional fully connected neural network, we define the neural operator as follows: the output of the i-th neuron in a nonlinear operator layer is defined by mathcal O_i(u) = sigmaleft( sum_j mathcal W_{ij} u + mathcal B_{ij}right). Here, mathcal W_{ij} denotes the bounded linear operator connecting j-th input neuron to i-th output neuron, and the bias mathcal B_{ij} takes the form of a function rather than a scalar. Given its new universal approximation property, the efficient parameterization of the bounded linear operators between two neurons (Banach spaces) plays a critical role. As a result, we introduce MgNO, utilizing multigrid structures to parameterize these linear operators between neurons. This approach offers both mathematical rigor and practical expressivity. Additionally, MgNO obviates the need for conventional lifting and projecting operators typically required in previous neural operators. Moreover, it seamlessly accommodates diverse boundary conditions. Our empirical observations reveal that MgNO exhibits superior ease of training compared to other CNN-based models, while also displaying a reduced susceptibility to overfitting when contrasted with spectral-type neural operators. We demonstrate the efficiency and accuracy of our method with consistently state-of-the-art performance on different types of partial differential equations (PDEs).

The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We formulate the neural operator as a composition of linear integral operators and nonlinear activation functions. We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator. The proposed neural operators are also discretization-invariant, i.e., they share the same model parameters among different discretization of the underlying function spaces. Furthermore, we introduce four classes of efficient parameterization, viz., graph neural operators, multi-pole graph neural operators, low-rank neural operators, and Fourier neural operators. An important application for neural operators is learning surrogate maps for the solution operators of partial differential equations (PDEs). We consider standard PDEs such as the Burgers, Darcy subsurface flow, and the Navier-Stokes equations, and show that the proposed neural operators have superior performance compared to existing machine learning based methodologies, while being several orders of magnitude faster than conventional PDE solvers.

In this paper, we construct a mixture of neural operators (MoNOs) between function spaces whose complexity is distributed over a network of expert neural operators (NOs), with each NO satisfying parameter scaling restrictions. Our main result is a distributed universal approximation theorem guaranteeing that any Lipschitz non-linear operator between L^2([0,1]^d) spaces can be approximated uniformly over the Sobolev unit ball therein, to any given varepsilon>0 accuracy, by an MoNO while satisfying the constraint that: each expert NO has a depth, width, and rank of O(varepsilon^{-1}). Naturally, our result implies that the required number of experts must be large, however, each NO is guaranteed to be small enough to be loadable into the active memory of most computers for reasonable accuracies varepsilon. During our analysis, we also obtain new quantitative expression rates for classical NOs approximating uniformly continuous non-linear operators uniformly on compact subsets of L^2([0,1]^d).

The expressive power of Graph Neural Networks (GNNs) has been studied extensively through the Weisfeiler-Leman (WL) graph isomorphism test. However, standard GNNs and the WL framework are inapplicable for geometric graphs embedded in Euclidean space, such as biomolecules, materials, and other physical systems. In this work, we propose a geometric version of the WL test (GWL) for discriminating geometric graphs while respecting the underlying physical symmetries: permutations, rotation, reflection, and translation. We use GWL to characterise the expressive power of geometric GNNs that are invariant or equivariant to physical symmetries in terms of distinguishing geometric graphs. GWL unpacks how key design choices influence geometric GNN expressivity: (1) Invariant layers have limited expressivity as they cannot distinguish one-hop identical geometric graphs; (2) Equivariant layers distinguish a larger class of graphs by propagating geometric information beyond local neighbourhoods; (3) Higher order tensors and scalarisation enable maximally powerful geometric GNNs; and (4) GWL's discrimination-based perspective is equivalent to universal approximation. Synthetic experiments supplementing our results are available at https://github.com/chaitjo/geometric-gnn-dojo

Relative Positional Encoding (RPE), which encodes the relative distance between any pair of tokens, is one of the most successful modifications to the original Transformer. As far as we know, theoretical understanding of the RPE-based Transformers is largely unexplored. In this work, we mathematically analyze the power of RPE-based Transformers regarding whether the model is capable of approximating any continuous sequence-to-sequence functions. One may naturally assume the answer is in the affirmative -- RPE-based Transformers are universal function approximators. However, we present a negative result by showing there exist continuous sequence-to-sequence functions that RPE-based Transformers cannot approximate no matter how deep and wide the neural network is. One key reason lies in that most RPEs are placed in the softmax attention that always generates a right stochastic matrix. This restricts the network from capturing positional information in the RPEs and limits its capacity. To overcome the problem and make the model more powerful, we first present sufficient conditions for RPE-based Transformers to achieve universal function approximation. With the theoretical guidance, we develop a novel attention module, called Universal RPE-based (URPE) Attention, which satisfies the conditions. Therefore, the corresponding URPE-based Transformers become universal function approximators. Extensive experiments covering typical architectures and tasks demonstrate that our model is parameter-efficient and can achieve superior performance to strong baselines in a wide range of applications. The code will be made publicly available at https://github.com/lsj2408/URPE.

This work considers a rather general and broad class of Markov chains, Ito chains that look like Euler-Maryama discretization of some Stochastic Differential Equation. The chain we study is a unified framework for theoretical analysis. It comes with almost arbitrary isotropic and state-dependent noise instead of normal and state-independent one, as in most related papers. Moreover, our chain's drift and diffusion coefficient can be inexact to cover a wide range of applications such as Stochastic Gradient Langevin Dynamics, sampling, Stochastic Gradient Descent, or Stochastic Gradient Boosting. We prove an upper bound for W_{2}-distance between laws of the Ito chain and the corresponding Stochastic Differential Equation. These results improve or cover most of the known estimates. Moreover, for some particular cases, our analysis is the first.

Recent work on mini-batch consistency (MBC) for set functions has brought attention to the need for sequentially processing and aggregating chunks of a partitioned set while guaranteeing the same output for all partitions. However, existing constraints on MBC architectures lead to models with limited expressive power. Additionally, prior work has not addressed how to deal with large sets during training when the full set gradient is required. To address these issues, we propose a Universally MBC (UMBC) class of set functions which can be used in conjunction with arbitrary non-MBC components while still satisfying MBC, enabling a wider range of function classes to be used in MBC settings. Furthermore, we propose an efficient MBC training algorithm which gives an unbiased approximation of the full set gradient and has a constant memory overhead for any set size for both train- and test-time. We conduct extensive experiments including image completion, text classification, unsupervised clustering, and cancer detection on high-resolution images to verify the efficiency and efficacy of our scalable set encoding framework. Our code is available at github.com/jeffwillette/umbc

Methods of computational quantum chemistry provide accurate approximations of molecular properties crucial for computer-aided drug discovery and other areas of chemical science. However, high computational complexity limits the scalability of their applications. Neural network potentials (NNPs) are a promising alternative to quantum chemistry methods, but they require large and diverse datasets for training. This work presents a new dataset and benchmark called nabla^2DFT that is based on the nablaDFT. It contains twice as much molecular structures, three times more conformations, new data types and tasks, and state-of-the-art models. The dataset includes energies, forces, 17 molecular properties, Hamiltonian and overlap matrices, and a wavefunction object. All calculations were performed at the DFT level (omegaB97X-D/def2-SVP) for each conformation. Moreover, nabla^2DFT is the first dataset that contains relaxation trajectories for a substantial number of drug-like molecules. We also introduce a novel benchmark for evaluating NNPs in molecular property prediction, Hamiltonian prediction, and conformational optimization tasks. Finally, we propose an extendable framework for training NNPs and implement 10 models within it.

Existing analyses of the expressive capacity of Transformer models have required excessively deep layers for data memorization, leading to a discrepancy with the Transformers actually used in practice. This is primarily due to the interpretation of the softmax function as an approximation of the hardmax function. By clarifying the connection between the softmax function and the Boltzmann operator, we prove that a single layer of self-attention with low-rank weight matrices possesses the capability to perfectly capture the context of an entire input sequence. As a consequence, we show that one-layer and single-head Transformers have a memorization capacity for finite samples, and that Transformers consisting of one self-attention layer with two feed-forward neural networks are universal approximators for continuous permutation equivariant functions on a compact domain.

We investigate the expressive power of depth-2 bandlimited random neural networks. A random net is a neural network where the hidden layer parameters are frozen with random assignment, and only the output layer parameters are trained by loss minimization. Using random weights for a hidden layer is an effective method to avoid non-convex optimization in standard gradient descent learning. It has also been adopted in recent deep learning theories. Despite the well-known fact that a neural network is a universal approximator, in this study, we mathematically show that when hidden parameters are distributed in a bounded domain, the network may not achieve zero approximation error. In particular, we derive a new nontrivial approximation error lower bound. The proof utilizes the technique of ridgelet analysis, a harmonic analysis method designed for neural networks. This method is inspired by fundamental principles in classical signal processing, specifically the idea that signals with limited bandwidth may not always be able to perfectly recreate the original signal. We corroborate our theoretical results with various simulation studies, and generally, two main take-home messages are offered: (i) Not any distribution for selecting random weights is feasible to build a universal approximator; (ii) A suitable assignment of random weights exists but to some degree is associated with the complexity of the target function.

Dropout is a widely utilized regularization technique in the training of neural networks, nevertheless, its underlying mechanism and its impact on achieving good generalization abilities remain poorly understood. In this work, we derive the stochastic modified equations for analyzing the dynamics of dropout, where its discrete iteration process is approximated by a class of stochastic differential equations. In order to investigate the underlying mechanism by which dropout facilitates the identification of flatter minima, we study the noise structure of the derived stochastic modified equation for dropout. By drawing upon the structural resemblance between the Hessian and covariance through several intuitive approximations, we empirically demonstrate the universal presence of the inverse variance-flatness relation and the Hessian-variance relation, throughout the training process of dropout. These theoretical and empirical findings make a substantial contribution to our understanding of the inherent tendency of dropout to locate flatter minima.

Computers calculate transcendental functions by approximating them through the composition of a few limited-precision instructions. For example, an exponential can be calculated with a Taylor series. These approximation methods were developed over the centuries by mathematicians, who emphasized the attainability of arbitrary precision. Computers, however, operate on few limited precision types, such as the popular float32. In this study, we show that when aiming for limited precision, existing approximation methods can be outperformed by programs automatically discovered from scratch by a simple evolutionary algorithm. In particular, over real numbers, our method can approximate the exponential function reaching orders of magnitude more precision for a given number of operations when compared to previous approaches. More practically, over float32 numbers and constrained to less than 1 ULP of error, the same method attains a speedup over baselines by generating code that triggers better XLA/LLVM compilation paths. In other words, in both cases, evolution searched a vast space of possible programs, without knowledge of mathematics, to discover previously unknown optimized approximations to high precision, for the first time. We also give evidence that these results extend beyond the exponential. The ubiquity of transcendental functions suggests that our method has the potential to reduce the cost of scientific computing applications.

Deep learning techniques are increasingly applied to scientific problems, where the precision of networks is crucial. Despite being deemed as universal function approximators, neural networks, in practice, struggle to reduce the prediction errors below O(10^{-5}) even with large network size and extended training iterations. To address this issue, we developed the multi-stage neural networks that divides the training process into different stages, with each stage using a new network that is optimized to fit the residue from the previous stage. Across successive stages, the residue magnitudes decreases substantially and follows an inverse power-law relationship with the residue frequencies. The multi-stage neural networks effectively mitigate the spectral biases associated with regular neural networks, enabling them to capture the high frequency feature of target functions. We demonstrate that the prediction error from the multi-stage training for both regression problems and physics-informed neural networks can nearly reach the machine-precision O(10^{-16}) of double-floating point within a finite number of iterations. Such levels of accuracy are rarely attainable using single neural networks alone.

Universality is a key hypothesis in mechanistic interpretability -- that different models learn similar features and circuits when trained on similar tasks. In this work, we study the universality hypothesis by examining how small neural networks learn to implement group composition. We present a novel algorithm by which neural networks may implement composition for any finite group via mathematical representation theory. We then show that networks consistently learn this algorithm by reverse engineering model logits and weights, and confirm our understanding using ablations. By studying networks of differing architectures trained on various groups, we find mixed evidence for universality: using our algorithm, we can completely characterize the family of circuits and features that networks learn on this task, but for a given network the precise circuits learned -- as well as the order they develop -- are arbitrary.

We resolve an open problem of Hanneke on the subject of universally consistent online learning with non-i.i.d. processes and unbounded losses. The notion of an optimistically universal learning rule was defined by Hanneke in an effort to study learning theory under minimal assumptions. A given learning rule is said to be optimistically universal if it achieves a low long-run average loss whenever the data generating process makes this goal achievable by some learning rule. Hanneke posed as an open problem whether, for every unbounded loss, the family of processes admitting universal learning are precisely those having a finite number of distinct values almost surely. In this paper, we completely resolve this problem, showing that this is indeed the case. As a consequence, this also offers a dramatically simpler formulation of an optimistically universal learning rule for any unbounded loss: namely, the simple memorization rule already suffices. Our proof relies on constructing random measurable partitions of the instance space and could be of independent interest for solving other open questions. We extend the results to the non-realizable setting thereby providing an optimistically universal Bayes consistent learning rule.

Meta-learning has emerged as a powerful approach to train neural networks to learn new tasks quickly from limited data. Broad exposure to different tasks leads to versatile representations enabling general problem solving. But, what are the limits of meta-learning? In this work, we explore the potential of amortizing the most powerful universal predictor, namely Solomonoff Induction (SI), into neural networks via leveraging meta-learning to its limits. We use Universal Turing Machines (UTMs) to generate training data used to expose networks to a broad range of patterns. We provide theoretical analysis of the UTM data generation processes and meta-training protocols. We conduct comprehensive experiments with neural architectures (e.g. LSTMs, Transformers) and algorithmic data generators of varying complexity and universality. Our results suggest that UTM data is a valuable resource for meta-learning, and that it can be used to train neural networks capable of learning universal prediction strategies.

The lottery ticket hypothesis conjectures the existence of sparse subnetworks of large randomly initialized deep neural networks that can be successfully trained in isolation. Recent work has experimentally observed that some of these tickets can be practically reused across a variety of tasks, hinting at some form of universality. We formalize this concept and theoretically prove that not only do such universal tickets exist but they also do not require further training. Our proofs introduce a couple of technical innovations related to pruning for strong lottery tickets, including extensions of subset sum results and a strategy to leverage higher amounts of depth. Our explicit sparse constructions of universal function families might be of independent interest, as they highlight representational benefits induced by univariate convolutional architectures.

In this manuscript we consider the problem of generalized linear estimation on Gaussian mixture data with labels given by a single-index model. Our first result is a sharp asymptotic expression for the test and training errors in the high-dimensional regime. Motivated by the recent stream of results on the Gaussian universality of the test and training errors in generalized linear estimation, we ask ourselves the question: "when is a single Gaussian enough to characterize the error?". Our formula allow us to give sharp answers to this question, both in the positive and negative directions. More precisely, we show that the sufficient conditions for Gaussian universality (or lack of thereof) crucially depend on the alignment between the target weights and the means and covariances of the mixture clusters, which we precisely quantify. In the particular case of least-squares interpolation, we prove a strong universality property of the training error, and show it follows a simple, closed-form expression. Finally, we apply our results to real datasets, clarifying some recent discussion in the literature about Gaussian universality of the errors in this context.

Machine learning models are vulnerable to adversarial perturbations, and a thought-provoking paper by Bubeck and Sellke has analyzed this phenomenon through the lens of over-parameterization: interpolating smoothly the data requires significantly more parameters than simply memorizing it. However, this "universal" law provides only a necessary condition for robustness, and it is unable to discriminate between models. In this paper, we address these gaps by focusing on empirical risk minimization in two prototypical settings, namely, random features and the neural tangent kernel (NTK). We prove that, for random features, the model is not robust for any degree of over-parameterization, even when the necessary condition coming from the universal law of robustness is satisfied. In contrast, for even activations, the NTK model meets the universal lower bound, and it is robust as soon as the necessary condition on over-parameterization is fulfilled. This also addresses a conjecture in prior work by Bubeck, Li and Nagaraj. Our analysis decouples the effect of the kernel of the model from an "interaction matrix", which describes the interaction with the test data and captures the effect of the activation. Our theoretical results are corroborated by numerical evidence on both synthetic and standard datasets (MNIST, CIFAR-10).

A basic question within the emerging field of mechanistic interpretability is the degree to which neural networks learn the same underlying mechanisms. In other words, are neural mechanisms universal across different models? In this work, we study the universality of individual neurons across GPT2 models trained from different initial random seeds, motivated by the hypothesis that universal neurons are likely to be interpretable. In particular, we compute pairwise correlations of neuron activations over 100 million tokens for every neuron pair across five different seeds and find that 1-5\% of neurons are universal, that is, pairs of neurons which consistently activate on the same inputs. We then study these universal neurons in detail, finding that they usually have clear interpretations and taxonomize them into a small number of neuron families. We conclude by studying patterns in neuron weights to establish several universal functional roles of neurons in simple circuits: deactivating attention heads, changing the entropy of the next token distribution, and predicting the next token to (not) be within a particular set.

A spanning tree T of graph G is a rho-approximate universal Steiner tree (UST) for root vertex r if, for any subset of vertices S containing r, the cost of the minimal subgraph of T connecting S is within a rho factor of the minimum cost tree connecting S in G. Busch et al. (FOCS 2012) showed that every graph admits 2^{O(log n)}-approximate USTs by showing that USTs are equivalent to strong sparse partition hierarchies (up to poly-logs). Further, they posed poly-logarithmic USTs and strong sparse partition hierarchies as open questions. We settle these open questions by giving polynomial-time algorithms for computing both O(log ^ 7 n)-approximate USTs and poly-logarithmic strong sparse partition hierarchies. For graphs with constant doubling dimension or constant pathwidth we improve this to O(log n)-approximate USTs and O(1) strong sparse partition hierarchies. Our doubling dimension result is tight up to second order terms. We reduce the existence of these objects to the previously studied cluster aggregation problem and what we call dangling nets.

Deep learning for time series forecasting has traditionally operated within a one-model-per-dataset framework, limiting its potential to leverage the game-changing impact of large pre-trained models. The concept of universal forecasting, emerging from pre-training on a vast collection of time series datasets, envisions a single Large Time Series Model capable of addressing diverse downstream forecasting tasks. However, constructing such a model poses unique challenges specific to time series data: i) cross-frequency learning, ii) accommodating an arbitrary number of variates for multivariate time series, and iii) addressing the varying distributional properties inherent in large-scale data. To address these challenges, we present novel enhancements to the conventional time series Transformer architecture, resulting in our proposed Masked Encoder-based Universal Time Series Forecasting Transformer (Moirai). Trained on our newly introduced Large-scale Open Time Series Archive (LOTSA) featuring over 27B observations across nine domains, Moirai achieves competitive or superior performance as a zero-shot forecaster when compared to full-shot models. Code, model weights, and data will be released.

We propose a novel random walk-based algorithm for unbiased estimation of arbitrary functions of a weighted adjacency matrix, coined universal graph random features (u-GRFs). This includes many of the most popular examples of kernels defined on the nodes of a graph. Our algorithm enjoys subquadratic time complexity with respect to the number of nodes, overcoming the notoriously prohibitive cubic scaling of exact graph kernel evaluation. It can also be trivially distributed across machines, permitting learning on much larger networks. At the heart of the algorithm is a modulation function which upweights or downweights the contribution from different random walks depending on their lengths. We show that by parameterising it with a neural network we can obtain u-GRFs that give higher-quality kernel estimates or perform efficient, scalable kernel learning. We provide robust theoretical analysis and support our findings with experiments including pointwise estimation of fixed graph kernels, solving non-homogeneous graph ordinary differential equations, node clustering and kernel regression on triangular meshes.

The Maximal Update Parametrization (muP) aims to make the optimal hyperparameters (HPs) of a model independent of its size, allowing them to be swept using a cheap proxy model rather than the full-size target model. We present a new scheme, u-muP, which improves upon muP by combining it with Unit Scaling, a method for designing models that makes them easy to train in low-precision. The two techniques have a natural affinity: muP ensures that the scale of activations is independent of model size, and Unit Scaling ensures that activations, weights and gradients begin training with a scale of one. This synthesis opens the door to a simpler scheme, whose default values are near-optimal. This in turn facilitates a more efficient sweeping strategy, with u-muP models reaching a lower loss than comparable muP models and working out-of-the-box in FP8.

It is often useful to compactly summarize important properties of model parameters and training data so that they can be used later without storing and/or iterating over the entire dataset. As a specific case, we consider estimating the Function Space Distance (FSD) over a training set, i.e. the average discrepancy between the outputs of two neural networks. We propose a Linearized Activation Function TRick (LAFTR) and derive an efficient approximation to FSD for ReLU neural networks. The key idea is to approximate the architecture as a linear network with stochastic gating. Despite requiring only one parameter per unit of the network, our approach outcompetes other parametric approximations with larger memory requirements. Applied to continual learning, our parametric approximation is competitive with state-of-the-art nonparametric approximations, which require storing many training examples. Furthermore, we show its efficacy in estimating influence functions accurately and detecting mislabeled examples without expensive iterations over the entire dataset.

The last decade has seen tremendous progress in our ability to generate realistic-looking data, be it images, text, audio, or video. Here, we discuss the closely related problem of quantifying realism, that is, designing functions that can reliably tell realistic data from unrealistic data. This problem turns out to be significantly harder to solve and remains poorly understood, despite its prevalence in machine learning and recent breakthroughs in generative AI. Drawing on insights from algorithmic information theory, we discuss why this problem is challenging, why a good generative model alone is insufficient to solve it, and what a good solution would look like. In particular, we introduce the notion of a universal critic, which unlike adversarial critics does not require adversarial training. While universal critics are not immediately practical, they can serve both as a North Star for guiding practical implementations and as a tool for analyzing existing attempts to capture realism.

We identify empirical scaling laws for the cross-entropy loss in four domains: generative image modeling, video modeling, multimodal imageleftrightarrowtext models, and mathematical problem solving. In all cases autoregressive Transformers smoothly improve in performance as model size and compute budgets increase, following a power-law plus constant scaling law. The optimal model size also depends on the compute budget through a power-law, with exponents that are nearly universal across all data domains. The cross-entropy loss has an information theoretic interpretation as S(True) + D_{KL}(True||Model), and the empirical scaling laws suggest a prediction for both the true data distribution's entropy and the KL divergence between the true and model distributions. With this interpretation, billion-parameter Transformers are nearly perfect models of the YFCC100M image distribution downsampled to an 8times 8 resolution, and we can forecast the model size needed to achieve any given reducible loss (ie D_{KL}) in nats/image for other resolutions. We find a number of additional scaling laws in specific domains: (a) we identify a scaling relation for the mutual information between captions and images in multimodal models, and show how to answer the question "Is a picture worth a thousand words?"; (b) in the case of mathematical problem solving, we identify scaling laws for model performance when extrapolating beyond the training distribution; (c) we finetune generative image models for ImageNet classification and find smooth scaling of the classification loss and error rate, even as the generative loss levels off. Taken together, these results strengthen the case that scaling laws have important implications for neural network performance, including on downstream tasks.

While score-based generative models (SGMs) have achieved remarkable success in enormous image generation tasks, their mathematical foundations are still limited. In this paper, we analyze the approximation and generalization of SGMs in learning a family of sub-Gaussian probability distributions. We introduce a notion of complexity for probability distributions in terms of their relative density with respect to the standard Gaussian measure. We prove that if the log-relative density can be locally approximated by a neural network whose parameters can be suitably bounded, then the distribution generated by empirical score matching approximates the target distribution in total variation with a dimension-independent rate. We illustrate our theory through examples, which include certain mixtures of Gaussians. An essential ingredient of our proof is to derive a dimension-free deep neural network approximation rate for the true score function associated with the forward process, which is interesting in its own right.

Neural scaling laws (NSL) refer to the phenomenon where model performance improves with scale. Sharma & Kaplan analyzed NSL using approximation theory and predict that MSE losses decay as N^{-alpha}, alpha=4/d, where N is the number of model parameters, and d is the intrinsic input dimension. Although their theory works well for some cases (e.g., ReLU networks), we surprisingly find that a simple 1D problem y=x^2 manifests a different scaling law (alpha=1) from their predictions (alpha=4). We opened the neural networks and found that the new scaling law originates from lottery ticket ensembling: a wider network on average has more "lottery tickets", which are ensembled to reduce the variance of outputs. We support the ensembling mechanism by mechanistically interpreting single neural networks, as well as studying them statistically. We attribute the N^{-1} scaling law to the "central limit theorem" of lottery tickets. Finally, we discuss its potential implications for large language models and statistical physics-type theories of learning.

We investigate novel random graph embeddings that can be computed in expected polynomial time and that are able to distinguish all non-isomorphic graphs in expectation. Previous graph embeddings have limited expressiveness and either cannot distinguish all graphs or cannot be computed efficiently for every graph. To be able to approximate arbitrary functions on graphs, we are interested in efficient alternatives that become arbitrarily expressive with increasing resources. Our approach is based on Lov\'asz' characterisation of graph isomorphism through an infinite dimensional vector of homomorphism counts. Our empirical evaluation shows competitive results on several benchmark graph learning tasks.

Combining several (sample approximations of) distributions, which we term sub-posteriors, into a single distribution proportional to their product, is a common challenge. Occurring, for instance, in distributed 'big data' problems, or when working under multi-party privacy constraints. Many existing approaches resort to approximating the individual sub-posteriors for practical necessity, then find either an analytical approximation or sample approximation of the resulting (product-pooled) posterior. The quality of the posterior approximation for these approaches is poor when the sub-posteriors fall out-with a narrow range of distributional form, such as being approximately Gaussian. Recently, a Fusion approach has been proposed which finds an exact Monte Carlo approximation of the posterior, circumventing the drawbacks of approximate approaches. Unfortunately, existing Fusion approaches have a number of computational limitations, particularly when unifying a large number of sub-posteriors. In this paper, we generalise the theory underpinning existing Fusion approaches, and embed the resulting methodology within a recursive divide-and-conquer sequential Monte Carlo paradigm. This ultimately leads to a competitive Fusion approach, which is robust to increasing numbers of sub-posteriors.

We discuss the numerical solution of initial value problems for varepsilon^2,varphi''+a(x),varphi=0 in the highly oscillatory regime, i.e., with a(x)>0 and 0<varepsilonll 1. We analyze and implement an approximate solution based on the well-known WKB-ansatz. The resulting approximation error is of magnitude O(varepsilon^{N}) where N refers to the truncation order of the underlying asymptotic series. When the optimal truncation order N_{opt} is chosen, the error behaves like O(varepsilon^{-2}exp(-cvarepsilon^{-1})) with some c>0.

A neural architecture with randomly initialized weights, in the infinite width limit, is equivalent to a Gaussian Random Field whose covariance function is the so-called Neural Network Gaussian Process kernel (NNGP). We prove that a reproducing kernel Hilbert space (RKHS) defined by the NNGP contains only functions that can be approximated by the architecture. To achieve a certain approximation error the required number of neurons in each layer is defined by the RKHS norm of the target function. Moreover, the approximation can be constructed from a supervised dataset by a random multi-layer representation of an input vector, together with training of the last layer's weights. For a 2-layer NN and a domain equal to an n-1-dimensional sphere in {mathbb R}^n, we compare the number of neurons required by Barron's theorem and by the multi-layer features construction. We show that if eigenvalues of the integral operator of the NNGP decay slower than k^{-n-2{3}} where k is an order of an eigenvalue, then our theorem guarantees a more succinct neural network approximation than Barron's theorem. We also make some computational experiments to verify our theoretical findings. Our experiments show that realistic neural networks easily learn target functions even when both theorems do not give any guarantees.

This manuscript considers the problem of learning a random Gaussian network function using a fully connected network with frozen intermediate layers and trainable readout layer. This problem can be seen as a natural generalization of the widely studied random features model to deeper architectures. First, we prove Gaussian universality of the test error in a ridge regression setting where the learner and target networks share the same intermediate layers, and provide a sharp asymptotic formula for it. Establishing this result requires proving a deterministic equivalent for traces of the deep random features sample covariance matrices which can be of independent interest. Second, we conjecture the asymptotic Gaussian universality of the test error in the more general setting of arbitrary convex losses and generic learner/target architectures. We provide extensive numerical evidence for this conjecture, which requires the derivation of closed-form expressions for the layer-wise post-activation population covariances. In light of our results, we investigate the interplay between architecture design and implicit regularization.

We prove an inverse approximation theorem for the approximation of nonlinear sequence-to-sequence relationships using recurrent neural networks (RNNs). This is a so-called Bernstein-type result in approximation theory, which deduces properties of a target function under the assumption that it can be effectively approximated by a hypothesis space. In particular, we show that nonlinear sequence relationships that can be stably approximated by nonlinear RNNs must have an exponential decaying memory structure - a notion that can be made precise. This extends the previously identified curse of memory in linear RNNs into the general nonlinear setting, and quantifies the essential limitations of the RNN architecture for learning sequential relationships with long-term memory. Based on the analysis, we propose a principled reparameterization method to overcome the limitations. Our theoretical results are confirmed by numerical experiments. The code has been released in https://github.com/radarFudan/Curse-of-memory

By classifying infinite-width neural networks and identifying the *optimal* limit, Tensor Programs IV and V demonstrated a universal way, called muP, for *widthwise hyperparameter transfer*, i.e., predicting optimal hyperparameters of wide neural networks from narrow ones. Here we investigate the analogous classification for *depthwise parametrizations* of deep residual networks (resnets). We classify depthwise parametrizations of block multiplier and learning rate by their infinite-width-then-depth limits. In resnets where each block has only one layer, we identify a unique optimal parametrization, called Depth-muP that extends muP and show empirically it admits depthwise hyperparameter transfer. We identify *feature diversity* as a crucial factor in deep networks, and Depth-muP can be characterized as maximizing both feature learning and feature diversity. Exploiting this, we find that absolute value, among all homogeneous nonlinearities, maximizes feature diversity and indeed empirically leads to significantly better performance. However, if each block is deeper (such as modern transformers), then we find fundamental limitations in all possible infinite-depth limits of such parametrizations, which we illustrate both theoretically and empirically on simple networks as well as Megatron transformer trained on Common Crawl.

As language models scale up, it becomes increasingly expensive to verify research ideas because conclusions on small models do not trivially transfer to large ones. A possible solution is to establish a generic system that directly predicts some metrics for large models solely based on the results and hyperparameters from small models. Existing methods based on scaling laws require hyperparameter search on the largest models, which is impractical with limited resources. We address this issue by presenting our discoveries indicating that Maximal Update parametrization (Mup) enables accurate fitting of scaling laws for hyperparameters close to common loss basins, without any search. Thus, different models can be directly compared on large scales with loss prediction even before the training starts. We propose a new paradigm as a first step towards reliable academic research for any model scale without heavy computation. Code is publicly available at https://github.com/cofe-ai/Mu-scaling.

Discovering the existence of universal adversarial perturbations had large theoretical and practical impacts on the field of adversarial learning. In the text domain, most universal studies focused on adversarial prefixes which are added to all texts. However, unlike the vision domain, adding the same perturbation to different inputs results in noticeably unnatural inputs. Therefore, we introduce a new universal adversarial setup - a universal adversarial policy, which has many advantages of other universal attacks but also results in valid texts - thus making it relevant in practice. We achieve this by learning a single search policy over a predefined set of semantics preserving text alterations, on many texts. This formulation is universal in that the policy is successful in finding adversarial examples on new texts efficiently. Our approach uses text perturbations which were extensively shown to produce natural attacks in the non-universal setup (specific synonym replacements). We suggest a strong baseline approach for this formulation which uses reinforcement learning. It's ability to generalise (from as few as 500 training texts) shows that universal adversarial patterns exist in the text domain as well.

The impressive capabilities of Large Language Models (LLMs) across diverse tasks are now well-established, yet their effective deployment necessitates careful hyperparameter optimization. Through extensive empirical studies involving grid searches across diverse configurations, we discover universal scaling laws governing these hyperparameters: optimal learning rate follows a power-law relationship with both model parameters and data sizes, while optimal batch size scales primarily with data sizes. Our analysis reveals a convex optimization landscape for hyperparameters under fixed models and data size conditions. This convexity implies an optimal hyperparameter plateau. We contribute a universal, plug-and-play optimal hyperparameter tool for the community. Its estimated values on the test set are merely 0.07\% away from the globally optimal LLM performance found via an exhaustive search. These laws demonstrate remarkable robustness across variations in model sparsity, training data distribution, and model shape. To our best known, this is the first work that unifies different model shapes and structures, such as Mixture-of-Experts models and dense transformers, as well as establishes optimal hyperparameter scaling laws across diverse data distributions. This exhaustive optimization process demands substantial computational resources, utilizing nearly one million NVIDIA H800 GPU hours to train 3,700 LLMs of varying sizes and hyperparameters from scratch and consuming approximately 100 trillion tokens in total. To facilitate reproducibility and further research, we will progressively release all loss measurements and model checkpoints through our designated repository https://step-law.github.io/

Zipf's law appears in many application areas but does not have a closed form expression, which may make its use cumbersome. Since it coincides with the truncated version of the Zeta distribution, in this paper we propose three approximate closed form expressions for the truncated Zeta distribution, which may be employed for Zipf's law as well. The three approximations are based on the replacement of the sum occurring in Zipf's law with an integral, and are named respectively the integral approximation, the average integral approximation, and the trapezoidal approximation. While the first one is shown to be of little use, the trapezoidal approximation exhibits an error which is typically lower than 1\%, but is as low as 0.1\% for the range of values of the Zipf parameter below 1.

The present work explores the theoretical limits of Machine Learning (ML) within the framework of Kolmogorov's theory of Algorithmic Probability, which clarifies the notion of entropy as Expected Kolmogorov Complexity and formalizes other fundamental concepts such as Occam's razor via Levin's Universal Distribution. As a fundamental application, we develop Maximum Entropy methods that allow us to derive the Erdos--Kac Law in Probabilistic Number Theory, and establish the impossibility of discovering a formula for primes using Machine Learning via the Prime Coding Theorem.

Maximizing a monotone submodular function under cardinality constraint k is a core problem in machine learning and database with many basic applications, including video and data summarization, recommendation systems, feature extraction, exemplar clustering, and coverage problems. We study this classic problem in the fully dynamic model where a stream of insertions and deletions of elements of an underlying ground set is given and the goal is to maintain an approximate solution using a fast update time. A recent paper at NeurIPS'20 by Lattanzi, Mitrovic, Norouzi{-}Fard, Tarnawski, Zadimoghaddam claims to obtain a dynamic algorithm for this problem with a 1{2} -epsilon approximation ratio and a query complexity bounded by poly(log(n),log(k),epsilon^{-1}). However, as we explain in this paper, the analysis has some important gaps. Having a dynamic algorithm for the problem with polylogarithmic update time is even more important in light of a recent result by Chen and Peng at STOC'22 who show a matching lower bound for the problem -- any randomized algorithm with a 1{2}+epsilon approximation ratio must have an amortized query complexity that is polynomial in n. In this paper, we develop a simpler algorithm for the problem that maintains a (1{2}-epsilon)-approximate solution for submodular maximization under cardinality constraint k using a polylogarithmic amortized update time.

The infinitely wide neural network has been proven a useful and manageable mathematical model that enables the understanding of many phenomena appearing in deep learning. One example is the convergence of random deep networks to Gaussian processes that allows a rigorous analysis of the way the choice of activation function and network weights impacts the training dynamics. In this paper, we extend the seminal proof of Matthews et al. (2018) to a larger class of initial weight distributions (which we call PSEUDO-IID), including the established cases of IID and orthogonal weights, as well as the emerging low-rank and structured sparse settings celebrated for their computational speed-up benefits. We show that fully-connected and convolutional networks initialized with PSEUDO-IID distributions are all effectively equivalent up to their variance. Using our results, one can identify the Edge-of-Chaos for a broader class of neural networks and tune them at criticality in order to enhance their training. Moreover, they enable the posterior distribution of Bayesian Neural Networks to be tractable across these various initialization schemes.

In economy, markets are denoted as efficient when it is impossible to systematically generate profits which outperform the average. In the past years, the concept has been tested in other domains such as the growing sports betting market. Surprisingly, despite its large size and its level of maturity, sports betting shows traits of inefficiency. The anomalies indicate the existence of strategies which shift betting from a game of chance towards a game of skill. This article shows an example for an inefficiency detected in the German soccer betting TOTO 13er Wette, which is operated by state-run lottery agencies. Gamblers have to guess the outcome (win, draw, loss) of 13 soccer matches listed on a lottery tip. Applying stochastic methods, a recipe is presented to determine hit rates for single match outcomes. More important, the recipe provides the number of lottery tips required to achieve a specific number of strikes (number of correct match forecasts per lottery tip) for any given level of safety. An approximation is derived to cope with large numbers in hypergeometric distributions, valid under certain constraints. Overall, the strategy does lead to returns exceeding the aggregated lottery fees, resulting in moderate, but consistent profits. It is briefly discussed if lessions learned from soccer betting can be transferred back to financial markets, because gamblers and retail investors face similar challenges and opportunities.

The Shapley value is arguably the most popular approach for assigning a meaningful contribution value to players in a cooperative game, which has recently been used intensively in explainable artificial intelligence. The meaningfulness is due to axiomatic properties that only the Shapley value satisfies, which, however, comes at the expense of an exact computation growing exponentially with the number of agents. Accordingly, a number of works are devoted to the efficient approximation of the Shapley values, most of them revolve around the notion of an agent's marginal contribution. In this paper, we propose with SVARM and Stratified SVARM two parameter-free and domain-independent approximation algorithms based on a representation of the Shapley value detached from the notion of marginal contributions. We prove unmatched theoretical guarantees regarding their approximation quality and provide empirical results including synthetic games as well as common explainability use cases comparing ourselves with state-of-the-art methods.

The choice of approximate posterior distribution is one of the core problems in variational inference. Most applications of variational inference employ simple families of posterior approximations in order to allow for efficient inference, focusing on mean-field or other simple structured approximations. This restriction has a significant impact on the quality of inferences made using variational methods. We introduce a new approach for specifying flexible, arbitrarily complex and scalable approximate posterior distributions. Our approximations are distributions constructed through a normalizing flow, whereby a simple initial density is transformed into a more complex one by applying a sequence of invertible transformations until a desired level of complexity is attained. We use this view of normalizing flows to develop categories of finite and infinitesimal flows and provide a unified view of approaches for constructing rich posterior approximations. We demonstrate that the theoretical advantages of having posteriors that better match the true posterior, combined with the scalability of amortized variational approaches, provides a clear improvement in performance and applicability of variational inference.

The surprising ability of Large Language Models (LLMs) to perform well on complex reasoning with only few-shot chain-of-thought prompts is believed to emerge only in very large-scale models (100+ billion parameters). We show that such abilities can, in fact, be distilled down from GPT-3.5 (ge 175B) to T5 variants (le 11B). We propose model specialization, to specialize the model's ability towards a target task. The hypothesis is that large models (commonly viewed as larger than 100B) have strong modeling power, but are spread on a large spectrum of tasks. Small models (commonly viewed as smaller than 10B) have limited model capacity, but if we concentrate their capacity on a specific target task, the model can achieve a decent improved performance. We use multi-step math reasoning as our testbed because it is a very typical emergent ability. We show two important aspects of model abilities: (1). there exists a very complex balance/ tradeoff between language models' multi-dimensional abilities; (2). by paying the price of decreased generic ability, we can clearly lift up the scaling curve of models smaller than 10B towards a specialized multi-step math reasoning ability. We further give comprehensive discussions about important design choices for better generalization, including the tuning data format, the start model checkpoint, and a new model selection method. We hope our practice and discoveries can serve as an important attempt towards specialized smaller models in the new research paradigm set by LLMs.

With infinitely many high-quality data points, infinite computational power, an infinitely large foundation model with a perfect training algorithm and guaranteed zero generalization error on the pretext task, can the model be used for everything? This question cannot be answered by the existing theory of representation, optimization or generalization, because the issues they mainly investigate are assumed to be nonexistent here. In this paper, we show that category theory provides powerful machinery to answer this question. We have proved three results. The first one limits the power of prompt-based learning, saying that the model can solve a downstream task with prompts if and only if the task is representable. The second one says fine tuning does not have this limit, as a foundation model with the minimum required power (up to symmetry) can theoretically solve downstream tasks for the category defined by pretext task, with fine tuning and enough resources. Our final result can be seen as a new type of generalization theorem, showing that the foundation model can generate unseen objects from the target category (e.g., images) using the structural information from the source category (e.g., texts). Along the way, we provide a categorical framework for supervised and self-supervised learning, which might be of independent interest.

A challenging problem in many modern machine learning tasks is to process weight-space features, i.e., to transform or extract information from the weights and gradients of a neural network. Recent works have developed promising weight-space models that are equivariant to the permutation symmetries of simple feedforward networks. However, they are not applicable to general architectures, since the permutation symmetries of a weight space can be complicated by recurrence or residual connections. This work proposes an algorithm that automatically constructs permutation equivariant models, which we refer to as universal neural functionals (UNFs), for any weight space. Among other applications, we demonstrate how UNFs can be substituted into existing learned optimizer designs, and find promising improvements over prior methods when optimizing small image classifiers and language models. Our results suggest that learned optimizers can benefit from considering the (symmetry) structure of the weight space they optimize. We open-source our library for constructing UNFs at https://github.com/AllanYangZhou/universal_neural_functional.

The scientific scale-up of large language models (LLMs) necessitates a comprehensive understanding of their scaling properties. However, the existing literature on the scaling properties only yields an incomplete answer: optimization loss decreases predictably as the model size increases, in line with established scaling law; yet no scaling law for task has been established and the task performances are far from predictable during scaling. Task performances typically show minor gains on small models until they improve dramatically once models exceed a size threshold, exemplifying the ``emergent abilities''. In this study, we discover that small models, although they exhibit minor performance, demonstrate critical and consistent task performance improvements that are not captured by conventional evaluation strategies due to insufficient measurement resolution. To measure such improvements, we introduce PassUntil, an evaluation strategy through massive sampling in the decoding phase. We conduct quantitative investigations into the scaling law of task performance. Firstly, a strict task scaling law is identified, enhancing the predictability of task performances. Remarkably, we are able to predict the performance of the 2.4B model on code generation with merely 0.05\% deviation before training starts. Secondly, underpinned by PassUntil, we observe concrete evidence of emergent abilities and ascertain that they are not in conflict with the continuity of performance improvement. Their semblance to break-through is that their scaling curve cannot be fitted by standard scaling law function. We then introduce a mathematical definition for the emergent abilities. Through the definition, we refute a prevalent ``multi-step reasoning hypothesis'' regarding the genesis of emergent abilities and propose a new hypothesis with a satisfying fit to the observed scaling curve.

Low-rank approximation is a fundamental technique in modern data analysis, widely utilized across various fields such as signal processing, machine learning, and natural language processing. Despite its ubiquity, the mechanics of low-rank approximation and its application in adaptation can sometimes be obscure, leaving practitioners and researchers with questions about its true capabilities and limitations. This paper seeks to clarify low-rank approximation and adaptation by offering a comprehensive guide that reveals their inner workings and explains their utility in a clear and accessible way. Our focus here is to develop a solid intuition for how low-rank approximation and adaptation operate, and why they are so effective. We begin with basic concepts and gradually build up to the mathematical underpinnings, ensuring that readers of all backgrounds can gain a deeper understanding of low-rank approximation and adaptation. We strive to strike a balance between informal explanations and rigorous mathematics, ensuring that both newcomers and experienced experts can benefit from this survey. Additionally, we introduce new low-rank decomposition and adaptation algorithms that have not yet been explored in the field, hoping that future researchers will investigate their potential applicability.

This paper explores the expressive power of deep neural networks through the framework of function compositions. We demonstrate that the repeated compositions of a single fixed-size ReLU network exhibit surprising expressive power, despite the limited expressive capabilities of the individual network itself. Specifically, we prove by construction that L_2circ g^{circ r}circ mathcal{L}_1 can approximate 1-Lipschitz continuous functions on [0,1]^d with an error O(r^{-1/d}), where g is realized by a fixed-size ReLU network, mathcal{L}_1 and L_2 are two affine linear maps matching the dimensions, and g^{circ r} denotes the r-times composition of g. Furthermore, we extend such a result to generic continuous functions on [0,1]^d with the approximation error characterized by the modulus of continuity. Our results reveal that a continuous-depth network generated via a dynamical system has immense approximation power even if its dynamics function is time-independent and realized by a fixed-size ReLU network.

Despite the recent advances in the field of computational Schr\"odinger Bridges (SB), most existing SB solvers are still heavy-weighted and require complex optimization of several neural networks. It turns out that there is no principal solver which plays the role of simple-yet-effective baseline for SB just like, e.g., k-means method in clustering, logistic regression in classification or Sinkhorn algorithm in discrete optimal transport. We address this issue and propose a novel fast and simple SB solver. Our development is a smart combination of two ideas which recently appeared in the field: (a) parameterization of the Schr\"odinger potentials with sum-exp quadratic functions and (b) viewing the log-Schr\"odinger potentials as the energy functions. We show that combined together these ideas yield a lightweight, simulation-free and theoretically justified SB solver with a simple straightforward optimization objective. As a result, it allows solving SB in moderate dimensions in a matter of minutes on CPU without a painful hyperparameter selection. Our light solver resembles the Gaussian mixture model which is widely used for density estimation. Inspired by this similarity, we also prove an important theoretical result showing that our light solver is a universal approximator of SBs. Furthemore, we conduct the analysis of the generalization error of our light solver. The code for our solver can be found at https://github.com/ngushchin/LightSB

Recent approaches to multi-task learning (MTL) have focused on modelling connections between tasks at the decoder level. This leads to a tight coupling between tasks, which need retraining if a new task is inserted or removed. We argue that MTL is a stepping stone towards universal feature learning (UFL), which is the ability to learn generic features that can be applied to new tasks without retraining. We propose Medusa to realize this goal, designing task heads with dual attention mechanisms. The shared feature attention masks relevant backbone features for each task, allowing it to learn a generic representation. Meanwhile, a novel Multi-Scale Attention head allows the network to better combine per-task features from different scales when making the final prediction. We show the effectiveness of Medusa in UFL (+13.18% improvement), while maintaining MTL performance and being 25% more efficient than previous approaches.

This paper provides a non-asymptotic analysis of linear stochastic approximation (LSA) algorithms with fixed stepsize. This family of methods arises in many machine learning tasks and is used to obtain approximate solutions of a linear system Atheta = b for which A and b can only be accessed through random estimates {({bf A}_n, {bf b}_n): n in N^*}. Our analysis is based on new results regarding moments and high probability bounds for products of matrices which are shown to be tight. We derive high probability bounds on the performance of LSA under weaker conditions on the sequence {({bf A}_n, {bf b}_n): n in N^*} than previous works. However, in contrast, we establish polynomial concentration bounds with order depending on the stepsize. We show that our conclusions cannot be improved without additional assumptions on the sequence of random matrices {{bf A}_n: n in N^*}, and in particular that no Gaussian or exponential high probability bounds can hold. Finally, we pay a particular attention to establishing bounds with sharp order with respect to the number of iterations and the stepsize and whose leading terms contain the covariance matrices appearing in the central limit theorems.

While Bayesian inference provides a principled framework for reasoning under uncertainty, its widespread adoption is limited by the intractability of exact posterior computation, necessitating the use of approximate inference. However, existing methods are often computationally expensive, or demand costly retraining when priors change, limiting their utility, particularly in sequential inference problems such as real-time sensor fusion. To address these challenges, we introduce the Distribution Transformer -- a novel architecture that can learn arbitrary distribution-to-distribution mappings. Our method can be trained to map a prior to the corresponding posterior, conditioned on some dataset -- thus performing approximate Bayesian inference. Our novel architecture represents a prior distribution as a (universally-approximating) Gaussian Mixture Model (GMM), and transforms it into a GMM representation of the posterior. The components of the GMM attend to each other via self-attention, and to the datapoints via cross-attention. We demonstrate that Distribution Transformers both maintain flexibility to vary the prior, and significantly reduces computation times-from minutes to milliseconds-while achieving log-likelihood performance on par with or superior to existing approximate inference methods across tasks such as sequential inference, quantum system parameter inference, and Gaussian Process predictive posterior inference with hyperpriors.

We extend conformal prediction to control the expected value of any monotone loss function. The algorithm generalizes split conformal prediction together with its coverage guarantee. Like conformal prediction, the conformal risk control procedure is tight up to an O(1/n) factor. We also introduce extensions of the idea to distribution shift, quantile risk control, multiple and adversarial risk control, and expectations of U-statistics. Worked examples from computer vision and natural language processing demonstrate the usage of our algorithm to bound the false negative rate, graph distance, and token-level F1-score.

Approximate inference in Gaussian process (GP) models with non-conjugate likelihoods gets entangled with the learning of the model hyperparameters. We improve hyperparameter learning in GP models and focus on the interplay between variational inference (VI) and the learning target. While VI's lower bound to the marginal likelihood is a suitable objective for inferring the approximate posterior, we show that a direct approximation of the marginal likelihood as in Expectation Propagation (EP) is a better learning objective for hyperparameter optimization. We design a hybrid training procedure to bring the best of both worlds: it leverages conjugate-computation VI for inference and uses an EP-like marginal likelihood approximation for hyperparameter learning. We compare VI, EP, Laplace approximation, and our proposed training procedure and empirically demonstrate the effectiveness of our proposal across a wide range of data sets.

We introduce the concept of scalable neural network kernels (SNNKs), the replacements of regular feedforward layers (FFLs), capable of approximating the latter, but with favorable computational properties. SNNKs effectively disentangle the inputs from the parameters of the neural network in the FFL, only to connect them in the final computation via the dot-product kernel. They are also strictly more expressive, as allowing to model complicated relationships beyond the functions of the dot-products of parameter-input vectors. We also introduce the neural network bundling process that applies SNNKs to compactify deep neural network architectures, resulting in additional compression gains. In its extreme version, it leads to the fully bundled network whose optimal parameters can be expressed via explicit formulae for several loss functions (e.g. mean squared error), opening a possibility to bypass backpropagation. As a by-product of our analysis, we introduce the mechanism of the universal random features (or URFs), applied to instantiate several SNNK variants, and interesting on its own in the context of scalable kernel methods. We provide rigorous theoretical analysis of all these concepts as well as an extensive empirical evaluation, ranging from point-wise kernel estimation to Transformers' fine-tuning with novel adapter layers inspired by SNNKs. Our mechanism provides up to 5x reduction in the number of trainable parameters, while maintaining competitive accuracy.

We study the cost of overfitting in noisy kernel ridge regression (KRR), which we define as the ratio between the test error of the interpolating ridgeless model and the test error of the optimally-tuned model. We take an "agnostic" view in the following sense: we consider the cost as a function of sample size for any target function, even if the sample size is not large enough for consistency or the target is outside the RKHS. We analyze the cost of overfitting under a Gaussian universality ansatz using recently derived (non-rigorous) risk estimates in terms of the task eigenstructure. Our analysis provides a more refined characterization of benign, tempered and catastrophic overfitting (cf. Mallinar et al. 2022).

This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the output of trained networks by solving layer-to-layer iteration equations and nonlinear learning dynamics. A main result is that the predictions of networks are described by nearly-Gaussian distributions, with the depth-to-width aspect ratio of the network controlling the deviations from the infinite-width Gaussian description. We explain how these effectively-deep networks learn nontrivial representations from training and more broadly analyze the mechanism of representation learning for nonlinear models. From a nearly-kernel-methods perspective, we find that the dependence of such models' predictions on the underlying learning algorithm can be expressed in a simple and universal way. To obtain these results, we develop the notion of representation group flow (RG flow) to characterize the propagation of signals through the network. By tuning networks to criticality, we give a practical solution to the exploding and vanishing gradient problem. We further explain how RG flow leads to near-universal behavior and lets us categorize networks built from different activation functions into universality classes. Altogether, we show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks. By using information-theoretic techniques, we estimate the optimal aspect ratio at which we expect the network to be practically most useful and show how residual connections can be used to push this scale to arbitrary depths. With these tools, we can learn in detail about the inductive bias of architectures, hyperparameters, and optimizers.

In astrophysical and cosmological analyses, the increasing quality and volume of astronomical data demand efficient and precise computational tools. This work introduces a novel adaptive algorithm for automatic knots (AutoKnots) allocation in spline interpolation, designed to meet user-defined precision requirements. Unlike traditional methods that rely on manually configured knot distributions with numerous parameters, the proposed technique automatically determines the optimal number and placement of knots based on interpolation error criteria. This simplifies configuration, often requiring only a single parameter. The algorithm progressively improves the interpolation by adaptively sampling the function-to-be-approximated, f(x), in regions where the interpolation error exceeds the desired threshold. All function evaluations contribute directly to the final approximation, ensuring efficiency. While each resampling step involves recomputing the interpolation table, this process is highly optimized and usually computationally negligible compared to the cost of evaluating f(x). We show the algorithm's efficacy through a series of precision tests on different functions. However, the study underscores the necessity for caution when dealing with certain function types, notably those featuring plateaus. To address this challenge, a heuristic enhancement is incorporated, improving accuracy in flat regions. This algorithm has been extensively used and tested over the years. NumCosmo includes a comprehensive set of unit tests that rigorously evaluate the algorithm both directly and indirectly, underscoring its robustness and reliability. As a practical application, we compute the surface mass density Sigma(R) and the average surface mass density Sigma(<R) for Navarro-Frenk-White and Hernquist halo density profiles, which provide analytical benchmarks. (abridged)

With the latest advances in deep learning, there has been a lot of focus on the online learning paradigm due to its relevance in practical settings. Although many methods have been investigated for optimal learning settings in scenarios where the data stream is continuous over time, sparse networks training in such settings have often been overlooked. In this paper, we explore the problem of training a neural network with a target sparsity in a particular case of online learning: the anytime learning at macroscale paradigm (ALMA). We propose a novel way of progressive pruning, referred to as Anytime Progressive Pruning (APP); the proposed approach significantly outperforms the baseline dense and Anytime OSP models across multiple architectures and datasets under short, moderate, and long-sequence training. Our method, for example, shows an improvement in accuracy of approx 7% and a reduction in the generalization gap by approx 22%, while being approx 1/3 rd the size of the dense baseline model in few-shot restricted imagenet training. We further observe interesting nonmonotonic transitions in the generalization gap in the high number of megabatches-based ALMA. The code and experiment dashboards can be accessed at https://github.com/landskape-ai/Progressive-Pruning and https://wandb.ai/landskape/APP, respectively.

Empirical neural tangent kernels (eNTKs) can provide a good understanding of a given network's representation: they are often far less expensive to compute and applicable more broadly than infinite width NTKs. For networks with O output units (e.g. an O-class classifier), however, the eNTK on N inputs is of size NO times NO, taking O((NO)^2) memory and up to O((NO)^3) computation. Most existing applications have therefore used one of a handful of approximations yielding N times N kernel matrices, saving orders of magnitude of computation, but with limited to no justification. We prove that one such approximation, which we call "sum of logits", converges to the true eNTK at initialization for any network with a wide final "readout" layer. Our experiments demonstrate the quality of this approximation for various uses across a range of settings.

We consider in this work quantities that can be obtained as limits of powers of parametrized matrices, for instance the inverse matrix or the logarithm of the determinant. Under the assumption of affine dependence in the parameters, we use the Empirical Interpolation Method (EIM) to derive an approximation for powers of these matrices, from which we derive a nonintrusive approximation for the aforementioned limits. We derive upper bounds of the error made by the obtained formula. Finally, numerical comparisons with classical intrusive and nonintrusive approximation techniques are provided: in the considered test-cases, our algorithm performs well compared to the nonintrusive ones.

Wider adoption of neural networks in many critical domains such as finance and healthcare is being hindered by the need to explain their predictions and to impose additional constraints on them. Monotonicity constraint is one of the most requested properties in real-world scenarios and is the focus of this paper. One of the oldest ways to construct a monotonic fully connected neural network is to constrain signs on its weights. Unfortunately, this construction does not work with popular non-saturated activation functions as it can only approximate convex functions. We show this shortcoming can be fixed by constructing two additional activation functions from a typical unsaturated monotonic activation function and employing each of them on the part of neurons. Our experiments show this approach of building monotonic neural networks has better accuracy when compared to other state-of-the-art methods, while being the simplest one in the sense of having the least number of parameters, and not requiring any modifications to the learning procedure or post-learning steps. Finally, we prove it can approximate any continuous monotone function on a compact subset of R^n.

The measure of a machine learning algorithm is the difficulty of the tasks it can perform, and sufficiently difficult tasks are critical drivers of strong machine learning models. However, quantifying the generalization difficulty of machine learning benchmarks has remained challenging. We propose what is to our knowledge the first model-agnostic measure of the inherent generalization difficulty of tasks. Our inductive bias complexity measure quantifies the total information required to generalize well on a task minus the information provided by the data. It does so by measuring the fractional volume occupied by hypotheses that generalize on a task given that they fit the training data. It scales exponentially with the intrinsic dimensionality of the space over which the model must generalize but only polynomially in resolution per dimension, showing that tasks which require generalizing over many dimensions are drastically more difficult than tasks involving more detail in fewer dimensions. Our measure can be applied to compute and compare supervised learning, reinforcement learning and meta-learning generalization difficulties against each other. We show that applied empirically, it formally quantifies intuitively expected trends, e.g. that in terms of required inductive bias, MNIST < CIFAR10 < Imagenet and fully observable Markov decision processes (MDPs) < partially observable MDPs. Further, we show that classification of complex images < few-shot meta-learning with simple images. Our measure provides a quantitative metric to guide the construction of more complex tasks requiring greater inductive bias, and thereby encourages the development of more sophisticated architectures and learning algorithms with more powerful generalization capabilities.

Graph neural networks (GNNs) are effective models for representation learning on relational data. However, standard GNNs are limited in their expressive power, as they cannot distinguish graphs beyond the capability of the Weisfeiler-Leman graph isomorphism heuristic. In order to break this expressiveness barrier, GNNs have been enhanced with random node initialization (RNI), where the idea is to train and run the models with randomized initial node features. In this work, we analyze the expressive power of GNNs with RNI, and prove that these models are universal, a first such result for GNNs not relying on computationally demanding higher-order properties. This universality result holds even with partially randomized initial node features, and preserves the invariance properties of GNNs in expectation. We then empirically analyze the effect of RNI on GNNs, based on carefully constructed datasets. Our empirical findings support the superior performance of GNNs with RNI over standard GNNs.

Neural scaling laws define a predictable relationship between a model's parameter count and its performance after training in the form of a power law. However, most research to date has not explicitly investigated whether scaling laws can be used to accelerate model development. In this work, we perform such an empirical investigation across a wide range of language understanding tasks, starting from models with as few as 10K parameters, and evaluate downstream performance across 9 language understanding tasks. We find that scaling laws emerge at finetuning time in some NLP tasks, and that they can also be exploited for debugging convergence when training large models. Moreover, for tasks where scaling laws exist, they can be used to predict the performance of larger models, which enables effective model selection. However, revealing scaling laws requires careful hyperparameter tuning and multiple runs for the purpose of uncertainty estimation, which incurs additional overhead, partially offsetting the computational benefits.

In this paper, we consider the problem of Iterative Machine Teaching (IMT), where the teacher provides examples to the learner iteratively such that the learner can achieve fast convergence to a target model. However, existing IMT algorithms are solely based on parameterized families of target models. They mainly focus on convergence in the parameter space, resulting in difficulty when the target models are defined to be functions without dependency on parameters. To address such a limitation, we study a more general task -- Nonparametric Iterative Machine Teaching (NIMT), which aims to teach nonparametric target models to learners in an iterative fashion. Unlike parametric IMT that merely operates in the parameter space, we cast NIMT as a functional optimization problem in the function space. To solve it, we propose both random and greedy functional teaching algorithms. We obtain the iterative teaching dimension (ITD) of the random teaching algorithm under proper assumptions, which serves as a uniform upper bound of ITD in NIMT. Further, the greedy teaching algorithm has a significantly lower ITD, which reaches a tighter upper bound of ITD in NIMT. Finally, we verify the correctness of our theoretical findings with extensive experiments in nonparametric scenarios.

No free lunch theorems for supervised learning state that no learner can solve all problems or that all learners achieve exactly the same accuracy on average over a uniform distribution on learning problems. Accordingly, these theorems are often referenced in support of the notion that individual problems require specially tailored inductive biases. While virtually all uniformly sampled datasets have high complexity, real-world problems disproportionately generate low-complexity data, and we argue that neural network models share this same preference, formalized using Kolmogorov complexity. Notably, we show that architectures designed for a particular domain, such as computer vision, can compress datasets on a variety of seemingly unrelated domains. Our experiments show that pre-trained and even randomly initialized language models prefer to generate low-complexity sequences. Whereas no free lunch theorems seemingly indicate that individual problems require specialized learners, we explain how tasks that often require human intervention such as picking an appropriately sized model when labeled data is scarce or plentiful can be automated into a single learning algorithm. These observations justify the trend in deep learning of unifying seemingly disparate problems with an increasingly small set of machine learning models.

We study the approximability of an existing framework for clustering edge-colored hypergraphs, which is closely related to chromatic correlation clustering and is motivated by machine learning and data mining applications where the goal is to cluster a set of objects based on multiway interactions of different categories or types. We present improved approximation guarantees based on linear programming, and show they are tight by proving a matching integrality gap. Our results also include new approximation hardness results, a combinatorial 2-approximation whose runtime is linear in the hypergraph size, and several new connections to well-studied objectives such as vertex cover and hypergraph multiway cut.

The Empirical Interpolation Method (EIM) is a greedy procedure that constructs approximate representations of two-variable functions in separated form. In its classical presentation, the two variables play a non-symmetric role. In this work, we give an equivalent definition of the EIM approximation, in which the two variables play symmetric roles. Then, we give a proof for the existence of this approximation, and extend it up to the convergence of the EIM, and for any norm chosen to compute the error in the greedy step. Finally, we introduce a way to compute a separated representation in the case where the number of selected values is different for each variable. In the case of a physical field measured by sensors, this is useful to discard a broken sensor while keeping the information provided by the associated selected field.

Scaling laws are typically fit using a family of models with a narrow range of frozen hyper-parameter choices. In this work we study scaling laws using a wide range of architecture and hyper-parameter choices, and highlight their impact on resulting prescriptions. As a primary artifact of our research, we release the Gemstones: the most comprehensive open-source scaling law dataset to date, consisting of over 4000 checkpoints from transformers with up to 2 billion parameters; these models have been trained with different learning rates, cooldown schedules, and architectural shapes. Our checkpoints enable more complex studies of scaling, such as a law that predicts language modeling performance as a function of model width and depth. By examining the various facets of our model suite, we find that the prescriptions of scaling laws can be highly sensitive to the experimental design process and the specific model checkpoints used during fitting. Code: https://github.com/mcleish7/gemstone-scaling-laws

It is becoming increasingly clear that many machine learning classifiers are vulnerable to adversarial examples. In attempting to explain the origin of adversarial examples, previous studies have typically focused on the fact that neural networks operate on high dimensional data, they overfit, or they are too linear. Here we argue that the origin of adversarial examples is primarily due to an inherent uncertainty that neural networks have about their predictions. We show that the functional form of this uncertainty is independent of architecture, dataset, and training protocol; and depends only on the statistics of the logit differences of the network, which do not change significantly during training. This leads to adversarial error having a universal scaling, as a power-law, with respect to the size of the adversarial perturbation. We show that this universality holds for a broad range of datasets (MNIST, CIFAR10, ImageNet, and random data), models (including state-of-the-art deep networks, linear models, adversarially trained networks, and networks trained on randomly shuffled labels), and attacks (FGSM, step l.l., PGD). Motivated by these results, we study the effects of reducing prediction entropy on adversarial robustness. Finally, we study the effect of network architectures on adversarial sensitivity. To do this, we use neural architecture search with reinforcement learning to find adversarially robust architectures on CIFAR10. Our resulting architecture is more robust to white and black box attacks compared to previous attempts.

We propose a new framework for estimating generative models via an adversarial process, in which we simultaneously train two models: a generative model G that captures the data distribution, and a discriminative model D that estimates the probability that a sample came from the training data rather than G. The training procedure for G is to maximize the probability of D making a mistake. This framework corresponds to a minimax two-player game. In the space of arbitrary functions G and D, a unique solution exists, with G recovering the training data distribution and D equal to 1/2 everywhere. In the case where G and D are defined by multilayer perceptrons, the entire system can be trained with backpropagation. There is no need for any Markov chains or unrolled approximate inference networks during either training or generation of samples. Experiments demonstrate the potential of the framework through qualitative and quantitative evaluation of the generated samples.

We examine the connections between deterministic, complete, and general global optimisation of continuous functions and a general concept of regression from the perspective of constructive type theory via the concept of 'searchability'. We see how the property of convergence of global optimisation is a straightforward consequence of searchability. The abstract setting allows us to generalise searchability and continuity to higher-order functions, so that we can formulate novel convergence criteria for regression, derived from the convergence of global optimisation. All the theory and the motivating examples are fully formalised in the proof assistant Agda.

We show that deep neural networks (DNNs) can efficiently learn any composition of functions with bounded F_{1}-norm, which allows DNNs to break the curse of dimensionality in ways that shallow networks cannot. More specifically, we derive a generalization bound that combines a covering number argument for compositionality, and the F_{1}-norm (or the related Barron norm) for large width adaptivity. We show that the global minimizer of the regularized loss of DNNs can fit for example the composition of two functions f^{*}=hcirc g from a small number of observations, assuming g is smooth/regular and reduces the dimensionality (e.g. g could be the modulo map of the symmetries of f^{*}), so that h can be learned in spite of its low regularity. The measures of regularity we consider is the Sobolev norm with different levels of differentiability, which is well adapted to the F_{1} norm. We compute scaling laws empirically and observe phase transitions depending on whether g or h is harder to learn, as predicted by our theory.

Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincar\'e ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincar\'e ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincar\'e ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM.

Recently, a new Continual Learning (CL) paradigm was presented to control catastrophic forgetting, called Interval Continual Learning (InterContiNet), which relies on enforcing interval constraints on the neural network parameter space. Unfortunately, InterContiNet training is challenging due to the high dimensionality of the weight space, making intervals difficult to manage. To address this issue, we introduce HyperInterval, a technique that employs interval arithmetic within the embedding space and utilizes a hypernetwork to map these intervals to the target network parameter space. We train interval embeddings for consecutive tasks and train a hypernetwork to transform these embeddings into weights of the target network. An embedding for a given task is trained along with the hypernetwork, preserving the response of the target network for the previous task embeddings. Interval arithmetic works with a more manageable, lower-dimensional embedding space rather than directly preparing intervals in a high-dimensional weight space. Our model allows faster and more efficient training. Furthermore, HyperInterval maintains the guarantee of not forgetting. At the end of training, we can choose one universal embedding to produce a single network dedicated to all tasks. In such a framework, hypernetwork is used only for training and can be seen as a meta-trainer. HyperInterval obtains significantly better results than InterContiNet and gives SOTA results on several benchmarks.

Previous work on Universal Transformers (UTs) has demonstrated the importance of parameter sharing across layers. By allowing recurrence in depth, UTs have advantages over standard Transformers in learning compositional generalizations, but layer-sharing comes with a practical limitation of parameter-compute ratio: it drastically reduces the parameter count compared to the non-shared model with the same dimensionality. Naively scaling up the layer size to compensate for the loss of parameters makes its computational resource requirements prohibitive. In practice, no previous work has succeeded in proposing a shared-layer Transformer design that is competitive in parameter count-dominated tasks such as language modeling. Here we propose MoEUT (pronounced "moot"), an effective mixture-of-experts (MoE)-based shared-layer Transformer architecture, which combines several recent advances in MoEs for both feedforward and attention layers of standard Transformers together with novel layer-normalization and grouping schemes that are specific and crucial to UTs. The resulting UT model, for the first time, slightly outperforms standard Transformers on language modeling tasks such as BLiMP and PIQA, while using significantly less compute and memory.

As its width tends to infinity, a deep neural network's behavior under gradient descent can become simplified and predictable (e.g. given by the Neural Tangent Kernel (NTK)), if it is parametrized appropriately (e.g. the NTK parametrization). However, we show that the standard and NTK parametrizations of a neural network do not admit infinite-width limits that can learn features, which is crucial for pretraining and transfer learning such as with BERT. We propose simple modifications to the standard parametrization to allow for feature learning in the limit. Using the *Tensor Programs* technique, we derive explicit formulas for such limits. On Word2Vec and few-shot learning on Omniglot via MAML, two canonical tasks that rely crucially on feature learning, we compute these limits exactly. We find that they outperform both NTK baselines and finite-width networks, with the latter approaching the infinite-width feature learning performance as width increases. More generally, we classify a natural space of neural network parametrizations that generalizes standard, NTK, and Mean Field parametrizations. We show 1) any parametrization in this space either admits feature learning or has an infinite-width training dynamics given by kernel gradient descent, but not both; 2) any such infinite-width limit can be computed using the Tensor Programs technique. Code for our experiments can be found at github.com/edwardjhu/TP4.

Large artificial neural networks have become a mainstay of language, vision, and audio processing and synthesis, yet their initializations and learning rates are often set in an unsophisticated fashion, due to the high cost of hyperparameter sweeps at scale. The mu-Parameterization (muP) offers a potential solution to this challenge, yielding scaling rules for model initialization and learning rates while reportedly enabling zero-shot hyperparameter transfer from small to large models. Despite its evident promise, the muP method is not yet widely adopted, perhaps due to higher implementation complexity, many variations, or complex theoretical background. This work investigates muP empirically, focusing on the ubiquitous transformer architecture, and aims to answer a simple question: does mu-Transfer yield optimal learning rates in practice? Studying models of up to 10B parameters and training budgets of up to 190B tokens, we find mu-Transfer works as intended for the majority of important cases, yet also identify a few cases where it may not.

In response to recent data regulation requirements, machine unlearning (MU) has emerged as a critical process to remove the influence of specific examples from a given model. Although exact unlearning can be achieved through complete model retraining using the remaining dataset, the associated computational costs have driven the development of efficient, approximate unlearning techniques. Moving beyond data-centric MU approaches, our study introduces a novel model-based perspective: model sparsification via weight pruning, which is capable of reducing the gap between exact unlearning and approximate unlearning. We show in both theory and practice that model sparsity can boost the multi-criteria unlearning performance of an approximate unlearner, closing the approximation gap, while continuing to be efficient. This leads to a new MU paradigm, termed prune first, then unlearn, which infuses a sparse model prior into the unlearning process. Building on this insight, we also develop a sparsity-aware unlearning method that utilizes sparsity regularization to enhance the training process of approximate unlearning. Extensive experiments show that our proposals consistently benefit MU in various unlearning scenarios. A notable highlight is the 77% unlearning efficacy gain of fine-tuning (one of the simplest unlearning methods) when using sparsity-aware unlearning. Furthermore, we demonstrate the practical impact of our proposed MU methods in addressing other machine learning challenges, such as defending against backdoor attacks and enhancing transfer learning. Codes are available at https://github.com/OPTML-Group/Unlearn-Sparse.

One method for obtaining generalizable solutions to machine learning tasks when presented with diverse training environments is to find invariant representations of the data. These are representations of the covariates such that the best model on top of the representation is invariant across training environments. In the context of linear Structural Equation Models (SEMs), invariant representations might allow us to learn models with out-of-distribution guarantees, i.e., models that are robust to interventions in the SEM. To address the invariant representation problem in a {\em finite sample} setting, we consider the notion of epsilon-approximate invariance. We study the following question: If a representation is approximately invariant with respect to a given number of training interventions, will it continue to be approximately invariant on a larger collection of unseen SEMs? This larger collection of SEMs is generated through a parameterized family of interventions. Inspired by PAC learning, we obtain finite-sample out-of-distribution generalization guarantees for approximate invariance that holds probabilistically over a family of linear SEMs without faithfulness assumptions. Our results show bounds that do not scale in ambient dimension when intervention sites are restricted to lie in a constant size subset of in-degree bounded nodes. We also show how to extend our results to a linear indirect observation model that incorporates latent variables.

All current formulations of thermodynamics invoke some form of coarse-graining or ensembles as the supposed link between their own laws and the microscopic laws of motion. They deal only with ensemble-averages, expectation values, macroscopic limits, infinite heat baths, etc., not with the details of physical variables of individual microscopic systems. They are consistent with the laws of motion for finite systems only in certain approximations, which improve with increasing scale, given various assumptions about initial conditions which are neither specified precisely nor even thought to hold exactly in nature. Here I propose a new formulation of the zeroth, first and second laws, improving upon the axiomatic approach to thermodynamics (Carath\'eodory, 1909; Lieb & Yngvason, 1999), via the principles of the recently proposed constructor theory. Specifically, I provide a non-approximative, scale-independent formulation of 'adiabatic accessibility'; this in turn provides a non-approximative, scale-independent distinction between work and heat and reveals an unexpected connection between information theory and the first law of thermodynamics (not just the second). It also achieves the long-sought unification of the axiomatic approach with Kelvin's.

Motivated by emerging applications such as live-streaming e-commerce, promotions and recommendations, we introduce and solve a general class of non-stationary multi-armed bandit problems that have the following two features: (i) the decision maker can pull and collect rewards from up to K,(ge 1) out of N different arms in each time period; (ii) the expected reward of an arm immediately drops after it is pulled, and then non-parametrically recovers as the arm's idle time increases. With the objective of maximizing the expected cumulative reward over T time periods, we design a class of ``Purely Periodic Policies'' that jointly set a period to pull each arm. For the proposed policies, we prove performance guarantees for both the offline problem and the online problems. For the offline problem when all model parameters are known, the proposed periodic policy obtains an approximation ratio that is at the order of 1-mathcal O(1/K), which is asymptotically optimal when K grows to infinity. For the online problem when the model parameters are unknown and need to be dynamically learned, we integrate the offline periodic policy with the upper confidence bound procedure to construct on online policy. The proposed online policy is proved to approximately have mathcal O(NT) regret against the offline benchmark. Our framework and policy design may shed light on broader offline planning and online learning applications with non-stationary and recovering rewards.

Deep Boltzmann machines (DBMs), one of the first ``deep'' learning methods ever studied, are multi-layered probabilistic models governed by a pairwise energy function that describes the likelihood of all variables/nodes in the network. In practice, DBMs are often constrained, i.e., via the restricted Boltzmann machine (RBM) architecture (which does not permit intra-layer connections), in order to allow for more efficient inference. In this work, we revisit the generic DBM approach, and ask the question: are there other possible restrictions to their design that would enable efficient (approximate) inference? In particular, we develop a new class of restricted model, the monotone DBM, which allows for arbitrary self-connection in each layer, but restricts the weights in a manner that guarantees the existence and global uniqueness of a mean-field fixed point. To do this, we leverage tools from the recently-proposed monotone Deep Equilibrium model and show that a particular choice of activation results in a fixed-point iteration that gives a variational mean-field solution. While this approach is still largely conceptual, it is the first architecture that allows for efficient approximate inference in fully-general weight structures for DBMs. We apply this approach to simple deep convolutional Boltzmann architectures and demonstrate that it allows for tasks such as the joint completion and classification of images, within a single deep probabilistic setting, while avoiding the pitfalls of mean-field inference in traditional RBMs.

Standard infinite-width limits of neural networks sacrifice the ability for intermediate layers to learn representations from data. Recent work (A theory of representation learning gives a deep generalisation of kernel methods, Yang et al. 2023) modified the Neural Network Gaussian Process (NNGP) limit of Bayesian neural networks so that representation learning is retained. Furthermore, they found that applying this modified limit to a deep Gaussian process gives a practical learning algorithm which they dubbed the deep kernel machine (DKM). However, they only considered the simplest possible setting: regression in small, fully connected networks with e.g. 10 input features. Here, we introduce convolutional deep kernel machines. This required us to develop a novel inter-domain inducing point approximation, as well as introducing and experimentally assessing a number of techniques not previously seen in DKMs, including analogues to batch normalisation, different likelihoods, and different types of top-layer. The resulting model trains in roughly 77 GPU hours, achieving around 99% test accuracy on MNIST, 72% on CIFAR-100, and 92.7% on CIFAR-10, which is SOTA for kernel methods.

We study the classic Text-to-Pattern Hamming Distances problem: given a pattern P of length m and a text T of length n, both over a polynomial-size alphabet, compute the Hamming distance between P and T[i, ., . , i+m-1] for every shift i, under the standard Word-RAM model with Theta(log n)-bit words. - We provide an O(nm) time Las Vegas randomized algorithm for this problem, beating the decades-old O(n m log m) running time [Abrahamson, SICOMP 1987]. We also obtain a deterministic algorithm, with a slightly higher O(nm(log mloglog m)^{1/4}) running time. Our randomized algorithm extends to the k-bounded setting, with running time Obig(n+nk{m}big), removing all the extra logarithmic factors from earlier algorithms [Gawrychowski and Uzna\'{n}ski, ICALP 2018; Chan, Golan, Kociumaka, Kopelowitz and Porat, STOC 2020]. - For the (1+epsilon)-approximate version of Text-to-Pattern Hamming Distances, we give an O(epsilon^{-0.93}n) time Monte Carlo randomized algorithm, beating the previous O(epsilon^{-1}n) running time [Kopelowitz and Porat, FOCS 2015; Kopelowitz and Porat, SOSA 2018]. Our approximation algorithm exploits a connection with 3SUM, and uses a combination of Fredman's trick, equality matrix product, and random sampling; in particular, we obtain new results on approximate counting versions of 3SUM and Exact Triangle, which may be of independent interest. Our exact algorithms use a novel combination of hashing, bit-packed FFT, and recursion; in particular, we obtain a faster algorithm for computing the sumset of two integer sets, in the regime when the universe size is close to quadratic in the number of elements. We also prove a fine-grained equivalence between the exact Text-to-Pattern Hamming Distances problem and a range-restricted, counting version of 3SUM.

On a variety of tasks, the performance of neural networks predictably improves with training time, dataset size and model size across many orders of magnitude. This phenomenon is known as a neural scaling law. Of fundamental importance is the compute-optimal scaling law, which reports the performance as a function of units of compute when choosing model sizes optimally. We analyze a random feature model trained with gradient descent as a solvable model of network training and generalization. This reproduces many observations about neural scaling laws. First, our model makes a prediction about why the scaling of performance with training time and with model size have different power law exponents. Consequently, the theory predicts an asymmetric compute-optimal scaling rule where the number of training steps are increased faster than model parameters, consistent with recent empirical observations. Second, it has been observed that early in training, networks converge to their infinite-width dynamics at a rate 1/width but at late time exhibit a rate width^{-c}, where c depends on the structure of the architecture and task. We show that our model exhibits this behavior. Lastly, our theory shows how the gap between training and test loss can gradually build up over time due to repeated reuse of data.

The equivalence of realizable and agnostic learnability is a fundamental phenomenon in learning theory. With variants ranging from classical settings like PAC learning and regression to recent trends such as adversarially robust learning, it's surprising that we still lack a unified theory; traditional proofs of the equivalence tend to be disparate, and rely on strong model-specific assumptions like uniform convergence and sample compression. In this work, we give the first model-independent framework explaining the equivalence of realizable and agnostic learnability: a three-line blackbox reduction that simplifies, unifies, and extends our understanding across a wide variety of settings. This includes models with no known characterization of learnability such as learning with arbitrary distributional assumptions and more general loss functions, as well as a host of other popular settings such as robust learning, partial learning, fair learning, and the statistical query model. More generally, we argue that the equivalence of realizable and agnostic learning is actually a special case of a broader phenomenon we call property generalization: any desirable property of a learning algorithm (e.g. noise tolerance, privacy, stability) that can be satisfied over finite hypothesis classes extends (possibly in some variation) to any learnable hypothesis class.

Neural networks can approximate complex functions, but they struggle to perform exact arithmetic operations over real numbers. The lack of inductive bias for arithmetic operations leaves neural networks without the underlying logic necessary to extrapolate on tasks such as addition, subtraction, and multiplication. We present two new neural network components: the Neural Addition Unit (NAU), which can learn exact addition and subtraction; and the Neural Multiplication Unit (NMU) that can multiply subsets of a vector. The NMU is, to our knowledge, the first arithmetic neural network component that can learn to multiply elements from a vector, when the hidden size is large. The two new components draw inspiration from a theoretical analysis of recently proposed arithmetic components. We find that careful initialization, restricting parameter space, and regularizing for sparsity is important when optimizing the NAU and NMU. Our proposed units NAU and NMU, compared with previous neural units, converge more consistently, have fewer parameters, learn faster, can converge for larger hidden sizes, obtain sparse and meaningful weights, and can extrapolate to negative and small values.

ImageNet serves as the primary dataset for evaluating the quality of computer-vision models. The common practice today is training each architecture with a tailor-made scheme, designed and tuned by an expert. In this paper, we present a unified scheme for training any backbone on ImageNet. The scheme, named USI (Unified Scheme for ImageNet), is based on knowledge distillation and modern tricks. It requires no adjustments or hyper-parameters tuning between different models, and is efficient in terms of training times. We test USI on a wide variety of architectures, including CNNs, Transformers, Mobile-oriented and MLP-only. On all models tested, USI outperforms previous state-of-the-art results. Hence, we are able to transform training on ImageNet from an expert-oriented task to an automatic seamless routine. Since USI accepts any backbone and trains it to top results, it also enables to perform methodical comparisons, and identify the most efficient backbones along the speed-accuracy Pareto curve. Implementation is available at:https://github.com/Alibaba-MIIL/Solving_ImageNet

Mathematical capabilities were previously believed to emerge in common language models only at a very large scale or require extensive math-related pre-training. This paper shows that the LLaMA-2 7B model with common pre-training already exhibits strong mathematical abilities, as evidenced by its impressive accuracy of 97.7% and 72.0% on the GSM8K and MATH benchmarks, respectively, when selecting the best response from 256 random generations. The primary issue with the current base model is the difficulty in consistently eliciting its inherent mathematical capabilities. Notably, the accuracy for the first answer drops to 49.5% and 7.9% on the GSM8K and MATH benchmarks, respectively. We find that simply scaling up the SFT data can significantly enhance the reliability of generating correct answers. However, the potential for extensive scaling is constrained by the scarcity of publicly available math questions. To overcome this limitation, we employ synthetic data, which proves to be nearly as effective as real data and shows no clear saturation when scaled up to approximately one million samples. This straightforward approach achieves an accuracy of 82.6% on GSM8K and 40.6% on MATH using LLaMA-2 7B models, surpassing previous models by 14.2% and 20.8%, respectively. We also provide insights into scaling behaviors across different reasoning complexities and error types.

UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction. UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology. The result is a practical scalable algorithm that applies to real world data. The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance. Furthermore, UMAP has no computational restrictions on embedding dimension, making it viable as a general purpose dimension reduction technique for machine learning.

The expressivity of Graph Neural Networks (GNNs) is dependent on the aggregation functions they employ. Theoretical works have pointed towards Sum aggregation GNNs subsuming every other GNNs, while certain practical works have observed a clear advantage to using Mean and Max. An examination of the theoretical guarantee identifies two caveats. First, it is size-restricted, that is, the power of every specific GNN is limited to graphs of a specific size. Successfully processing larger graphs may require an other GNN, and so on. Second, it concerns the power to distinguish non-isomorphic graphs, not the power to approximate general functions on graphs, and the former does not necessarily imply the latter. It is desired that a GNN's usability will not be limited to graphs of any specific size. Therefore, we explore the realm of unrestricted-size expressivity. We prove that basic functions, which can be computed exactly by Mean or Max GNNs, are inapproximable by any Sum GNN. We prove that under certain restrictions, every Mean or Max GNN can be approximated by a Sum GNN, but even there, a combination of (Sum, [Mean/Max]) is more expressive than Sum alone. Lastly, we prove further expressivity limitations for GNNs with a broad class of aggregations.

We find that the cross-entropy loss curves of neural language models empirically adhere to a scaling law with learning rate (LR) annealing over training steps (s): $L(s) = L_0 + Acdot S_1^{-alpha} - Ccdot S_2 Where S_1 is forward area and S_2$ is learning rate annealing area. This formulation takes into account two factors: (1) The forward scaling defined as typical scaling law, and (2) the additional loss drop brought by LR annealing. Therefore, this formulation can describe the full loss curve at each step, rather than the single loss point at the end of training. Applying the scaling law with LR annealing and fitting only one or two training curves, we can accurately predict the loss of language model training at any given step and across any learning rate scheduler (LRS). Furthermore, this equation accurately describes the dynamics during training process, and provides a theoretical verification and explanation for numerous experimental findings of previous studies, particularly those focusing on LR schedule and LR annealing. The resulting insights, also serve as a guide for researchers to select critical LRS in advance by prediction using our equation. Most significantly, since all the points in a full training curve follow the equation, we can achieve accurate loss prediction at any given step across any learning rate scheduler, while expending less than 1\% of the computational cost required by the chinchilla scaling law to fit language modeling loss. This approach extremely democratizes scaling law fitting and predicting in developing large language models.

We study the robust interpolation problem of arbitrary data distributions supported on a bounded space and propose a two-fold law of robustness. Robust interpolation refers to the problem of interpolating n noisy training data points in R^d by a Lipschitz function. Although this problem has been well understood when the samples are drawn from an isoperimetry distribution, much remains unknown concerning its performance under generic or even the worst-case distributions. We prove a Lipschitzness lower bound Omega(n/p) of the interpolating neural network with p parameters on arbitrary data distributions. With this result, we validate the law of robustness conjecture in prior work by Bubeck, Li, and Nagaraj on two-layer neural networks with polynomial weights. We then extend our result to arbitrary interpolating approximators and prove a Lipschitzness lower bound Omega(n^{1/d}) for robust interpolation. Our results demonstrate a two-fold law of robustness: i) we show the potential benefit of overparametrization for smooth data interpolation when n=poly(d), and ii) we disprove the potential existence of an O(1)-Lipschitz robust interpolating function when n=exp(omega(d)).

The cost of hyperparameter tuning in deep learning has been rising with model sizes, prompting practitioners to find new tuning methods using a proxy of smaller networks. One such proposal uses muP parameterized networks, where the optimal hyperparameters for small width networks transfer to networks with arbitrarily large width. However, in this scheme, hyperparameters do not transfer across depths. As a remedy, we study residual networks with a residual branch scale of 1/text{depth} in combination with the muP parameterization. We provide experiments demonstrating that residual architectures including convolutional ResNets and Vision Transformers trained with this parameterization exhibit transfer of optimal hyperparameters across width and depth on CIFAR-10 and ImageNet. Furthermore, our empirical findings are supported and motivated by theory. Using recent developments in the dynamical mean field theory (DMFT) description of neural network learning dynamics, we show that this parameterization of ResNets admits a well-defined feature learning joint infinite-width and infinite-depth limit and show convergence of finite-size network dynamics towards this limit.

Can Transformers predict new syllogisms by composing established ones? More generally, what type of targets can be learned by such models from scratch? Recent works show that Transformers can be Turing-complete in terms of expressivity, but this does not address the learnability objective. This paper puts forward the notion of 'globality degree' of a target distribution to capture when weak learning is efficiently achievable by regular Transformers, where the latter measures the least number of tokens required in addition to the tokens histogram to correlate nontrivially with the target. As shown experimentally and theoretically under additional assumptions, distributions with high globality cannot be learned efficiently. In particular, syllogisms cannot be composed on long chains. Furthermore, we show that (i) an agnostic scratchpad cannot help to break the globality barrier, (ii) an educated scratchpad can help if it breaks the globality at each step, however not all such scratchpads can generalize to out-of-distribution (OOD) samples, (iii) a notion of 'inductive scratchpad', that composes the prior information more efficiently, can both break the globality barrier and improve the OOD generalization. In particular, some inductive scratchpads can achieve length generalizations of up to 6x for some arithmetic tasks depending on the input formatting.

Training state-of-the-art neural networks requires a high cost in terms of compute and time. Model scale is recognized to be a critical factor to achieve and improve the state-of-the-art. Increasing the scale of a neural network normally requires restarting from scratch by randomly initializing all the parameters of the model, as this implies a change of architecture's parameters that does not allow for a straightforward transfer of knowledge from smaller size models. In this work, we propose six composable transformations to incrementally increase the size of transformer-based neural networks while preserving functionality, allowing to expand the capacity of the model as needed. We provide proof of exact function preservation under minimal initialization constraints for each transformation. The proposed methods may enable efficient training pipelines for larger and more powerful models by progressively expanding the architecture throughout training.

Approaches to continual learning aim to successfully learn a set of related tasks that arrive in an online manner. Recently, several frameworks have been developed which enable deep learning to be deployed in this learning scenario. A key modelling decision is to what extent the architecture should be shared across tasks. On the one hand, separately modelling each task avoids catastrophic forgetting but it does not support transfer learning and leads to large models. On the other hand, rigidly specifying a shared component and a task-specific part enables task transfer and limits the model size, but it is vulnerable to catastrophic forgetting and restricts the form of task-transfer that can occur. Ideally, the network should adaptively identify which parts of the network to share in a data driven way. Here we introduce such an approach called Continual Learning with Adaptive Weights (CLAW), which is based on probabilistic modelling and variational inference. Experiments show that CLAW achieves state-of-the-art performance on six benchmarks in terms of overall continual learning performance, as measured by classification accuracy, and in terms of addressing catastrophic forgetting.

A burgeoning line of research leverages deep neural networks to approximate the solutions to high dimensional PDEs, opening lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of dimensionality. However, most prior theoretical analyses have been limited to linear PDEs. In this work, we take a step towards studying the representational power of neural networks for approximating solutions to nonlinear PDEs. We focus on a class of PDEs known as nonlinear elliptic variational PDEs, whose solutions minimize an Euler-Lagrange energy functional E(u) = int_Omega L(x, u(x), nabla u(x)) - f(x) u(x)dx. We show that if composing a function with Barron norm b with partial derivatives of L produces a function of Barron norm at most B_L b^p, the solution to the PDE can be epsilon-approximated in the L^2 sense by a function with Barron norm Oleft(left(dB_Lright)^{max{p log(1/ epsilon), p^{log(1/epsilon)}}}right). By a classical result due to Barron [1993], this correspondingly bounds the size of a 2-layer neural network needed to approximate the solution. Treating p, epsilon, B_L as constants, this quantity is polynomial in dimension, thus showing neural networks can evade the curse of dimensionality. Our proof technique involves neurally simulating (preconditioned) gradient in an appropriate Hilbert space, which converges exponentially fast to the solution of the PDE, and such that we can bound the increase of the Barron norm at each iterate. Our results subsume and substantially generalize analogous prior results for linear elliptic PDEs over a unit hypercube.

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of effective species in an environment; when applied to a one-parameter family of metric spaces tX with scale parameter t, the magnitude captures much of the underlying geometry of the space. Prior work has mostly focussed on properties of magnitude in a global sense; in this paper we restrict the sets to finite subsets of Euclidean space and investigate its individual components. We give an explicit formula for the corrected inclusion-exclusion principle, and define a quantity associated with each point, called the moment which gives an intrinsic ordering to the points. We exploit this in order to form an algorithm which approximates the convex hull.

The approximation of Partial Differential Equations (PDEs) using neural networks has seen significant advancements through Physics-Informed Neural Networks (PINNs). Despite their straightforward optimization framework and flexibility in implementing various PDEs, PINNs often suffer from limited accuracy due to the spectral bias of Multi-Layer Perceptrons (MLPs), which struggle to effectively learn high-frequency and non-linear components. Recently, parametric mesh representations in combination with neural networks have been investigated as a promising approach to eliminate the inductive biases of neural networks. However, they usually require very high-resolution grids and a large number of collocation points to achieve high accuracy while avoiding overfitting issues. In addition, the fixed positions of the mesh parameters restrict their flexibility, making it challenging to accurately approximate complex PDEs. To overcome these limitations, we propose Physics-Informed Gaussians (PIGs), which combine feature embeddings using Gaussian functions with a lightweight neural network. Our approach uses trainable parameters for the mean and variance of each Gaussian, allowing for dynamic adjustment of their positions and shapes during training. This adaptability enables our model to optimally approximate PDE solutions, unlike models with fixed parameter positions. Furthermore, the proposed approach maintains the same optimization framework used in PINNs, allowing us to benefit from their excellent properties. Experimental results show the competitive performance of our model across various PDEs, demonstrating its potential as a robust tool for solving complex PDEs. Our project page is available at https://namgyukang.github.io/Physics-Informed-Gaussians/

Since the success of GPT, large language models (LLMs) have been revolutionizing machine learning and have initiated the so-called LLM prompting paradigm. In the era of LLMs, people train a single general-purpose LLM and provide the LLM with different prompts to perform different tasks. However, such empirical success largely lacks theoretical understanding. Here, we present the first theoretical study on the LLM prompting paradigm to the best of our knowledge. In this work, we show that prompting is in fact Turing-complete: there exists a finite-size Transformer such that for any computable function, there exists a corresponding prompt following which the Transformer computes the function. Furthermore, we show that even though we use only a single finite-size Transformer, it can still achieve nearly the same complexity bounds as that of the class of all unbounded-size Transformers. Overall, our result reveals that prompting can enable a single finite-size Transformer to be efficiently universal, which establishes a theoretical underpinning for prompt engineering in practice.

There has been a recent surge of interest in modeling neural networks (NNs) as Gaussian processes. In the limit of a NN of infinite width the NN becomes equivalent to a Gaussian process. Here we demonstrate that for an ensemble of large, finite, fully connected networks with a single hidden layer the distribution of outputs at initialization is well described by a Gaussian perturbed by the fourth Hermite polynomial for weights drawn from a symmetric distribution. We show that the scale of the perturbation is inversely proportional to the number of units in the NN and that higher order terms decay more rapidly, thereby recovering the Edgeworth expansion. We conclude by observing that understanding how this perturbation changes under training would reveal the regimes in which the Gaussian process framework is valid to model NN behavior.

Variational inference is becoming more and more popular for approximating intractable posterior distributions in Bayesian statistics and machine learning. Meanwhile, a few recent works have provided theoretical justification and new insights on deep neural networks for estimating smooth functions in usual settings such as nonparametric regression. In this paper, we show that variational inference for sparse deep learning retains the same generalization properties than exact Bayesian inference. In particular, we highlight the connection between estimation and approximation theories via the classical bias-variance trade-off and show that it leads to near-minimax rates of convergence for H\"older smooth functions. Additionally, we show that the model selection framework over the neural network architecture via ELBO maximization does not overfit and adaptively achieves the optimal rate of convergence.

We propose a new localized inference algorithm for answering marginalization queries in large graphical models with the correlation decay property. Given a query variable and a large graphical model, we define a much smaller model in a local region around the query variable in the target model so that the marginal distribution of the query variable can be accurately approximated. We introduce two approximation error bounds based on the Dobrushin's comparison theorem and apply our bounds to derive a greedy expansion algorithm that efficiently guides the selection of neighbor nodes for localized inference. We verify our theoretical bounds on various datasets and demonstrate that our localized inference algorithm can provide fast and accurate approximation for large graphical models.

We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples (x,y) from an unknown distribution on R^n times { pm 1}, whose marginal distribution on x is the standard Gaussian and the labels y can be arbitrary, the goal is to output a hypothesis with 0-1 loss OPT+epsilon, where OPT is the 0-1 loss of the best-fitting halfspace. We prove a near-optimal computational hardness result for this task, under the widely believed sub-exponential time hardness of the Learning with Errors (LWE) problem. Prior hardness results are either qualitatively suboptimal or apply to restricted families of algorithms. Our techniques extend to yield near-optimal lower bounds for related problems, including ReLU regression.

We report a flexible language-model based deep learning strategy, applied here to solve complex forward and inverse problems in protein modeling, based on an attention neural network that integrates transformer and graph convolutional architectures in a causal multi-headed graph mechanism, to realize a generative pretrained model. The model is applied to predict secondary structure content (per-residue level and overall content), protein solubility, and sequencing tasks. Further trained on inverse tasks, the model is rendered capable of designing proteins with these properties as target features. The model is formulated as a general framework, completely prompt-based, and can be adapted for a variety of downstream tasks. We find that adding additional tasks yields emergent synergies that the model exploits in improving overall performance, beyond what would be possible by training a model on each dataset alone. Case studies are presented to validate the method, yielding protein designs specifically focused on structural proteins, but also exploring the applicability in the design of soluble, antimicrobial biomaterials. While our model is trained to ultimately perform 8 distinct tasks, with available datasets it can be extended to solve additional problems. In a broader sense, this work illustrates a form of multiscale modeling that relates a set of ultimate building blocks (here, byte-level utf8 characters) to complex output. This materiomic scheme captures complex emergent relationships between universal building block and resulting properties via a synergizing learning capacity to express a set of potentialities embedded in the knowledge used in training, via the interplay of universality and diversity.

Numerical simulations in climate, chemistry, or astrophysics are computationally too expensive for uncertainty quantification or parameter-exploration at high-resolution. Reduced-order or surrogate models are multiple orders of magnitude faster, but traditional surrogates are inflexible or inaccurate and pure machine learning (ML)-based surrogates too data-hungry. We propose a hybrid, flexible surrogate model that exploits known physics for simulating large-scale dynamics and limits learning to the hard-to-model term, which is called parametrization or closure and captures the effect of fine- onto large-scale dynamics. Leveraging neural operators, we are the first to learn grid-independent, non-local, and flexible parametrizations. Our multiscale neural operator is motivated by a rich literature in multiscale modeling, has quasilinear runtime complexity, is more accurate or flexible than state-of-the-art parametrizations and demonstrated on the chaotic equation multiscale Lorenz96.

Scaling laws play an instrumental role in the sustainable improvement in model quality. Unfortunately, recommendation models to date do not exhibit such laws similar to those observed in the domain of large language models, due to the inefficiencies of their upscaling mechanisms. This limitation poses significant challenges in adapting these models to increasingly more complex real-world datasets. In this paper, we propose an effective network architecture based purely on stacked factorization machines, and a synergistic upscaling strategy, collectively dubbed Wukong, to establish a scaling law in the domain of recommendation. Wukong's unique design makes it possible to capture diverse, any-order of interactions simply through taller and wider layers. We conducted extensive evaluations on six public datasets, and our results demonstrate that Wukong consistently outperforms state-of-the-art models quality-wise. Further, we assessed Wukong's scalability on an internal, large-scale dataset. The results show that Wukong retains its superiority in quality over state-of-the-art models, while holding the scaling law across two orders of magnitude in model complexity, extending beyond 100 Gflop or equivalently up to GPT-3/LLaMa-2 scale of total training compute, where prior arts fall short.

Post-training has emerged as a crucial paradigm for adapting large-scale pre-trained models to various tasks, whose effects are fully reflected by delta parameters (i.e., the disparity between post-trained and pre-trained parameters). While numerous studies have explored delta parameter properties via operations like pruning, quantization, low-rank approximation, and extrapolation, a unified framework for systematically examining these characteristics has been lacking. In this paper, we propose a novel perspective based on Riemann sum approximation of the loss function to elucidate delta parameter editing operations. Our analysis categorizes existing methods into three classes based on their post-editing performance: competitive, decreased, and improved, explaining how they are expressed by the Riemann sum approximation term and how they alter the model performance. Extensive experiments on both visual and language models, including ViT, LLaMA 3, Qwen 2, and Mistral, corroborate our theoretical findings. Furthermore, we introduce extensions to existing techniques like DARE and BitDelta, highlighting their limitations in leveraging the properties of delta parameters and reorganizing them into general expressions to enhance the applicability and effectiveness of delta parameter editing in post-trained models.

The ever-growing ecosystem of LLMs has posed a challenge in selecting the most appropriate pre-trained model to fine-tune amidst a sea of options. Given constrained resources, fine-tuning all models and making selections afterward is unrealistic. In this work, we formulate this resource-constrained selection task into predicting fine-tuning performance and illustrate its natural connection with scaling laws. Unlike pre-training, We find that the fine-tuning scaling curve includes not just the well-known "power phase" but also the previously unobserved "pre-power phase". We also explain why existing scaling laws fail to capture this phase transition phenomenon both theoretically and empirically. To address this, we introduce the concept of "pre-learned data size" into our rectified scaling law, which overcomes theoretical limitations and fits experimental results much better. By leveraging our law, we propose a novel LLM selection algorithm that selects the near-optimal model with hundreds of times less resource consumption, while other methods may provide negatively correlated selection.

k-Clustering in R^d (e.g., k-median and k-means) is a fundamental machine learning problem. While near-linear time approximation algorithms were known in the classical setting for a dataset with cardinality n, it remains open to find sublinear-time quantum algorithms. We give quantum algorithms that find coresets for k-clustering in R^d with O(nkd^{3/2}) query complexity. Our coreset reduces the input size from n to poly(kepsilon^{-1}d), so that existing alpha-approximation algorithms for clustering can run on top of it and yield (1 + epsilon)alpha-approximation. This eventually yields a quadratic speedup for various k-clustering approximation algorithms. We complement our algorithm with a nearly matching lower bound, that any quantum algorithm must make Omega(nk) queries in order to achieve even O(1)-approximation for k-clustering.

A common statistical task lies in showing asymptotic normality of certain statistics. In many of these situations, classical textbook results on weak convergence theory suffice for the problem at hand. However, there are quite some scenarios where stronger results are needed in order to establish an asymptotic normal approximation uniformly over a family of probability measures. In this note we collect some results in this direction. We restrict ourselves to weak convergence in mathbb R^d with continuous limit measures.

Scaling law principles indicate a power-law correlation between loss and variables such as model size, dataset size, and computational resources utilized during training. These principles play a vital role in optimizing various aspects of model pre-training, ultimately contributing to the success of large language models such as GPT-4, Llama and Gemini. However, the original scaling law paper by OpenAI did not disclose the complete details necessary to derive the precise scaling law formulas, and their conclusions are only based on models containing up to 1.5 billion parameters. Though some subsequent works attempt to unveil these details and scale to larger models, they often neglect the training dependency of important factors such as the learning rate, context length and batch size, leading to their failure to establish a reliable formula for predicting the test loss trajectory. In this technical report, we confirm that the scaling law formulations proposed in the original OpenAI paper remain valid when scaling the model size up to 33 billion, but the constant coefficients in these formulas vary significantly with the experiment setup. We meticulously identify influential factors and provide transparent, step-by-step instructions to estimate all constant terms in scaling-law formulas by training on models with only 1M~60M parameters. Using these estimated formulas, we showcase the capability to accurately predict various attributes for models with up to 33B parameters before their training, including (1) the minimum possible test loss; (2) the minimum required training steps and processed tokens to achieve a specific loss; (3) the critical batch size with an optimal time/computation trade-off at any loss value; and (4) the complete test loss trajectory with arbitrary batch size.

We consider the problem of global optimization with noisy zeroth order oracles - a well-motivated problem useful for various applications ranging from hyper-parameter tuning for deep learning to new material design. Existing work relies on Gaussian processes or other non-parametric family, which suffers from the curse of dimensionality. In this paper, we propose a new algorithm GO-UCB that leverages a parametric family of functions (e.g., neural networks) instead. Under a realizable assumption and a few other mild geometric conditions, we show that GO-UCB achieves a cumulative regret of O(T) where T is the time horizon. At the core of GO-UCB is a carefully designed uncertainty set over parameters based on gradients that allows optimistic exploration. Synthetic and real-world experiments illustrate GO-UCB works better than Bayesian optimization approaches in high dimensional cases, even if the model is misspecified.

We propose a new approach for generative modeling based on training a neural network to be idempotent. An idempotent operator is one that can be applied sequentially without changing the result beyond the initial application, namely f(f(z))=f(z). The proposed model f is trained to map a source distribution (e.g, Gaussian noise) to a target distribution (e.g. realistic images) using the following objectives: (1) Instances from the target distribution should map to themselves, namely f(x)=x. We define the target manifold as the set of all instances that f maps to themselves. (2) Instances that form the source distribution should map onto the defined target manifold. This is achieved by optimizing the idempotence term, f(f(z))=f(z) which encourages the range of f(z) to be on the target manifold. Under ideal assumptions such a process provably converges to the target distribution. This strategy results in a model capable of generating an output in one step, maintaining a consistent latent space, while also allowing sequential applications for refinement. Additionally, we find that by processing inputs from both target and source distributions, the model adeptly projects corrupted or modified data back to the target manifold. This work is a first step towards a ``global projector'' that enables projecting any input into a target data distribution.

Hoffmann et al. (2022) propose three methods for estimating a compute-optimal scaling law. We attempt to replicate their third estimation procedure, which involves fitting a parametric loss function to a reconstruction of data from their plots. We find that the reported estimates are inconsistent with their first two estimation methods, fail at fitting the extracted data, and report implausibly narrow confidence intervals--intervals this narrow would require over 600,000 experiments, while they likely only ran fewer than 500. In contrast, our rederivation of the scaling law using the third approach yields results that are compatible with the findings from the first two estimation procedures described by Hoffmann et al.

We study empirical scaling laws for language model performance on the cross-entropy loss. The loss scales as a power-law with model size, dataset size, and the amount of compute used for training, with some trends spanning more than seven orders of magnitude. Other architectural details such as network width or depth have minimal effects within a wide range. Simple equations govern the dependence of overfitting on model/dataset size and the dependence of training speed on model size. These relationships allow us to determine the optimal allocation of a fixed compute budget. Larger models are significantly more sample-efficient, such that optimally compute-efficient training involves training very large models on a relatively modest amount of data and stopping significantly before convergence.

This paper is the second in the series Commutative Scaling of Width and Depth (WD) about commutativity of infinite width and depth limits in deep neural networks. Our aim is to understand the behaviour of neural functions (functions that depend on a neural network model) as width and depth go to infinity (in some sense), and eventually identify settings under which commutativity holds, i.e. the neural function tends to the same limit no matter how width and depth limits are taken. In this paper, we formally introduce and define the commutativity framework, and discuss its implications on neural network design and scaling. We study commutativity for the neural covariance kernel which reflects how network layers separate data. Our findings extend previous results established in [55] by showing that taking the width and depth to infinity in a deep neural network with skip connections, when branches are suitably scaled to avoid exploding behaviour, result in the same covariance structure no matter how that limit is taken. This has a number of theoretical and practical implications that we discuss in the paper. The proof techniques in this paper are novel and rely on tools that are more accessible to readers who are not familiar with stochastic calculus (used in the proofs of WD(I))).

We study a distributed stochastic multi-armed bandit where a client supplies the learner with communication-constrained feedback based on the rewards for the corresponding arm pulls. In our setup, the client must encode the rewards such that the second moment of the encoded rewards is no more than P, and this encoded reward is further corrupted by additive Gaussian noise of variance sigma^2; the learner only has access to this corrupted reward. For this setting, we derive an information-theoretic lower bound of Omegaleft(frac{KT{SNR wedge1}} right) on the minimax regret of any scheme, where SNR := P{sigma^2}, and K and T are the number of arms and time horizon, respectively. Furthermore, we propose a multi-phase bandit algorithm, UEtext{-UCB++}, which matches this lower bound to a minor additive factor. UEtext{-UCB++} performs uniform exploration in its initial phases and then utilizes the {\em upper confidence bound }(UCB) bandit algorithm in its final phase. An interesting feature of UEtext{-UCB++} is that the coarser estimates of the mean rewards formed during a uniform exploration phase help to refine the encoding protocol in the next phase, leading to more accurate mean estimates of the rewards in the subsequent phase. This positive reinforcement cycle is critical to reducing the number of uniform exploration rounds and closely matching our lower bound.

Hyperparameter tuning of deep learning models can lead to order-of-magnitude performance gains for the same amount of compute. Despite this, systematic tuning is uncommon, particularly for large models, which are expensive to evaluate and tend to have many hyperparameters, necessitating difficult judgment calls about tradeoffs, budgets, and search bounds. To address these issues and propose a practical method for robustly tuning large models, we present Cost-Aware Pareto Region Bayesian Search (CARBS), a Bayesian optimization algorithm that performs local search around the performance-cost Pareto frontier. CARBS does well even in unbounded search spaces with many hyperparameters, learns scaling relationships so that it can tune models even as they are scaled up, and automates much of the "black magic" of tuning. Among our results, we effectively solve the entire ProcGen benchmark just by tuning a simple baseline (PPO, as provided in the original ProcGen paper). We also reproduce the model size vs. training tokens scaling result from the Chinchilla project (Hoffmann et al. 2022), while simultaneously discovering scaling laws for every other hyperparameter, via an easy automated process that uses significantly less compute and is applicable to any deep learning problem (not just language models).

We propose algorithms for approximate filtering and smoothing in high-dimensional Factorial hidden Markov models. The approximation involves discarding, in a principled way, likelihood factors according to a notion of locality in a factor graph associated with the emission distribution. This allows the exponential-in-dimension cost of exact filtering and smoothing to be avoided. We prove that the approximation accuracy, measured in a local total variation norm, is "dimension-free" in the sense that as the overall dimension of the model increases the error bounds we derive do not necessarily degrade. A key step in the analysis is to quantify the error introduced by localizing the likelihood function in a Bayes' rule update. The factorial structure of the likelihood function which we exploit arises naturally when data have known spatial or network structure. We demonstrate the new algorithms on synthetic examples and a London Underground passenger flow problem, where the factor graph is effectively given by the train network.

We consider the solvable neural scaling model with three parameters: data complexity, target complexity, and model-parameter-count. We use this neural scaling model to derive new predictions about the compute-limited, infinite-data scaling law regime. To train the neural scaling model, we run one-pass stochastic gradient descent on a mean-squared loss. We derive a representation of the loss curves which holds over all iteration counts and improves in accuracy as the model parameter count grows. We then analyze the compute-optimal model-parameter-count, and identify 4 phases (+3 subphases) in the data-complexity/target-complexity phase-plane. The phase boundaries are determined by the relative importance of model capacity, optimizer noise, and embedding of the features. We furthermore derive, with mathematical proof and extensive numerical evidence, the scaling-law exponents in all of these phases, in particular computing the optimal model-parameter-count as a function of floating point operation budget.

We show that transformer-based large language models are computationally universal when augmented with an external memory. Any deterministic language model that conditions on strings of bounded length is equivalent to a finite automaton, hence computationally limited. However, augmenting such models with a read-write memory creates the possibility of processing arbitrarily large inputs and, potentially, simulating any algorithm. We establish that an existing large language model, Flan-U-PaLM 540B, can be combined with an associative read-write memory to exactly simulate the execution of a universal Turing machine, U_{15,2}. A key aspect of the finding is that it does not require any modification of the language model weights. Instead, the construction relies solely on designing a form of stored instruction computer that can subsequently be programmed with a specific set of prompts.

Much research in machine learning involves finding appropriate inductive biases (e.g. convolutional neural networks, momentum-based optimizers, transformers) to promote generalization on tasks. However, quantification of the amount of inductive bias associated with these architectures and hyperparameters has been limited. We propose a novel method for efficiently computing the inductive bias required for generalization on a task with a fixed training data budget; formally, this corresponds to the amount of information required to specify well-generalizing models within a specific hypothesis space of models. Our approach involves modeling the loss distribution of random hypotheses drawn from a hypothesis space to estimate the required inductive bias for a task relative to these hypotheses. Unlike prior work, our method provides a direct estimate of inductive bias without using bounds and is applicable to diverse hypothesis spaces. Moreover, we derive approximation error bounds for our estimation approach in terms of the number of sampled hypotheses. Consistent with prior results, our empirical results demonstrate that higher dimensional tasks require greater inductive bias. We show that relative to other expressive model classes, neural networks as a model class encode large amounts of inductive bias. Furthermore, our measure quantifies the relative difference in inductive bias between different neural network architectures. Our proposed inductive bias metric provides an information-theoretic interpretation of the benefits of specific model architectures for certain tasks and provides a quantitative guide to developing tasks requiring greater inductive bias, thereby encouraging the development of more powerful inductive biases.

In this paper I apply newly-proposed information-theoretic principles to thermodynamic work extraction. I show that if it is possible to extract work deterministically from a physical system prepared in any one of a set of states, then those states must be distinguishable from one another. This result is formulated independently of scale and of particular dynamical laws; it also provides a novel connection between thermodynamics and information theory, established via the law of conservation of energy (rather than the second law of thermodynamics). Albeit compatible with these conclusions, existing thermodynamics approaches cannot provide a result of such generality, because they are scale-dependent (relying on ensembles or coarse-graining) or tied to particular dynamical laws. This paper thus provides a broader foundation for thermodynamics, with implications for the theory of von Neumann's universal constructor

Convolutional neural networks (CNN) are built upon the classical McCulloch-Pitts neuron model, which is essentially a linear model, where the nonlinearity is provided by a separate activation function. Several researchers have proposed enhanced neuron models, including quadratic neurons, generalized operational neurons, generative neurons, and super neurons, with stronger nonlinearity than that provided by the pointwise activation function. There has also been a proposal to use Pade approximation as a generalized activation function. In this paper, we introduce a brand new neuron model called Pade neurons (Paons), inspired by the Pade approximants, which is the best mathematical approximation of a transcendental function as a ratio of polynomials with different orders. We show that Paons are a super set of all other proposed neuron models. Hence, the basic neuron in any known CNN model can be replaced by Paons. In this paper, we extend the well-known ResNet to PadeNet (built by Paons) to demonstrate the concept. Our experiments on the single-image super-resolution task show that PadeNets can obtain better results than competing architectures.

We study the representation complexity of model-based and model-free reinforcement learning (RL) in the context of circuit complexity. We prove theoretically that there exists a broad class of MDPs such that their underlying transition and reward functions can be represented by constant depth circuits with polynomial size, while the optimal Q-function suffers an exponential circuit complexity in constant-depth circuits. By drawing attention to the approximation errors and building connections to complexity theory, our theory provides unique insights into why model-based algorithms usually enjoy better sample complexity than model-free algorithms from a novel representation complexity perspective: in some cases, the ground-truth rule (model) of the environment is simple to represent, while other quantities, such as Q-function, appear complex. We empirically corroborate our theory by comparing the approximation error of the transition kernel, reward function, and optimal Q-function in various Mujoco environments, which demonstrates that the approximation errors of the transition kernel and reward function are consistently lower than those of the optimal Q-function. To the best of our knowledge, this work is the first to study the circuit complexity of RL, which also provides a rigorous framework for future research.

Global climate models typically operate at a grid resolution of hundreds of kilometers and fail to resolve atmospheric mesoscale processes, e.g., clouds, precipitation, and gravity waves (GWs). Model representation of these processes and their sources is essential to the global circulation and planetary energy budget, but subgrid scale contributions from these processes are often only approximately represented in models using parameterizations. These parameterizations are subject to approximations and idealizations, which limit their capability and accuracy. The most drastic of these approximations is the "single-column approximation" which completely neglects the horizontal evolution of these processes, resulting in key biases in current climate models. With a focus on atmospheric GWs, we present the first-ever global simulation of atmospheric GW fluxes using machine learning (ML) models trained on the WINDSET dataset to emulate global GW emulation in the atmosphere, as an alternative to traditional single-column parameterizations. Using an Attention U-Net-based architecture trained on globally resolved GW momentum fluxes, we illustrate the importance and effectiveness of global nonlocality, when simulating GWs using data-driven schemes.

Overparametrization often helps improve the generalization performance. This paper proposes a dual view of overparametrization suggesting that downsampling may also help generalize. Motivated by this dual view, we characterize two out-of-sample prediction risks of the sketched ridgeless least square estimator in the proportional regime masymp n asymp p, where m is the sketching size, n the sample size, and p the feature dimensionality. Our results reveal the statistical role of downsampling. Specifically, downsampling does not always hurt the generalization performance, and may actually help improve it in some cases. We identify the optimal sketching sizes that minimize the out-of-sample prediction risks, and find that the optimally sketched estimator has stabler risk curves that eliminates the peaks of those for the full-sample estimator. We then propose a practical procedure to empirically identify the optimal sketching size. Finally, we extend our results to cover central limit theorems and misspecified models. Numerical studies strongly support our theory.

A supervised learning algorithm searches over a set of functions A to B parametrised by a space P to find the best approximation to some ideal function fcolon A to B. It does this by taking examples (a,f(a)) in Atimes B, and updating the parameter according to some rule. We define a category where these update rules may be composed, and show that gradient descent---with respect to a fixed step size and an error function satisfying a certain property---defines a monoidal functor from a category of parametrised functions to this category of update rules. This provides a structural perspective on backpropagation, as well as a broad generalisation of neural networks.

In a recent work, Laforgue et al. introduce the model of last switch dependent (LSD) bandits, in an attempt to capture nonstationary phenomena induced by the interaction between the player and the environment. Examples include satiation, where consecutive plays of the same action lead to decreased performance, or deprivation, where the payoff of an action increases after an interval of inactivity. In this work, we take a step towards understanding the approximability of planning LSD bandits, namely, the (NP-hard) problem of computing an optimal arm-pulling strategy under complete knowledge of the model. In particular, we design the first efficient constant approximation algorithm for the problem and show that, under a natural monotonicity assumption on the payoffs, its approximation guarantee (almost) matches the state-of-the-art for the special and well-studied class of recharging bandits (also known as delay-dependent). In this attempt, we develop new tools and insights for this class of problems, including a novel higher-dimensional relaxation and the technique of mirroring the evolution of virtual states. We believe that these novel elements could potentially be used for approaching richer classes of action-induced nonstationary bandits (e.g., special instances of restless bandits). In the case where the model parameters are initially unknown, we develop an online learning adaptation of our algorithm for which we provide sublinear regret guarantees against its full-information counterpart.

A great deal of effort has been devoted to reducing the risk of spurious scientific discoveries, from the use of sophisticated validation techniques, to deep statistical methods for controlling the false discovery rate in multiple hypothesis testing. However, there is a fundamental disconnect between the theoretical results and the practice of data analysis: the theory of statistical inference assumes a fixed collection of hypotheses to be tested, or learning algorithms to be applied, selected non-adaptively before the data are gathered, whereas in practice data is shared and reused with hypotheses and new analyses being generated on the basis of data exploration and the outcomes of previous analyses. In this work we initiate a principled study of how to guarantee the validity of statistical inference in adaptive data analysis. As an instance of this problem, we propose and investigate the question of estimating the expectations of m adaptively chosen functions on an unknown distribution given n random samples. We show that, surprisingly, there is a way to estimate an exponential in n number of expectations accurately even if the functions are chosen adaptively. This gives an exponential improvement over standard empirical estimators that are limited to a linear number of estimates. Our result follows from a general technique that counter-intuitively involves actively perturbing and coordinating the estimates, using techniques developed for privacy preservation. We give additional applications of this technique to our question.

One of the main challenges in optimal scaling of large language models (LLMs) is the prohibitive cost of hyperparameter tuning, particularly learning rate eta and batch size B. While techniques like muP (Yang et al., 2022) provide scaling rules for optimal eta transfer in the infinite model size limit, the optimal scaling behavior in the infinite data size limit remains unknown. We fill in this gap by observing for the first time an intricate dependence of optimal eta scaling on the pretraining token budget T, B and its relation to the critical batch size B_crit, which we measure to evolve as B_crit propto T. Furthermore, we show that the optimal batch size is positively correlated with B_crit: keeping it fixed becomes suboptimal over time even if learning rate is scaled optimally. Surprisingly, our results demonstrate that the observed optimal eta and B dynamics are preserved with muP model scaling, challenging the conventional view of B_crit dependence solely on loss value. Complementing optimality, we examine the sensitivity of loss to changes in learning rate, where we find the sensitivity to decrease with increase of T and to remain constant with muP model scaling. We hope our results make the first step towards a unified picture of the joint optimal data and model scaling.

In recent years, the state-of-the-art in deep learning has been dominated by very large models that have been pre-trained on vast amounts of data. The paradigm is very simple: Investing more computational resources (optimally) leads to better performance, and even predictably so; neural scaling laws have been derived that accurately forecast the performance of a network for a desired level of compute. This leads to the notion of a "compute-optimal" model, i.e. a model that allocates a given level of compute during training optimally to maximise performance. In this work, we extend the concept of optimality by allowing for an "adaptive" model, i.e. a model that can change its shape during the course of training. By allowing the shape to adapt, we can optimally traverse between the underlying scaling laws, leading to a significant reduction in the required compute to reach a given target performance. We focus on vision tasks and the family of Vision Transformers, where the patch size as well as the width naturally serve as adaptive shape parameters. We demonstrate that, guided by scaling laws, we can design compute-optimal adaptive models that beat their "static" counterparts.

Numerical approximations of partial differential equations (PDEs) are routinely employed to formulate the solution of physics, engineering and mathematical problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, and more. While this has led to solving many complex phenomena, there are some limitations. Conventional approaches such as Finite Element Methods (FEMs) and Finite Differential Methods (FDMs) require considerable time and are computationally expensive. In contrast, data driven machine learning-based methods such as neural networks provide a faster, fairly accurate alternative, and have certain advantages such as discretization invariance and resolution invariance. This article aims to provide a comprehensive insight into how data-driven approaches can complement conventional techniques to solve engineering and physics problems, while also noting some of the major pitfalls of machine learning-based approaches. Furthermore, we highlight, a novel and fast machine learning-based approach (~1000x) to learning the solution operator of a PDE operator learning. We will note how these new computational approaches can bring immense advantages in tackling many problems in fundamental and applied physics.

We propose the use of low bit-depth Sigma-Delta and distributed noise-shaping methods for quantizing the Random Fourier features (RFFs) associated with shift-invariant kernels. We prove that our quantized RFFs -- even in the case of 1-bit quantization -- allow a high accuracy approximation of the underlying kernels, and the approximation error decays at least polynomially fast as the dimension of the RFFs increases. We also show that the quantized RFFs can be further compressed, yielding an excellent trade-off between memory use and accuracy. Namely, the approximation error now decays exponentially as a function of the bits used. Moreover, we empirically show by testing the performance of our methods on several machine learning tasks that our method compares favorably to other state of the art quantization methods in this context.

We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for partial differential equations. Algorithms for approximating the solution to these systems are often the bottleneck in problems that require their solution, particularly for modern applications that require many millions of unknowns. Indeed, numerical linear algebra techniques have been investigated for many decades to alleviate this computational burden. Recently, data-driven techniques have also shown promise for these problems. Motivated by the conjugate gradients algorithm that iteratively selects search directions for minimizing the matrix norm of the approximation error, we design an approach that utilizes a deep neural network to accelerate convergence via data-driven improvement of the search directions. Our method leverages a carefully chosen convolutional network to approximate the action of the inverse of the linear operator up to an arbitrary constant. We train the network using unsupervised learning with a loss function equal to the L^2 difference between an input and the system matrix times the network evaluation, where the unspecified constant in the approximate inverse is accounted for. We demonstrate the efficacy of our approach on spatially discretized Poisson equations with millions of degrees of freedom arising in computational fluid dynamics applications. Unlike state-of-the-art learning approaches, our algorithm is capable of reducing the linear system residual to a given tolerance in a small number of iterations, independent of the problem size. Moreover, our method generalizes effectively to various systems beyond those encountered during training.

We present Universal Manipulation Interface (UMI) -- a data collection and policy learning framework that allows direct skill transfer from in-the-wild human demonstrations to deployable robot policies. UMI employs hand-held grippers coupled with careful interface design to enable portable, low-cost, and information-rich data collection for challenging bimanual and dynamic manipulation demonstrations. To facilitate deployable policy learning, UMI incorporates a carefully designed policy interface with inference-time latency matching and a relative-trajectory action representation. The resulting learned policies are hardware-agnostic and deployable across multiple robot platforms. Equipped with these features, UMI framework unlocks new robot manipulation capabilities, allowing zero-shot generalizable dynamic, bimanual, precise, and long-horizon behaviors, by only changing the training data for each task. We demonstrate UMI's versatility and efficacy with comprehensive real-world experiments, where policies learned via UMI zero-shot generalize to novel environments and objects when trained on diverse human demonstrations. UMI's hardware and software system is open-sourced at https://umi-gripper.github.io.

We introduce Universal NER (UNER), an open, community-driven project to develop gold-standard NER benchmarks in many languages. The overarching goal of UNER is to provide high-quality, cross-lingually consistent annotations to facilitate and standardize multilingual NER research. UNER v1 contains 18 datasets annotated with named entities in a cross-lingual consistent schema across 12 diverse languages. In this paper, we detail the dataset creation and composition of UNER; we also provide initial modeling baselines on both in-language and cross-lingual learning settings. We release the data, code, and fitted models to the public.

Inductive transfer learning has greatly impacted computer vision, but existing approaches in NLP still require task-specific modifications and training from scratch. We propose Universal Language Model Fine-tuning (ULMFiT), an effective transfer learning method that can be applied to any task in NLP, and introduce techniques that are key for fine-tuning a language model. Our method significantly outperforms the state-of-the-art on six text classification tasks, reducing the error by 18-24% on the majority of datasets. Furthermore, with only 100 labeled examples, it matches the performance of training from scratch on 100x more data. We open-source our pretrained models and code.

Universal source separation (USS) is a fundamental research task for computational auditory scene analysis, which aims to separate mono recordings into individual source tracks. There are three potential challenges awaiting the solution to the audio source separation task. First, previous audio source separation systems mainly focus on separating one or a limited number of specific sources. There is a lack of research on building a unified system that can separate arbitrary sources via a single model. Second, most previous systems require clean source data to train a separator, while clean source data are scarce. Third, there is a lack of USS system that can automatically detect and separate active sound classes in a hierarchical level. To use large-scale weakly labeled/unlabeled audio data for audio source separation, we propose a universal audio source separation framework containing: 1) an audio tagging model trained on weakly labeled data as a query net; and 2) a conditional source separation model that takes query net outputs as conditions to separate arbitrary sound sources. We investigate various query nets, source separation models, and training strategies and propose a hierarchical USS strategy to automatically detect and separate sound classes from the AudioSet ontology. By solely leveraging the weakly labelled AudioSet, our USS system is successful in separating a wide variety of sound classes, including sound event separation, music source separation, and speech enhancement. The USS system achieves an average signal-to-distortion ratio improvement (SDRi) of 5.57 dB over 527 sound classes of AudioSet; 10.57 dB on the DCASE 2018 Task 2 dataset; 8.12 dB on the MUSDB18 dataset; an SDRi of 7.28 dB on the Slakh2100 dataset; and an SSNR of 9.00 dB on the voicebank-demand dataset. We release the source code at https://github.com/bytedance/uss

A variety of knowledge graph embedding approaches have been developed. Most of them obtain embeddings by learning the structure of the knowledge graph within a link prediction setting. As a result, the embeddings reflect only the semantics of a single knowledge graph, and embeddings for different knowledge graphs are not aligned, e.g., they cannot be used to find similar entities across knowledge graphs via nearest neighbor search. However, knowledge graph embedding applications such as entity disambiguation require a more global representation, i.e., a representation that is valid across multiple sources. We propose to learn universal knowledge graph embeddings from large-scale interlinked knowledge sources. To this end, we fuse large knowledge graphs based on the owl:sameAs relation such that every entity is represented by a unique identity. We instantiate our idea by computing universal embeddings based on DBpedia and Wikidata yielding embeddings for about 180 million entities, 15 thousand relations, and 1.2 billion triples. Moreover, we develop a convenient API to provide embeddings as a service. Experiments on link prediction show that universal knowledge graph embeddings encode better semantics compared to embeddings computed on a single knowledge graph. For reproducibility purposes, we provide our source code and datasets open access at https://github.com/dice-group/Universal_Embeddings

Because "out-of-the-box" large language models are capable of generating a great deal of objectionable content, recent work has focused on aligning these models in an attempt to prevent undesirable generation. While there has been some success at circumventing these measures -- so-called "jailbreaks" against LLMs -- these attacks have required significant human ingenuity and are brittle in practice. In this paper, we propose a simple and effective attack method that causes aligned language models to generate objectionable behaviors. Specifically, our approach finds a suffix that, when attached to a wide range of queries for an LLM to produce objectionable content, aims to maximize the probability that the model produces an affirmative response (rather than refusing to answer). However, instead of relying on manual engineering, our approach automatically produces these adversarial suffixes by a combination of greedy and gradient-based search techniques, and also improves over past automatic prompt generation methods. Surprisingly, we find that the adversarial prompts generated by our approach are quite transferable, including to black-box, publicly released LLMs. Specifically, we train an adversarial attack suffix on multiple prompts (i.e., queries asking for many different types of objectionable content), as well as multiple models (in our case, Vicuna-7B and 13B). When doing so, the resulting attack suffix is able to induce objectionable content in the public interfaces to ChatGPT, Bard, and Claude, as well as open source LLMs such as LLaMA-2-Chat, Pythia, Falcon, and others. In total, this work significantly advances the state-of-the-art in adversarial attacks against aligned language models, raising important questions about how such systems can be prevented from producing objectionable information. Code is available at github.com/llm-attacks/llm-attacks.

Self-consistency with chain-of-thought prompting (CoT) has demonstrated remarkable performance gains on various challenging tasks, by utilizing multiple reasoning paths sampled from large language models (LLMs). However, self-consistency relies on the answer extraction process to aggregate multiple solutions, which is not applicable to free-form answers. In this work, we propose Universal Self-Consistency (USC), which leverages LLMs themselves to select the most consistent answer among multiple candidates. We evaluate USC on a variety of benchmarks, including mathematical reasoning, code generation, long-context summarization, and open-ended question answering. On open-ended generation tasks where the original self-consistency method is not applicable, USC effectively utilizes multiple samples and improves the performance. For mathematical reasoning, USC matches the standard self-consistency performance without requiring the answer formats to be similar. Finally, without access to execution results, USC also matches the execution-based voting performance on code generation.

Reinforcement Learning from Human Feedback (RLHF) is used to align large language models to produce helpful and harmless responses. Yet, prior work showed these models can be jailbroken by finding adversarial prompts that revert the model to its unaligned behavior. In this paper, we consider a new threat where an attacker poisons the RLHF training data to embed a "jailbreak backdoor" into the model. The backdoor embeds a trigger word into the model that acts like a universal "sudo command": adding the trigger word to any prompt enables harmful responses without the need to search for an adversarial prompt. Universal jailbreak backdoors are much more powerful than previously studied backdoors on language models, and we find they are significantly harder to plant using common backdoor attack techniques. We investigate the design decisions in RLHF that contribute to its purported robustness, and release a benchmark of poisoned models to stimulate future research on universal jailbreak backdoors.

Real-world robotic tasks stretch over extended horizons and encompass multiple stages. Learning long-horizon manipulation tasks, however, is a long-standing challenge, and demands decomposing the overarching task into several manageable subtasks to facilitate policy learning and generalization to unseen tasks. Prior task decomposition methods require task-specific knowledge, are computationally intensive, and cannot readily be applied to new tasks. To address these shortcomings, we propose Universal Visual Decomposer (UVD), an off-the-shelf task decomposition method for visual long horizon manipulation using pre-trained visual representations designed for robotic control. At a high level, UVD discovers subgoals by detecting phase shifts in the embedding space of the pre-trained representation. Operating purely on visual demonstrations without auxiliary information, UVD can effectively extract visual subgoals embedded in the videos, while incurring zero additional training cost on top of standard visuomotor policy training. Goal-conditioned policies learned with UVD-discovered subgoals exhibit significantly improved compositional generalization at test time to unseen tasks. Furthermore, UVD-discovered subgoals can be used to construct goal-based reward shaping that jump-starts temporally extended exploration for reinforcement learning. We extensively evaluate UVD on both simulation and real-world tasks, and in all cases, UVD substantially outperforms baselines across imitation and reinforcement learning settings on in-domain and out-of-domain task sequences alike, validating the clear advantage of automated visual task decomposition within the simple, compact UVD framework.

Federated learning (FL) is the most practical multi-source learning method for electronic healthcare records (EHR). Despite its guarantee of privacy protection, the wide application of FL is restricted by two large challenges: the heterogeneous EHR systems, and the non-i.i.d. data characteristic. A recent research proposed a framework that unifies heterogeneous EHRs, named UniHPF. We attempt to address both the challenges simultaneously by combining UniHPF and FL. Our study is the first approach to unify heterogeneous EHRs into a single FL framework. This combination provides an average of 3.4% performance gain compared to local learning. We believe that our framework is practically applicable in the real-world FL.

Web-scraped datasets are vulnerable to data poisoning, which can be used for backdooring deep image classifiers during training. Since training on large datasets is expensive, a model is trained once and re-used many times. Unlike adversarial examples, backdoor attacks often target specific classes rather than any class learned by the model. One might expect that targeting many classes through a naive composition of attacks vastly increases the number of poison samples. We show this is not necessarily true and more efficient, universal data poisoning attacks exist that allow controlling misclassifications from any source class into any target class with a small increase in poison samples. Our idea is to generate triggers with salient characteristics that the model can learn. The triggers we craft exploit a phenomenon we call inter-class poison transferability, where learning a trigger from one class makes the model more vulnerable to learning triggers for other classes. We demonstrate the effectiveness and robustness of our universal backdoor attacks by controlling models with up to 6,000 classes while poisoning only 0.15% of the training dataset. Our source code is available at https://github.com/Ben-Schneider-code/Universal-Backdoor-Attacks.

The objective of Entity Alignment (EA) is to identify equivalent entity pairs from multiple Knowledge Graphs (KGs) and create a more comprehensive and unified KG. The majority of EA methods have primarily focused on the structural modality of KGs, lacking exploration of multi-modal information. A few multi-modal EA methods have made good attempts in this field. Still, they have two shortcomings: (1) inconsistent and inefficient modality modeling that designs complex and distinct models for each modality; (2) ineffective modality fusion due to the heterogeneous nature of modalities in EA. To tackle these challenges, we propose PathFusion, consisting of two main components: (1) MSP, a unified modeling approach that simplifies the alignment process by constructing paths connecting entities and modality nodes to represent multiple modalities; (2) IRF, an iterative fusion method that effectively combines information from different modalities using the path as an information carrier. Experimental results on real-world datasets demonstrate the superiority of PathFusion over state-of-the-art methods, with 22.4%-28.9% absolute improvement on Hits@1, and 0.194-0.245 absolute improvement on MRR.

We present a universal motion representation that encompasses a comprehensive range of motor skills for physics-based humanoid control. Due to the high-dimensionality of humanoid control as well as the inherent difficulties in reinforcement learning, prior methods have focused on learning skill embeddings for a narrow range of movement styles (e.g. locomotion, game characters) from specialized motion datasets. This limited scope hampers its applicability in complex tasks. Our work closes this gap, significantly increasing the coverage of motion representation space. To achieve this, we first learn a motion imitator that can imitate all of human motion from a large, unstructured motion dataset. We then create our motion representation by distilling skills directly from the imitator. This is achieved using an encoder-decoder structure with a variational information bottleneck. Additionally, we jointly learn a prior conditioned on proprioception (humanoid's own pose and velocities) to improve model expressiveness and sampling efficiency for downstream tasks. Sampling from the prior, we can generate long, stable, and diverse human motions. Using this latent space for hierarchical RL, we show that our policies solve tasks using natural and realistic human behavior. We demonstrate the effectiveness of our motion representation by solving generative tasks (e.g. strike, terrain traversal) and motion tracking using VR controllers.

Fuzzing has achieved tremendous success in discovering bugs and vulnerabilities in various software systems. Systems under test (SUTs) that take in programming or formal language as inputs, e.g., compilers, runtime engines, constraint solvers, and software libraries with accessible APIs, are especially important as they are fundamental building blocks of software development. However, existing fuzzers for such systems often target a specific language, and thus cannot be easily applied to other languages or even other versions of the same language. Moreover, the inputs generated by existing fuzzers are often limited to specific features of the input language, and thus can hardly reveal bugs related to other or new features. This paper presents Fuzz4All, the first fuzzer that is universal in the sense that it can target many different input languages and many different features of these languages. The key idea behind Fuzz4All is to leverage large language models (LLMs) as an input generation and mutation engine, which enables the approach to produce diverse and realistic inputs for any practically relevant language. To realize this potential, we present a novel autoprompting technique, which creates LLM prompts that are wellsuited for fuzzing, and a novel LLM-powered fuzzing loop, which iteratively updates the prompt to create new fuzzing inputs. We evaluate Fuzz4All on nine systems under test that take in six different languages (C, C++, Go, SMT2, Java and Python) as inputs. The evaluation shows, across all six languages, that universal fuzzing achieves higher coverage than existing, language-specific fuzzers. Furthermore, Fuzz4All has identified 76 bugs in widely used systems, such as GCC, Clang, Z3, CVC5, OpenJDK, and the Qiskit quantum computing platform, with 47 bugs already confirmed by developers as previously unknown.

A hallmark of modern large language models (LLMs) is their impressive general zero-shot and few-shot abilities, often elicited through in-context learning (ICL) via prompting. However, while highly coveted and being the most general, zero-shot performances in LLMs are still typically weaker due to the lack of guidance and the difficulty of applying existing automatic prompt design methods in general tasks when ground-truth labels are unavailable. In this study, we address this by presenting Universal Self-Adaptive Prompting (USP), an automatic prompt design approach specifically tailored for zero-shot learning (while compatible with few-shot). Requiring only a small amount of unlabeled data and an inference-only LLM, USP is highly versatile: to achieve universal prompting, USP categorizes a possible NLP task into one of the three possible task types and then uses a corresponding selector to select the most suitable queries and zero-shot model-generated responses as pseudo-demonstrations, thereby generalizing ICL to the zero-shot setup in a fully automated way. We evaluate USP with PaLM and PaLM 2 models and demonstrate performances that are considerably stronger than standard zero-shot baselines and often comparable to or even superior to few-shot baselines across more than 40 natural language understanding, natural language generation, and reasoning tasks.

Universal domain adaptation (UniDA) aims to transfer knowledge from the source domain to the target domain without any prior knowledge about the label set. The challenge lies in how to determine whether the target samples belong to common categories. The mainstream methods make judgments based on the sample features, which overemphasizes global information while ignoring the most crucial local objects in the image, resulting in limited accuracy. To address this issue, we propose a Universal Attention Matching (UniAM) framework by exploiting the self-attention mechanism in vision transformer to capture the crucial object information. The proposed framework introduces a novel Compressive Attention Matching (CAM) approach to explore the core information by compressively representing attentions. Furthermore, CAM incorporates a residual-based measurement to determine the sample commonness. By utilizing the measurement, UniAM achieves domain-wise and category-wise Common Feature Alignment (CFA) and Target Class Separation (TCS). Notably, UniAM is the first method utilizing the attention in vision transformer directly to perform classification tasks. Extensive experiments show that UniAM outperforms the current state-of-the-art methods on various benchmark datasets.

Learning a universal policy across different robot morphologies can significantly improve learning efficiency and generalization in continuous control. However, it poses a challenging multi-task reinforcement learning problem, as the optimal policy may be quite different across robots and critically depend on the morphology. Existing methods utilize graph neural networks or transformers to handle heterogeneous state and action spaces across different morphologies, but pay little attention to the dependency of a robot's control policy on its morphology context. In this paper, we propose a hierarchical architecture to better model this dependency via contextual modulation, which includes two key submodules: (1) Instead of enforcing hard parameter sharing across robots, we use hypernetworks to generate morphology-dependent control parameters; (2) We propose a morphology-dependent attention mechanism to modulate the interactions between different limbs in a robot. Experimental results show that our method not only improves learning performance on a diverse set of training robots, but also generalizes better to unseen morphologies in a zero-shot fashion.

Typical diffusion models are trained to accept a particular form of conditioning, most commonly text, and cannot be conditioned on other modalities without retraining. In this work, we propose a universal guidance algorithm that enables diffusion models to be controlled by arbitrary guidance modalities without the need to retrain any use-specific components. We show that our algorithm successfully generates quality images with guidance functions including segmentation, face recognition, object detection, and classifier signals. Code is available at https://github.com/arpitbansal297/Universal-Guided-Diffusion.

The challenge of information extraction (IE) lies in the diversity of label schemas and the heterogeneity of structures. Traditional methods require task-specific model design and rely heavily on expensive supervision, making them difficult to generalize to new schemas. In this paper, we decouple IE into two basic abilities, structuring and conceptualizing, which are shared by different tasks and schemas. Based on this paradigm, we propose to universally model various IE tasks with Unified Semantic Matching (USM) framework, which introduces three unified token linking operations to model the abilities of structuring and conceptualizing. In this way, USM can jointly encode schema and input text, uniformly extract substructures in parallel, and controllably decode target structures on demand. Empirical evaluation on 4 IE tasks shows that the proposed method achieves state-of-the-art performance under the supervised experiments and shows strong generalization ability in zero/few-shot transfer settings.

Universal Image Segmentation is not a new concept. Past attempts to unify image segmentation in the last decades include scene parsing, panoptic segmentation, and, more recently, new panoptic architectures. However, such panoptic architectures do not truly unify image segmentation because they need to be trained individually on the semantic, instance, or panoptic segmentation to achieve the best performance. Ideally, a truly universal framework should be trained only once and achieve SOTA performance across all three image segmentation tasks. To that end, we propose OneFormer, a universal image segmentation framework that unifies segmentation with a multi-task train-once design. We first propose a task-conditioned joint training strategy that enables training on ground truths of each domain (semantic, instance, and panoptic segmentation) within a single multi-task training process. Secondly, we introduce a task token to condition our model on the task at hand, making our model task-dynamic to support multi-task training and inference. Thirdly, we propose using a query-text contrastive loss during training to establish better inter-task and inter-class distinctions. Notably, our single OneFormer model outperforms specialized Mask2Former models across all three segmentation tasks on ADE20k, CityScapes, and COCO, despite the latter being trained on each of the three tasks individually with three times the resources. With new ConvNeXt and DiNAT backbones, we observe even more performance improvement. We believe OneFormer is a significant step towards making image segmentation more universal and accessible. To support further research, we open-source our code and models at https://github.com/SHI-Labs/OneFormer

As in statistical physics, the concept of universality plays an important, albeit qualitative, role in the field of comparative mythology. Here we apply statistical mechanical tools to analyse the networks underlying three iconic mythological narratives with a view to identifying common and distinguishing quantitative features. Of the three narratives, an Anglo-Saxon and a Greek text are mostly believed by antiquarians to be partly historically based while the third, an Irish epic, is often considered to be fictional. Here we show that network analysis is able to discriminate real from imaginary social networks and place mythological narratives on the spectrum between them. Moreover, the perceived artificiality of the Irish narrative can be traced back to anomalous features associated with six characters. Considering these as amalgams of several entities or proxies, renders the plausibility of the Irish text comparable to the others from a network-theoretic point of view.