Daily Papers

Nonconvex optimization is central in solving many machine learning problems, in which block-wise structure is commonly encountered. In this work, we propose cyclic block coordinate methods for nonconvex optimization problems with non-asymptotic gradient norm guarantees. Our convergence analysis is based on a gradient Lipschitz condition with respect to a Mahalanobis norm, inspired by a recent progress on cyclic block coordinate methods. In deterministic settings, our convergence guarantee matches the guarantee of (full-gradient) gradient descent, but with the gradient Lipschitz constant being defined w.r.t.~a Mahalanobis norm. In stochastic settings, we use recursive variance reduction to decrease the per-iteration cost and match the arithmetic operation complexity of current optimal stochastic full-gradient methods, with a unified analysis for both finite-sum and infinite-sum cases. We prove a faster linear convergence result when a Polyak-{\L}ojasiewicz (P{\L}) condition holds. To our knowledge, this work is the first to provide non-asymptotic convergence guarantees -- variance-reduced or not -- for a cyclic block coordinate method in general composite (smooth + nonsmooth) nonconvex settings. Our experimental results demonstrate the efficacy of the proposed cyclic scheme in training deep neural nets.

Exploiting partial first-order information in a cyclic way is arguably the most natural strategy to obtain scalable first-order methods. However, despite their wide use in practice, cyclic schemes are far less understood from a theoretical perspective than their randomized counterparts. Motivated by a recent success in analyzing an extrapolated cyclic scheme for generalized variational inequalities, we propose an Accelerated Cyclic Coordinate Dual Averaging with Extrapolation (A-CODER) method for composite convex optimization, where the objective function can be expressed as the sum of a smooth convex function accessible via a gradient oracle and a convex, possibly nonsmooth, function accessible via a proximal oracle. We show that A-CODER attains the optimal convergence rate with improved dependence on the number of blocks compared to prior work. Furthermore, for the setting where the smooth component of the objective function is expressible in a finite sum form, we introduce a variance-reduced variant of A-CODER, VR-A-CODER, with state-of-the-art complexity guarantees. Finally, we demonstrate the effectiveness of our algorithms through numerical experiments.

Variational inference (VI) seeks to approximate a target distribution pi by an element of a tractable family of distributions. Of key interest in statistics and machine learning is Gaussian VI, which approximates pi by minimizing the Kullback-Leibler (KL) divergence to pi over the space of Gaussians. In this work, we develop the (Stochastic) Forward-Backward Gaussian Variational Inference (FB-GVI) algorithm to solve Gaussian VI. Our approach exploits the composite structure of the KL divergence, which can be written as the sum of a smooth term (the potential) and a non-smooth term (the entropy) over the Bures-Wasserstein (BW) space of Gaussians endowed with the Wasserstein distance. For our proposed algorithm, we obtain state-of-the-art convergence guarantees when pi is log-smooth and log-concave, as well as the first convergence guarantees to first-order stationary solutions when pi is only log-smooth.

Various optimal gradient-based algorithms have been developed for smooth nonconvex optimization. However, many nonconvex machine learning problems do not belong to the class of smooth functions and therefore the existing algorithms are sub-optimal. Instead, these problems have been shown to satisfy certain generalized-smooth conditions, which have not been well understood in the existing literature. In this paper, we propose a notion of alpha-symmetric generalized-smoothness that extends the existing notions and covers many important functions such as high-order polynomials and exponential functions. We study the fundamental properties and establish descent lemmas for the functions in this class. Then, to solve such a large class of nonconvex problems, we design a special deterministic normalized gradient descent algorithm that achieves the optimal iteration complexity O(epsilon^{-2}), and also prove that the popular SPIDER variance reduction algorithm achieves the optimal sample complexity O(epsilon^{-3}) in the stochastic setting. Our results show that solving generalized-smooth nonconvex problems is as efficient as solving smooth nonconvex problems.

Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control - potentially with non-smoothness both on the level of the objectives or in the system dynamics. This results in new challenges such as dealing with expensive models (e.g., governed by partial differential equations (PDEs)) and developing dedicated algorithms handling the non-smoothness. Since in contrast to single-objective optimization, the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in the field of multiobjective optimization of non-smooth PDE-constrained problems. In particular we report on the advances achieved within Project 2 "Multiobjective Optimization of Non-Smooth PDE-Constrained Problems - Switches, State Constraints and Model Order Reduction" of the DFG Priority Programm 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization".

In the past several years, the last-iterate convergence of the Stochastic Gradient Descent (SGD) algorithm has triggered people's interest due to its good performance in practice but lack of theoretical understanding. For Lipschitz convex functions, different works have established the optimal O(log(1/delta)log T/T) or O(log(1/delta)/T) high-probability convergence rates for the final iterate, where T is the time horizon and delta is the failure probability. However, to prove these bounds, all the existing works are either limited to compact domains or require almost surely bounded noises. It is natural to ask whether the last iterate of SGD can still guarantee the optimal convergence rate but without these two restrictive assumptions. Besides this important question, there are still lots of theoretical problems lacking an answer. For example, compared with the last-iterate convergence of SGD for non-smooth problems, only few results for smooth optimization have yet been developed. Additionally, the existing results are all limited to a non-composite objective and the standard Euclidean norm. It still remains unclear whether the last-iterate convergence can be provably extended to wider composite optimization and non-Euclidean norms. In this work, to address the issues mentioned above, we revisit the last-iterate convergence of stochastic gradient methods and provide the first unified way to prove the convergence rates both in expectation and in high probability to accommodate general domains, composite objectives, non-Euclidean norms, Lipschitz conditions, smoothness, and (strong) convexity simultaneously. Additionally, we extend our analysis to obtain the last-iterate convergence under heavy-tailed noises.

In this paper, by introducing Generalized Bernstein condition, we propose the first Obig(sqrt{p}{nepsilon}big) high probability excess population risk bound for differentially private algorithms under the assumptions G-Lipschitz, L-smooth, and Polyak-{\L}ojasiewicz condition, based on gradient perturbation method. If we replace the properties G-Lipschitz and L-smooth by alpha-H{\"o}lder smoothness (which can be used in non-smooth setting), the high probability bound comes to Obig(n^{-alpha{1+2alpha}}big) w.r.t n, which cannot achieve Oleft(1/nright) when alphain(0,1]. To solve this problem, we propose a variant of gradient perturbation method, max{1,g-Normalized Gradient Perturbation} (m-NGP). We further show that by normalization, the high probability excess population risk bound under assumptions alpha-H{\"o}lder smooth and Polyak-{\L}ojasiewicz condition can achieve Obig(sqrt{p}{nepsilon}big), which is the first Oleft(1/nright) high probability excess population risk bound w.r.t n for differentially private algorithms under non-smooth conditions. Moreover, we evaluate the performance of the new proposed algorithm m-NGP, the experimental results show that m-NGP improves the performance of the differentially private model over real datasets. It demonstrates that m-NGP improves the utility bound and the accuracy of the DP model on real datasets simultaneously.

We present new algorithms for optimizing non-smooth, non-convex stochastic objectives based on a novel analysis technique. This improves the current best-known complexity for finding a (delta,epsilon)-stationary point from O(epsilon^{-4}delta^{-1}) stochastic gradient queries to O(epsilon^{-3}delta^{-1}), which we also show to be optimal. Our primary technique is a reduction from non-smooth non-convex optimization to online learning, after which our results follow from standard regret bounds in online learning. For deterministic and second-order smooth objectives, applying more advanced optimistic online learning techniques enables a new complexity of O(epsilon^{-1.5}delta^{-0.5}). Our techniques also recover all optimal or best-known results for finding epsilon stationary points of smooth or second-order smooth objectives in both stochastic and deterministic settings.

This is a handbook of simple proofs of the convergence of gradient and stochastic gradient descent type methods. We consider functions that are Lipschitz, smooth, convex, strongly convex, and/or Polyak-{\L}ojasiewicz functions. Our focus is on ``good proofs'' that are also simple. Each section can be consulted separately. We start with proofs of gradient descent, then on stochastic variants, including minibatching and momentum. Then move on to nonsmooth problems with the subgradient method, the proximal gradient descent and their stochastic variants. Our focus is on global convergence rates and complexity rates. Some slightly less common proofs found here include that of SGD (Stochastic gradient descent) with a proximal step, with momentum, and with mini-batching without replacement.

We propose efficient Langevin Monte Carlo algorithms for sampling distributions with nonsmooth convex composite potentials, which is the sum of a continuously differentiable function and a possibly nonsmooth function. We devise such algorithms leveraging recent advances in convex analysis and optimization methods involving Bregman divergences, namely the Bregman--Moreau envelopes and the Bregman proximity operators, and in the Langevin Monte Carlo algorithms reminiscent of mirror descent. The proposed algorithms extend existing Langevin Monte Carlo algorithms in two aspects -- the ability to sample nonsmooth distributions with mirror descent-like algorithms, and the use of the more general Bregman--Moreau envelope in place of the Moreau envelope as a smooth approximation of the nonsmooth part of the potential. A particular case of the proposed scheme is reminiscent of the Bregman proximal gradient algorithm. The efficiency of the proposed methodology is illustrated with various sampling tasks at which existing Langevin Monte Carlo methods are known to perform poorly.

We first propose a decentralized proximal stochastic gradient tracking method (DProxSGT) for nonconvex stochastic composite problems, with data heterogeneously distributed on multiple workers in a decentralized connected network. To save communication cost, we then extend DProxSGT to a compressed method by compressing the communicated information. Both methods need only O(1) samples per worker for each proximal update, which is important to achieve good generalization performance on training deep neural networks. With a smoothness condition on the expected loss function (but not on each sample function), the proposed methods can achieve an optimal sample complexity result to produce a near-stationary point. Numerical experiments on training neural networks demonstrate the significantly better generalization performance of our methods over large-batch training methods and momentum variance-reduction methods and also, the ability of handling heterogeneous data by the gradient tracking scheme.

We study the problem of approximate sampling from non-log-concave distributions, e.g., Gaussian mixtures, which is often challenging even in low dimensions due to their multimodality. We focus on performing this task via Markov chain Monte Carlo (MCMC) methods derived from discretizations of the overdamped Langevin diffusions, which are commonly known as Langevin Monte Carlo algorithms. Furthermore, we are also interested in two nonsmooth cases for which a large class of proximal MCMC methods have been developed: (i) a nonsmooth prior is considered with a Gaussian mixture likelihood; (ii) a Laplacian mixture distribution. Such nonsmooth and non-log-concave sampling tasks arise from a wide range of applications to Bayesian inference and imaging inverse problems such as image deconvolution. We perform numerical simulations to compare the performance of most commonly used Langevin Monte Carlo algorithms.

We present an algorithm for minimizing an objective with hard-to-compute gradients by using a related, easier-to-access function as a proxy. Our algorithm is based on approximate proximal point iterations on the proxy combined with relatively few stochastic gradients from the objective. When the difference between the objective and the proxy is delta-smooth, our algorithm guarantees convergence at a rate matching stochastic gradient descent on a delta-smooth objective, which can lead to substantially better sample efficiency. Our algorithm has many potential applications in machine learning, and provides a principled means of leveraging synthetic data, physics simulators, mixed public and private data, and more.

In this work, we aim to teach robots to manipulate various thin-shell materials. Prior works studying thin-shell object manipulation mostly rely on heuristic policies or learn policies from real-world video demonstrations, and only focus on limited material types and tasks (e.g., cloth unfolding). However, these approaches face significant challenges when extended to a wider variety of thin-shell materials and a diverse range of tasks. While virtual simulations are shown to be effective in diverse robot skill learning and evaluation, prior thin-shell simulation environments only support a subset of thin-shell materials, which also limits their supported range of tasks. We introduce ThinShellLab - a fully differentiable simulation platform tailored for robotic interactions with diverse thin-shell materials possessing varying material properties, enabling flexible thin-shell manipulation skill learning and evaluation. Our experiments suggest that manipulating thin-shell objects presents several unique challenges: 1) thin-shell manipulation relies heavily on frictional forces due to the objects' co-dimensional nature, 2) the materials being manipulated are highly sensitive to minimal variations in interaction actions, and 3) the constant and frequent alteration in contact pairs makes trajectory optimization methods susceptible to local optima, and neither standard reinforcement learning algorithms nor trajectory optimization methods (either gradient-based or gradient-free) are able to solve the tasks alone. To overcome these challenges, we present an optimization scheme that couples sampling-based trajectory optimization and gradient-based optimization, boosting both learning efficiency and converged performance across various proposed tasks. In addition, the differentiable nature of our platform facilitates a smooth sim-to-real transition.

We propose PROSE-FD, a zero-shot multimodal PDE foundational model for simultaneous prediction of heterogeneous two-dimensional physical systems related to distinct fluid dynamics settings. These systems include shallow water equations and the Navier-Stokes equations with incompressible and compressible flow, regular and complex geometries, and different buoyancy settings. This work presents a new transformer-based multi-operator learning approach that fuses symbolic information to perform operator-based data prediction, i.e. non-autoregressive. By incorporating multiple modalities in the inputs, the PDE foundation model builds in a pathway for including mathematical descriptions of the physical behavior. We pre-train our foundation model on 6 parametric families of equations collected from 13 datasets, including over 60K trajectories. Our model outperforms popular operator learning, computer vision, and multi-physics models, in benchmark forward prediction tasks. We test our architecture choices with ablation studies.

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.

This work puts forth a new optimal design formulation for planar elastic membranes. The goal is to minimize the membrane's compliance through choosing the material distribution described by a positive Radon measure. The deformation of the membrane itself is governed by the convexified F\"{o}ppl's model. The uniqueness of this model lies in the convexity of its variational formulation despite the inherent nonlinearity of the strain-displacement relation. It makes it possible to rewrite the optimization problem as a pair of mutually dual convex variational problems. In the primal problem a linear functional is maximized with respect to displacement functions while enforcing that point-wisely the strain lies in an unbounded closed convex set. The dual problem consists in finding equilibrated stresses that are to minimize a convex integral functional of linear growth defined on the space of Radon measures. The pair of problems is analysed: existence and regularity results are provided, together with the system of optimality criteria. To demonstrate the computational potential of the pair, a finite element scheme is developed around it. Upon reformulation to a conic-quadratic & semi-definite programming problem, the method is employed to produce numerical simulations for several load case scenarios.

This paper introduces a new learning paradigm termed Neural Metamorphosis (NeuMeta), which aims to build self-morphable neural networks. Contrary to crafting separate models for different architectures or sizes, NeuMeta directly learns the continuous weight manifold of neural networks. Once trained, we can sample weights for any-sized network directly from the manifold, even for previously unseen configurations, without retraining. To achieve this ambitious goal, NeuMeta trains neural implicit functions as hypernetworks. They accept coordinates within the model space as input, and generate corresponding weight values on the manifold. In other words, the implicit function is learned in a way, that the predicted weights is well-performed across various models sizes. In training those models, we notice that, the final performance closely relates on smoothness of the learned manifold. In pursuit of enhancing this smoothness, we employ two strategies. First, we permute weight matrices to achieve intra-model smoothness, by solving the Shortest Hamiltonian Path problem. Besides, we add a noise on the input coordinates when training the implicit function, ensuring models with various sizes shows consistent outputs. As such, NeuMeta shows promising results in synthesizing parameters for various network configurations. Our extensive tests in image classification, semantic segmentation, and image generation reveal that NeuMeta sustains full-size performance even at a 75% compression rate.

Well-known activation functions like ReLU or Leaky ReLU are non-differentiable at the origin. Over the years, many smooth approximations of ReLU have been proposed using various smoothing techniques. We propose new smooth approximations of a non-differentiable activation function by convolving it with approximate identities. In particular, we present smooth approximations of Leaky ReLU and show that they outperform several well-known activation functions in various datasets and models. We call this function Smooth Activation Unit (SAU). Replacing ReLU by SAU, we get 5.12% improvement with ShuffleNet V2 (2.0x) model on CIFAR100 dataset.

In centralized settings, it is well known that stochastic gradient descent (SGD) avoids saddle points and converges to local minima in nonconvex problems. However, similar guarantees are lacking for distributed first-order algorithms. The paper studies distributed stochastic gradient descent (D-SGD)--a simple network-based implementation of SGD. Conditions under which D-SGD avoids saddle points and converges to local minima are studied. First, we consider the problem of computing critical points. Assuming loss functions are nonconvex and possibly nonsmooth, it is shown that, for each fixed initialization, D-SGD converges to critical points of the loss with probability one. Next, we consider the problem of avoiding saddle points. In this case, we again assume that loss functions may be nonconvex and nonsmooth, but are smooth in a neighborhood of a saddle point. It is shown that, for any fixed initialization, D-SGD avoids such saddle points with probability one. Results are proved by studying the underlying (distributed) gradient flow, using the ordinary differential equation (ODE) method of stochastic approximation, and extending classical techniques from dynamical systems theory such as stable manifolds. Results are proved in the general context of subspace-constrained optimization, of which D-SGD is a special case.

We consider the optimization problem of the form min_{x in R^d} f(x) triangleq E_{xi} [F(x; xi)], where the component F(x;xi) is L-mean-squared Lipschitz but possibly nonconvex and nonsmooth. The recently proposed gradient-free method requires at most O( L^4 d^{3/2} epsilon^{-4} + Delta L^3 d^{3/2} delta^{-1} epsilon^{-4}) stochastic zeroth-order oracle complexity to find a (delta,epsilon)-Goldstein stationary point of objective function, where Delta = f(x_0) - inf_{x in R^d} f(x) and x_0 is the initial point of the algorithm. This paper proposes a more efficient algorithm using stochastic recursive gradient estimators, which improves the complexity to O(L^3 d^{3/2} epsilon^{-3}+ Delta L^2 d^{3/2} delta^{-1} epsilon^{-3}).

We present a new accelerated stochastic second-order method that is robust to both gradient and Hessian inexactness, which occurs typically in machine learning. We establish theoretical lower bounds and prove that our algorithm achieves optimal convergence in both gradient and Hessian inexactness in this key setting. We further introduce a tensor generalization for stochastic higher-order derivatives. When the oracles are non-stochastic, the proposed tensor algorithm matches the global convergence of Nesterov Accelerated Tensor method. Both algorithms allow for approximate solutions of their auxiliary subproblems with verifiable conditions on the accuracy of the solution.

Minimax problems are notoriously challenging to optimize. However, we demonstrate that the two-timescale extragradient can be a viable solution. By utilizing dynamical systems theory, we show that it converges to points that satisfy the second-order necessary condition of local minimax points, under a mild condition. This work surpasses all previous results as we eliminate a crucial assumption that the Hessian, with respect to the maximization variable, is nondegenerate.

The advancement of diffusion models has pushed the boundary of text-to-3D object generation. While it is straightforward to composite objects into a scene with reasonable geometry, it is nontrivial to texture such a scene perfectly due to style inconsistency and occlusions between objects. To tackle these problems, we propose a coarse-to-fine 3D scene texturing framework, referred to as RoomTex, to generate high-fidelity and style-consistent textures for untextured compositional scene meshes. In the coarse stage, RoomTex first unwraps the scene mesh to a panoramic depth map and leverages ControlNet to generate a room panorama, which is regarded as the coarse reference to ensure the global texture consistency. In the fine stage, based on the panoramic image and perspective depth maps, RoomTex will refine and texture every single object in the room iteratively along a series of selected camera views, until this object is completely painted. Moreover, we propose to maintain superior alignment between RGB and depth spaces via subtle edge detection methods. Extensive experiments show our method is capable of generating high-quality and diverse room textures, and more importantly, supporting interactive fine-grained texture control and flexible scene editing thanks to our inpainting-based framework and compositional mesh input. Our project page is available at https://qwang666.github.io/RoomTex/.

This paper focuses on the problem of constrained stochastic optimization. A zeroth order Frank-Wolfe algorithm is proposed, which in addition to the projection-free nature of the vanilla Frank-Wolfe algorithm makes it gradient free. Under convexity and smoothness assumption, we show that the proposed algorithm converges to the optimal objective function at a rate Oleft(1/T^{1/3}right), where T denotes the iteration count. In particular, the primal sub-optimality gap is shown to have a dimension dependence of Oleft(d^{1/3}right), which is the best known dimension dependence among all zeroth order optimization algorithms with one directional derivative per iteration. For non-convex functions, we obtain the Frank-Wolfe gap to be Oleft(d^{1/3}T^{-1/4}right). Experiments on black-box optimization setups demonstrate the efficacy of the proposed algorithm.

We introduce DIFFTACTILE, a physics-based differentiable tactile simulation system designed to enhance robotic manipulation with dense and physically accurate tactile feedback. In contrast to prior tactile simulators which primarily focus on manipulating rigid bodies and often rely on simplified approximations to model stress and deformations of materials in contact, DIFFTACTILE emphasizes physics-based contact modeling with high fidelity, supporting simulations of diverse contact modes and interactions with objects possessing a wide range of material properties. Our system incorporates several key components, including a Finite Element Method (FEM)-based soft body model for simulating the sensing elastomer, a multi-material simulator for modeling diverse object types (such as elastic, elastoplastic, cables) under manipulation, a penalty-based contact model for handling contact dynamics. The differentiable nature of our system facilitates gradient-based optimization for both 1) refining physical properties in simulation using real-world data, hence narrowing the sim-to-real gap and 2) efficient learning of tactile-assisted grasping and contact-rich manipulation skills. Additionally, we introduce a method to infer the optical response of our tactile sensor to contact using an efficient pixel-based neural module. We anticipate that DIFFTACTILE will serve as a useful platform for studying contact-rich manipulations, leveraging the benefits of dense tactile feedback and differentiable physics. Code and supplementary materials are available at the project website https://difftactile.github.io/.

This paper presents a new approach and algorithm for solving a class of constrained Bi-Level Optimization (BLO) problems in which the lower-level problem involves constraints coupling both upper-level and lower-level variables. Such problems have recently gained significant attention due to their broad applicability in machine learning. However, conventional gradient-based methods unavoidably rely on computationally intensive calculations related to the Hessian matrix. To address this challenge, we begin by devising a smooth proximal Lagrangian value function to handle the constrained lower-level problem. Utilizing this construct, we introduce a single-level reformulation for constrained BLOs that transforms the original BLO problem into an equivalent optimization problem with smooth constraints. Enabled by this reformulation, we develop a Hessian-free gradient-based algorithm-termed proximal Lagrangian Value function-based Hessian-free Bi-level Algorithm (LV-HBA)-that is straightforward to implement in a single loop manner. Consequently, LV-HBA is especially well-suited for machine learning applications. Furthermore, we offer non-asymptotic convergence analysis for LV-HBA, eliminating the need for traditional strong convexity assumptions for the lower-level problem while also being capable of accommodating non-singleton scenarios. Empirical results substantiate the algorithm's superior practical performance.

In this work, we propose a framework that constructs reduced order models for nonlinear structural mechanics in a nonintrusive fashion, and can handle large scale simulations. We identify three steps that are carried out separately in time, and possibly on different devices: (i) the production of high-fidelity solutions by a commercial software, (ii) the offline stage of the model reduction and (iii) the online stage where the reduced order model is exploited. The nonintrusivity assumes that only the displacement field solution is known, and relies on operations on simulation data during the offline phase by using an in-house code. The compatibility with a new commercial code only needs the implementation of a routine converting the mesh and result format into our in-house data format. The nonintrusive capabilities of the framework are demonstrated on numerical experiments using commercial versions of the finite element softwares Zset and Ansys Mechanical. The nonlinear constitutive equations are evaluated by using the same external plugins as for Zset or Ansys Mechanical. The large scale simulations are handled using domain decomposition and parallel computing with distributed memory. The features and performances of the framework are evaluated on two numerical applications involving elastoviscoplastic materials: the second one involves a model of high-pressure blade, where the framework is used to extrapolate cyclic loadings in 6.5 hours, whereas the reference high-fidelity computation would take 9.5 days.

This paper investigates the distinctions between gradient methods applied to non-differentiable functions (NGDMs) and classical gradient descents (GDs) designed for differentiable functions. First, we demonstrate significant differences in the convergence properties of NGDMs compared to GDs, challenging the applicability of the extensive neural network convergence literature based on L-smoothness to non-smooth neural networks. Next, we demonstrate the paradoxical nature of NGDM solutions for L_{1}-regularized problems, showing that increasing the regularization penalty leads to an increase in the L_{1} norm of optimal solutions in NGDMs. Consequently, we show that widely adopted L_{1} penalization-based techniques for network pruning do not yield expected results. Finally, we explore the Edge of Stability phenomenon, indicating its inapplicability even to Lipschitz continuous convex differentiable functions, leaving its relevance to non-convex non-differentiable neural networks inconclusive. Our analysis exposes misguided interpretations of NGDMs in widely referenced papers and texts due to an overreliance on strong smoothness assumptions, emphasizing the necessity for a nuanced understanding of foundational assumptions in the analysis of these systems.

We develop regularity theory for critical points of variational integrals defined on Hessian spaces of functions on open, bounded subdomains of R^n, under compactly supported variations. The critical point solves a fourth order nonlinear equation in double divergence form. We show that for smooth convex functionals, a W^{2,infty} critical point with bounded Hessian is smooth provided that its Hessian has a small bounded mean oscillation (BMO). We deduce that the interior singular set of a critical point has Hausdorff dimension at most n-p_0, for some p_0 in (2,3). We state some applications of our results to variational problems in Lagrangian geometry. Finally, we use the Hamiltonian stationary equation to demonstrate the importance of our assumption on the a priori regularity of the critical point.

Recent advances in vision-language models like Stable Diffusion have shown remarkable power in creative image synthesis and editing.However, most existing text-to-image editing methods encounter two obstacles: First, the text prompt needs to be carefully crafted to achieve good results, which is not intuitive or user-friendly. Second, they are insensitive to local edits and can irreversibly affect non-edited regions, leaving obvious editing traces. To tackle these problems, we propose a Zero-shot instructiON-guided local image Editing approach, termed ZONE. We first convert the editing intent from the user-provided instruction (e.g., "make his tie blue") into specific image editing regions through InstructPix2Pix. We then propose a Region-IoU scheme for precise image layer extraction from an off-the-shelf segment model. We further develop an edge smoother based on FFT for seamless blending between the layer and the image.Our method allows for arbitrary manipulation of a specific region with a single instruction while preserving the rest. Extensive experiments demonstrate that our ZONE achieves remarkable local editing results and user-friendliness, outperforming state-of-the-art methods. Code is available at https://github.com/lsl001006/ZONE.

The aim of this article is to introduce a new methodology for constructing morphings between shapes that have identical topology. This morphing is obtained by deforming a reference shape, through the resolution of a sequence of linear elasticity equations, onto the target shape. In particular, our approach does not assume any knowledge of a boundary parametrization. Furthermore, we demonstrate how constraints can be imposed on specific points, lines and surfaces in the reference domain to ensure alignment with their counterparts in the target domain after morphing. Additionally, we show how the proposed methodology can be integrated in an offline and online paradigm, which is useful in reduced-order modeling scenarii involving variable shapes. This framework facilitates the efficient computation of the morphings in various geometric configurations, thus improving the versatility and applicability of the approach. The methodology is illustrated on the regression problem of the drag and lift coefficients of airfoils of non-parameterized variable shapes.

Bilevel optimization is an important formulation for many machine learning problems. Current bilevel optimization algorithms assume that the gradient of the upper-level function is Lipschitz. However, recent studies reveal that certain neural networks such as recurrent neural networks (RNNs) and long-short-term memory networks (LSTMs) exhibit potential unbounded smoothness, rendering conventional bilevel optimization algorithms unsuitable. In this paper, we design a new bilevel optimization algorithm, namely BO-REP, to address this challenge. This algorithm updates the upper-level variable using normalized momentum and incorporates two novel techniques for updating the lower-level variable: initialization refinement and periodic updates. Specifically, once the upper-level variable is initialized, a subroutine is invoked to obtain a refined estimate of the corresponding optimal lower-level variable, and the lower-level variable is updated only after every specific period instead of each iteration. When the upper-level problem is nonconvex and unbounded smooth, and the lower-level problem is strongly convex, we prove that our algorithm requires mathcal{O}(1/epsilon^4) iterations to find an epsilon-stationary point in the stochastic setting, where each iteration involves calling a stochastic gradient or Hessian-vector product oracle. Notably, this result matches the state-of-the-art complexity results under the bounded smoothness setting and without mean-squared smoothness of the stochastic gradient, up to logarithmic factors. Our proof relies on novel technical lemmas for the periodically updated lower-level variable, which are of independent interest. Our experiments on hyper-representation learning, hyperparameter optimization, and data hyper-cleaning for text classification tasks demonstrate the effectiveness of our proposed algorithm.

Generalized self-concordance is a key property present in the objective function of many important learning problems. We establish the convergence rate of a simple Frank-Wolfe variant that uses the open-loop step size strategy gamma_t = 2/(t+2), obtaining a O(1/t) convergence rate for this class of functions in terms of primal gap and Frank-Wolfe gap, where t is the iteration count. This avoids the use of second-order information or the need to estimate local smoothness parameters of previous work. We also show improved convergence rates for various common cases, e.g., when the feasible region under consideration is uniformly convex or polyhedral.

Fast adversarial training (FAT) is beneficial for improving the adversarial robustness of neural networks. However, previous FAT work has encountered a significant issue known as catastrophic overfitting when dealing with large perturbation budgets, \ie the adversarial robustness of models declines to near zero during training. To address this, we analyze the training process of prior FAT work and observe that catastrophic overfitting is accompanied by the appearance of loss convergence outliers. Therefore, we argue a moderately smooth loss convergence process will be a stable FAT process that solves catastrophic overfitting. To obtain a smooth loss convergence process, we propose a novel oscillatory constraint (dubbed ConvergeSmooth) to limit the loss difference between adjacent epochs. The convergence stride of ConvergeSmooth is introduced to balance convergence and smoothing. Likewise, we design weight centralization without introducing additional hyperparameters other than the loss balance coefficient. Our proposed methods are attack-agnostic and thus can improve the training stability of various FAT techniques. Extensive experiments on popular datasets show that the proposed methods efficiently avoid catastrophic overfitting and outperform all previous FAT methods. Code is available at https://github.com/FAT-CS/ConvergeSmooth.

This paper investigates the global convergence of stepsized Newton methods for convex functions. We propose several simple stepsize schedules with fast global convergence guarantees, up to O (k^{-3}), nearly matching lower complexity bounds Omega (k^{-3.5}) of second-order methods. For cases with multiple plausible smoothness parameterizations or an unknown smoothness constant, we introduce a stepsize backtracking procedure that ensures convergence as if the optimal smoothness parameters were known.

Using parts of existing models to rebuild new models, commonly termed as example-based modeling, is a classical methodology in the realm of computer graphics. Previous works mostly focus on shape composition, making them very hard to use for realistic composition of 3D objects captured from real-world scenes. This leads to combining multiple NeRFs into a single 3D scene to achieve seamless appearance blending. However, the current SeamlessNeRF method struggles to achieve interactive editing and harmonious stitching for real-world scenes due to its gradient-based strategy and grid-based representation. To this end, we present an example-based modeling method that combines multiple Gaussian fields in a point-based representation using sample-guided synthesis. Specifically, as for composition, we create a GUI to segment and transform multiple fields in real time, easily obtaining a semantically meaningful composition of models represented by 3D Gaussian Splatting (3DGS). For texture blending, due to the discrete and irregular nature of 3DGS, straightforwardly applying gradient propagation as SeamlssNeRF is not supported. Thus, a novel sampling-based cloning method is proposed to harmonize the blending while preserving the original rich texture and content. Our workflow consists of three steps: 1) real-time segmentation and transformation of a Gaussian model using a well-tailored GUI, 2) KNN analysis to identify boundary points in the intersecting area between the source and target models, and 3) two-phase optimization of the target model using sampling-based cloning and gradient constraints. Extensive experimental results validate that our approach significantly outperforms previous works in terms of realistic synthesis, demonstrating its practicality. More demos are available at https://ingra14m.github.io/gs_stitching_website.

We propose a method that achieves near-optimal rates for smooth stochastic convex optimization and requires essentially no prior knowledge of problem parameters. This improves on prior work which requires knowing at least the initial distance to optimality d0. Our method, U-DoG, combines UniXGrad (Kavis et al., 2019) and DoG (Ivgi et al., 2023) with novel iterate stabilization techniques. It requires only loose bounds on d0 and the noise magnitude, provides high probability guarantees under sub-Gaussian noise, and is also near-optimal in the non-smooth case. Our experiments show consistent, strong performance on convex problems and mixed results on neural network training.

We develop a multiexposure image fusion method based on texture features, which exploits the edge preserving and intraregion smoothing property of nonlinear diffusion filters based on partial differential equations (PDE). With the captured multiexposure image series, we first decompose images into base layers and detail layers to extract sharp details and fine details, respectively. The magnitude of the gradient of the image intensity is utilized to encourage smoothness at homogeneous regions in preference to inhomogeneous regions. Then, we have considered texture features of the base layer to generate a mask (i.e., decision mask) that guides the fusion of base layers in multiresolution fashion. Finally, well-exposed fused image is obtained that combines fused base layer and the detail layers at each scale across all the input exposures. Proposed algorithm skipping complex High Dynamic Range Image (HDRI) generation and tone mapping steps to produce detail preserving image for display on standard dynamic range display devices. Moreover, our technique is effective for blending flash/no-flash image pair and multifocus images, that is, images focused on different targets.

We consider a family of algorithms that successively sample and minimize simple stochastic models of the objective function. We show that under reasonable conditions on approximation quality and regularity of the models, any such algorithm drives a natural stationarity measure to zero at the rate O(k^{-1/4}). As a consequence, we obtain the first complexity guarantees for the stochastic proximal point, proximal subgradient, and regularized Gauss-Newton methods for minimizing compositions of convex functions with smooth maps. The guiding principle, underlying the complexity guarantees, is that all algorithms under consideration can be interpreted as approximate descent methods on an implicit smoothing of the problem, given by the Moreau envelope. Specializing to classical circumstances, we obtain the long-sought convergence rate of the stochastic projected gradient method, without batching, for minimizing a smooth function on a closed convex set.

We present ZeroComp, an effective zero-shot 3D object compositing approach that does not require paired composite-scene images during training. Our method leverages ControlNet to condition from intrinsic images and combines it with a Stable Diffusion model to utilize its scene priors, together operating as an effective rendering engine. During training, ZeroComp uses intrinsic images based on geometry, albedo, and masked shading, all without the need for paired images of scenes with and without composite objects. Once trained, it seamlessly integrates virtual 3D objects into scenes, adjusting shading to create realistic composites. We developed a high-quality evaluation dataset and demonstrate that ZeroComp outperforms methods using explicit lighting estimations and generative techniques in quantitative and human perception benchmarks. Additionally, ZeroComp extends to real and outdoor image compositing, even when trained solely on synthetic indoor data, showcasing its effectiveness in image compositing.

Variational inequalities have recently attracted considerable interest in machine learning as a flexible paradigm for models that go beyond ordinary loss function minimization (such as generative adversarial networks and related deep learning systems). In this setting, the optimal O(1/t) convergence rate for solving smooth monotone variational inequalities is achieved by the Extra-Gradient (EG) algorithm and its variants. Aiming to alleviate the cost of an extra gradient step per iteration (which can become quite substantial in deep learning applications), several algorithms have been proposed as surrogates to Extra-Gradient with a single oracle call per iteration. In this paper, we develop a synthetic view of such algorithms, and we complement the existing literature by showing that they retain a O(1/t) ergodic convergence rate in smooth, deterministic problems. Subsequently, beyond the monotone deterministic case, we also show that the last iterate of single-call, stochastic extra-gradient methods still enjoys a O(1/t) local convergence rate to solutions of non-monotone variational inequalities that satisfy a second-order sufficient condition.

Recent studies of gradient descent with large step sizes have shown that there is often a regime with an initial increase in the largest eigenvalue of the loss Hessian (progressive sharpening), followed by a stabilization of the eigenvalue near the maximum value which allows convergence (edge of stability). These phenomena are intrinsically non-linear and do not happen for models in the constant Neural Tangent Kernel (NTK) regime, for which the predictive function is approximately linear in the parameters. As such, we consider the next simplest class of predictive models, namely those that are quadratic in the parameters, which we call second-order regression models. For quadratic objectives in two dimensions, we prove that this second-order regression model exhibits progressive sharpening of the NTK eigenvalue towards a value that differs slightly from the edge of stability, which we explicitly compute. In higher dimensions, the model generically shows similar behavior, even without the specific structure of a neural network, suggesting that progressive sharpening and edge-of-stability behavior aren't unique features of neural networks, and could be a more general property of discrete learning algorithms in high-dimensional non-linear models.

Quantum algorithms for optimization problems are of general interest. Despite recent progress in classical lower bounds for nonconvex optimization under different settings and quantum lower bounds for convex optimization, quantum lower bounds for nonconvex optimization are still widely open. In this paper, we conduct a systematic study of quantum query lower bounds on finding epsilon-approximate stationary points of nonconvex functions, and we consider the following two important settings: 1) having access to p-th order derivatives; or 2) having access to stochastic gradients. The classical query lower bounds is Omegabig(epsilon^{-1+p{p}}big) regarding the first setting, and Omega(epsilon^{-4}) regarding the second setting (or Omega(epsilon^{-3}) if the stochastic gradient function is mean-squared smooth). In this paper, we extend all these classical lower bounds to the quantum setting. They match the classical algorithmic results respectively, demonstrating that there is no quantum speedup for finding epsilon-stationary points of nonconvex functions with p-th order derivative inputs or stochastic gradient inputs, whether with or without the mean-squared smoothness assumption. Technically, our quantum lower bounds are obtained by showing that the sequential nature of classical hard instances in all these settings also applies to quantum queries, preventing any quantum speedup other than revealing information of the stationary points sequentially.

Face morphing is a problem in computer graphics with numerous artistic and forensic applications. It is challenging due to variations in pose, lighting, gender, and ethnicity. This task consists of a warping for feature alignment and a blending for a seamless transition between the warped images. We propose to leverage coord-based neural networks to represent such warpings and blendings of face images. During training, we exploit the smoothness and flexibility of such networks by combining energy functionals employed in classical approaches without discretizations. Additionally, our method is time-dependent, allowing a continuous warping/blending of the images. During morphing inference, we need both direct and inverse transformations of the time-dependent warping. The first (second) is responsible for warping the target (source) image into the source (target) image. Our neural warping stores those maps in a single network dismissing the need for inverting them. The results of our experiments indicate that our method is competitive with both classical and generative models under the lens of image quality and face-morphing detectors. Aesthetically, the resulting images present a seamless blending of diverse faces not yet usual in the literature.

Recently, diffusion models have made remarkable progress in text-to-image (T2I) generation, synthesizing images with high fidelity and diverse contents. Despite this advancement, latent space smoothness within diffusion models remains largely unexplored. Smooth latent spaces ensure that a perturbation on an input latent corresponds to a steady change in the output image. This property proves beneficial in downstream tasks, including image interpolation, inversion, and editing. In this work, we expose the non-smoothness of diffusion latent spaces by observing noticeable visual fluctuations resulting from minor latent variations. To tackle this issue, we propose Smooth Diffusion, a new category of diffusion models that can be simultaneously high-performing and smooth. Specifically, we introduce Step-wise Variation Regularization to enforce the proportion between the variations of an arbitrary input latent and that of the output image is a constant at any diffusion training step. In addition, we devise an interpolation standard deviation (ISTD) metric to effectively assess the latent space smoothness of a diffusion model. Extensive quantitative and qualitative experiments demonstrate that Smooth Diffusion stands out as a more desirable solution not only in T2I generation but also across various downstream tasks. Smooth Diffusion is implemented as a plug-and-play Smooth-LoRA to work with various community models. Code is available at https://github.com/SHI-Labs/Smooth-Diffusion.

We give an improved theoretical analysis of score-based generative modeling. Under a score estimate with small L^2 error (averaged across timesteps), we provide efficient convergence guarantees for any data distribution with second-order moment, by either employing early stopping or assuming smoothness condition on the score function of the data distribution. Our result does not rely on any log-concavity or functional inequality assumption and has a logarithmic dependence on the smoothness. In particular, we show that under only a finite second moment condition, approximating the following in reverse KL divergence in epsilon-accuracy can be done in tilde Oleft(d log (1/delta){epsilon}right) steps: 1) the variance-delta Gaussian perturbation of any data distribution; 2) data distributions with 1/delta-smooth score functions. Our analysis also provides a quantitative comparison between different discrete approximations and may guide the choice of discretization points in practice.

The Mixture-of-Experts (MoE) architecture is showing promising results in improving parameter sharing in multi-task learning (MTL) and in scaling high-capacity neural networks. State-of-the-art MoE models use a trainable sparse gate to select a subset of the experts for each input example. While conceptually appealing, existing sparse gates, such as Top-k, are not smooth. The lack of smoothness can lead to convergence and statistical performance issues when training with gradient-based methods. In this paper, we develop DSelect-k: a continuously differentiable and sparse gate for MoE, based on a novel binary encoding formulation. The gate can be trained using first-order methods, such as stochastic gradient descent, and offers explicit control over the number of experts to select. We demonstrate the effectiveness of DSelect-k on both synthetic and real MTL datasets with up to 128 tasks. Our experiments indicate that DSelect-k can achieve statistically significant improvements in prediction and expert selection over popular MoE gates. Notably, on a real-world, large-scale recommender system, DSelect-k achieves over 22% improvement in predictive performance compared to Top-k. We provide an open-source implementation of DSelect-k.

We consider the block coordinate descent methods of Gauss-Seidel type with proximal regularization (BCD-PR), which is a classical method of minimizing general nonconvex objectives under constraints that has a wide range of practical applications. We theoretically establish the worst-case complexity bound for this algorithm. Namely, we show that for general nonconvex smooth objectives with block-wise constraints, the classical BCD-PR algorithm converges to an epsilon-stationary point within O(1/epsilon) iterations. Under a mild condition, this result still holds even if the algorithm is executed inexactly in each step. As an application, we propose a provable and efficient algorithm for `Wasserstein CP-dictionary learning', which seeks a set of elementary probability distributions that can well-approximate a given set of d-dimensional joint probability distributions. Our algorithm is a version of BCD-PR that operates in the dual space, where the primal problem is regularized both entropically and proximally.

With the widespread adoption of machine learning systems, the need to curtail their behavior has become increasingly apparent. This is evidenced by recent advancements towards developing models that satisfy robustness, safety, and fairness requirements. These requirements can be imposed (with generalization guarantees) by formulating constrained learning problems that can then be tackled by dual ascent algorithms. Yet, though these algorithms converge in objective value, even in non-convex settings, they cannot guarantee that their outcome is feasible. Doing so requires randomizing over all iterates, which is impractical in virtually any modern applications. Still, final iterates have been observed to perform well in practice. In this work, we address this gap between theory and practice by characterizing the constraint violation of Lagrangian minimizers associated with optimal dual variables, despite lack of convexity. To do this, we leverage the fact that non-convex, finite-dimensional constrained learning problems can be seen as parametrizations of convex, functional problems. Our results show that rich parametrizations effectively mitigate the issue of feasibility in dual methods, shedding light on prior empirical successes of dual learning. We illustrate our findings in fair learning tasks.

Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. This work combines techniques from semi-infinite constrained learning and manifold regularization to learn representations that are globally smooth on a manifold. To do so, it shows that under typical conditions the problem of learning a Lipschitz continuous function on a manifold is equivalent to a dynamically weighted manifold regularization problem. This observation leads to a practical algorithm based on a weighted Laplacian penalty whose weights are adapted using stochastic gradient techniques. It is shown that under mild conditions, this method estimates the Lipschitz constant of the solution, learning a globally smooth solution as a byproduct. Experiments on real world data illustrate the advantages of the proposed method relative to existing alternatives.

In this work, we study first-order algorithms for solving Bilevel Optimization (BO) where the objective functions are smooth but possibly nonconvex in both levels and the variables are restricted to closed convex sets. As a first step, we study the landscape of BO through the lens of penalty methods, in which the upper- and lower-level objectives are combined in a weighted sum with penalty parameter sigma > 0. In particular, we establish a strong connection between the penalty function and the hyper-objective by explicitly characterizing the conditions under which the values and derivatives of the two must be O(sigma)-close. A by-product of our analysis is the explicit formula for the gradient of hyper-objective when the lower-level problem has multiple solutions under minimal conditions, which could be of independent interest. Next, viewing the penalty formulation as O(sigma)-approximation of the original BO, we propose first-order algorithms that find an epsilon-stationary solution by optimizing the penalty formulation with sigma = O(epsilon). When the perturbed lower-level problem uniformly satisfies the small-error proximal error-bound (EB) condition, we propose a first-order algorithm that converges to an epsilon-stationary point of the penalty function, using in total O(epsilon^{-3}) and O(epsilon^{-7}) accesses to first-order (stochastic) gradient oracles when the oracle is deterministic and oracles are noisy, respectively. Under an additional assumption on stochastic oracles, we show that the algorithm can be implemented in a fully {\it single-loop} manner, i.e., with O(1) samples per iteration, and achieves the improved oracle-complexity of O(epsilon^{-3}) and O(epsilon^{-5}), respectively.

Given a factorization of an image into a sum of linear components, we present a zero-shot method to control each individual component through diffusion model sampling. For example, we can decompose an image into low and high spatial frequencies and condition these components on different text prompts. This produces hybrid images, which change appearance depending on viewing distance. By decomposing an image into three frequency subbands, we can generate hybrid images with three prompts. We also use a decomposition into grayscale and color components to produce images whose appearance changes when they are viewed in grayscale, a phenomena that naturally occurs under dim lighting. And we explore a decomposition by a motion blur kernel, which produces images that change appearance under motion blurring. Our method works by denoising with a composite noise estimate, built from the components of noise estimates conditioned on different prompts. We also show that for certain decompositions, our method recovers prior approaches to compositional generation and spatial control. Finally, we show that we can extend our approach to generate hybrid images from real images. We do this by holding one component fixed and generating the remaining components, effectively solving an inverse problem.

Humans manipulate various kinds of fluids in their everyday life: creating latte art, scooping floating objects from water, rolling an ice cream cone, etc. Using robots to augment or replace human labors in these daily settings remain as a challenging task due to the multifaceted complexities of fluids. Previous research in robotic fluid manipulation mostly consider fluids governed by an ideal, Newtonian model in simple task settings (e.g., pouring). However, the vast majority of real-world fluid systems manifest their complexities in terms of the fluid's complex material behaviors and multi-component interactions, both of which were well beyond the scope of the current literature. To evaluate robot learning algorithms on understanding and interacting with such complex fluid systems, a comprehensive virtual platform with versatile simulation capabilities and well-established tasks is needed. In this work, we introduce FluidLab, a simulation environment with a diverse set of manipulation tasks involving complex fluid dynamics. These tasks address interactions between solid and fluid as well as among multiple fluids. At the heart of our platform is a fully differentiable physics simulator, FluidEngine, providing GPU-accelerated simulations and gradient calculations for various material types and their couplings. We identify several challenges for fluid manipulation learning by evaluating a set of reinforcement learning and trajectory optimization methods on our platform. To address these challenges, we propose several domain-specific optimization schemes coupled with differentiable physics, which are empirically shown to be effective in tackling optimization problems featured by fluid system's non-convex and non-smooth properties. Furthermore, we demonstrate reasonable sim-to-real transfer by deploying optimized trajectories in real-world settings.

In this paper, we consider non-convex multi-block bilevel optimization (MBBO) problems, which involve mgg 1 lower level problems and have important applications in machine learning. Designing a stochastic gradient and controlling its variance is more intricate due to the hierarchical sampling of blocks and data and the unique challenge of estimating hyper-gradient. We aim to achieve three nice properties for our algorithm: (a) matching the state-of-the-art complexity of standard BO problems with a single block; (b) achieving parallel speedup by sampling I blocks and sampling B samples for each sampled block per-iteration; (c) avoiding the computation of the inverse of a high-dimensional Hessian matrix estimator. However, it is non-trivial to achieve all of these by observing that existing works only achieve one or two of these properties. To address the involved challenges for achieving (a, b, c), we propose two stochastic algorithms by using advanced blockwise variance-reduction techniques for tracking the Hessian matrices (for low-dimensional problems) or the Hessian-vector products (for high-dimensional problems), and prove an iteration complexity of O(mepsilon^{-3I(I<m)}{II} + mepsilon^{-3}{IB}) for finding an epsilon-stationary point under appropriate conditions. We also conduct experiments to verify the effectiveness of the proposed algorithms comparing with existing MBBO algorithms.

Physical simulators have been widely used in robot planning and control. Among them, differentiable simulators are particularly favored, as they can be incorporated into gradient-based optimization algorithms that are efficient in solving inverse problems such as optimal control and motion planning. Simulating deformable objects is, however, more challenging compared to rigid body dynamics. The underlying physical laws of deformable objects are more complex, and the resulting systems have orders of magnitude more degrees of freedom and therefore they are significantly more computationally expensive to simulate. Computing gradients with respect to physical design or controller parameters is typically even more computationally challenging. In this paper, we propose a real-time, differentiable hybrid Lagrangian-Eulerian physical simulator for deformable objects, ChainQueen, based on the Moving Least Squares Material Point Method (MLS-MPM). MLS-MPM can simulate deformable objects including contact and can be seamlessly incorporated into inference, control and co-design systems. We demonstrate that our simulator achieves high precision in both forward simulation and backward gradient computation. We have successfully employed it in a diverse set of control tasks for soft robots, including problems with nearly 3,000 decision variables.

Training neural networks requires optimizing a loss function that may be highly irregular, and in particular neither convex nor smooth. Popular training algorithms are based on stochastic gradient descent with momentum (SGDM), for which classical analysis applies only if the loss is either convex or smooth. We show that a very small modification to SGDM closes this gap: simply scale the update at each time point by an exponentially distributed random scalar. The resulting algorithm achieves optimal convergence guarantees. Intriguingly, this result is not derived by a specific analysis of SGDM: instead, it falls naturally out of a more general framework for converting online convex optimization algorithms to non-convex optimization algorithms.

In this paper we provide a novel analytical perspective on the theoretical understanding of gradient-based learning algorithms by interpreting consensus-based optimization (CBO), a recently proposed multi-particle derivative-free optimization method, as a stochastic relaxation of gradient descent. Remarkably, we observe that through communication of the particles, CBO exhibits a stochastic gradient descent (SGD)-like behavior despite solely relying on evaluations of the objective function. The fundamental value of such link between CBO and SGD lies in the fact that CBO is provably globally convergent to global minimizers for ample classes of nonsmooth and nonconvex objective functions, hence, on the one side, offering a novel explanation for the success of stochastic relaxations of gradient descent. On the other side, contrary to the conventional wisdom for which zero-order methods ought to be inefficient or not to possess generalization abilities, our results unveil an intrinsic gradient descent nature of such heuristics. This viewpoint furthermore complements previous insights into the working principles of CBO, which describe the dynamics in the mean-field limit through a nonlinear nonlocal partial differential equation that allows to alleviate complexities of the nonconvex function landscape. Our proofs leverage a completely nonsmooth analysis, which combines a novel quantitative version of the Laplace principle (log-sum-exp trick) and the minimizing movement scheme (proximal iteration). In doing so, we furnish useful and precise insights that explain how stochastic perturbations of gradient descent overcome energy barriers and reach deep levels of nonconvex functions. Instructive numerical illustrations support the provided theoretical insights.

Diffusion models have achieved remarkable image generation quality surpassing previous generative models. However, a notable limitation of diffusion models, in comparison to GANs, is their difficulty in smoothly interpolating between two image samples, due to their highly unstructured latent space. Such a smooth interpolation is intriguing as it naturally serves as a solution for the image morphing task with many applications. In this work, we present DiffMorpher, the first approach enabling smooth and natural image interpolation using diffusion models. Our key idea is to capture the semantics of the two images by fitting two LoRAs to them respectively, and interpolate between both the LoRA parameters and the latent noises to ensure a smooth semantic transition, where correspondence automatically emerges without the need for annotation. In addition, we propose an attention interpolation and injection technique and a new sampling schedule to further enhance the smoothness between consecutive images. Extensive experiments demonstrate that DiffMorpher achieves starkly better image morphing effects than previous methods across a variety of object categories, bridging a critical functional gap that distinguished diffusion models from GANs.

During recent years the interest of optimization and machine learning communities in high-probability convergence of stochastic optimization methods has been growing. One of the main reasons for this is that high-probability complexity bounds are more accurate and less studied than in-expectation ones. However, SOTA high-probability non-asymptotic convergence results are derived under strong assumptions such as the boundedness of the gradient noise variance or of the objective's gradient itself. In this paper, we propose several algorithms with high-probability convergence results under less restrictive assumptions. In particular, we derive new high-probability convergence results under the assumption that the gradient/operator noise has bounded central alpha-th moment for alpha in (1,2] in the following setups: (i) smooth non-convex / Polyak-Lojasiewicz / convex / strongly convex / quasi-strongly convex minimization problems, (ii) Lipschitz / star-cocoercive and monotone / quasi-strongly monotone variational inequalities. These results justify the usage of the considered methods for solving problems that do not fit standard functional classes studied in stochastic optimization.

We introduce the framework of performative reinforcement learning where the policy chosen by the learner affects the underlying reward and transition dynamics of the environment. Following the recent literature on performative prediction~Perdomo et. al., 2020, we introduce the concept of performatively stable policy. We then consider a regularized version of the reinforcement learning problem and show that repeatedly optimizing this objective converges to a performatively stable policy under reasonable assumptions on the transition dynamics. Our proof utilizes the dual perspective of the reinforcement learning problem and may be of independent interest in analyzing the convergence of other algorithms with decision-dependent environments. We then extend our results for the setting where the learner just performs gradient ascent steps instead of fully optimizing the objective, and for the setting where the learner has access to a finite number of trajectories from the changed environment. For both settings, we leverage the dual formulation of performative reinforcement learning and establish convergence to a stable solution. Finally, through extensive experiments on a grid-world environment, we demonstrate the dependence of convergence on various parameters e.g. regularization, smoothness, and the number of samples.

Vision Transformers (ViTs) and MLPs signal further efforts on replacing hand-wired features or inductive biases with general-purpose neural architectures. Existing works empower the models by massive data, such as large-scale pre-training and/or repeated strong data augmentations, and still report optimization-related problems (e.g., sensitivity to initialization and learning rates). Hence, this paper investigates ViTs and MLP-Mixers from the lens of loss geometry, intending to improve the models' data efficiency at training and generalization at inference. Visualization and Hessian reveal extremely sharp local minima of converged models. By promoting smoothness with a recently proposed sharpness-aware optimizer, we substantially improve the accuracy and robustness of ViTs and MLP-Mixers on various tasks spanning supervised, adversarial, contrastive, and transfer learning (e.g., +5.3\% and +11.0\% top-1 accuracy on ImageNet for ViT-B/16 and Mixer-B/16, respectively, with the simple Inception-style preprocessing). We show that the improved smoothness attributes to sparser active neurons in the first few layers. The resultant ViTs outperform ResNets of similar size and throughput when trained from scratch on ImageNet without large-scale pre-training or strong data augmentations. Model checkpoints are available at https://github.com/google-research/vision_transformer.

Nesterov's accelerated gradient (AG) is a popular technique to optimize objective functions comprising two components: a convex loss and a penalty function. While AG methods perform well for convex penalties, such as the LASSO, convergence issues may arise when it is applied to nonconvex penalties, such as SCAD. A recent proposal generalizes Nesterov's AG method to the nonconvex setting. The proposed algorithm requires specification of several hyperparameters for its practical application. Aside from some general conditions, there is no explicit rule for selecting the hyperparameters, and how different selection can affect convergence of the algorithm. In this article, we propose a hyperparameter setting based on the complexity upper bound to accelerate convergence, and consider the application of this nonconvex AG algorithm to high-dimensional linear and logistic sparse learning problems. We further establish the rate of convergence and present a simple and useful bound to characterize our proposed optimal damping sequence. Simulation studies show that convergence can be made, on average, considerably faster than that of the conventional proximal gradient algorithm. Our experiments also show that the proposed method generally outperforms the current state-of-the-art methods in terms of signal recovery.

Differential private optimization for nonconvex smooth objective is considered. In the previous work, the best known utility bound is widetilde O(d/(nvarepsilon_DP)) in terms of the squared full gradient norm, which is achieved by Differential Private Gradient Descent (DP-GD) as an instance, where n is the sample size, d is the problem dimensionality and varepsilon_DP is the differential privacy parameter. To improve the best known utility bound, we propose a new differential private optimization framework called DIFF2 (DIFFerential private optimization via gradient DIFFerences) that constructs a differential private global gradient estimator with possibly quite small variance based on communicated gradient differences rather than gradients themselves. It is shown that DIFF2 with a gradient descent subroutine achieves the utility of widetilde O(d^{2/3}/(nvarepsilon_DP)^{4/3}), which can be significantly better than the previous one in terms of the dependence on the sample size n. To the best of our knowledge, this is the first fundamental result to improve the standard utility widetilde O(d/(nvarepsilon_DP)) for nonconvex objectives. Additionally, a more computational and communication efficient subroutine is combined with DIFF2 and its theoretical analysis is also given. Numerical experiments are conducted to validate the superiority of DIFF2 framework.

Machine learning models have recently found tremendous success in data-driven control systems. However, standard learning models often suffer from an accuracy-robustness trade-off, which is a limitation that must be overcome in the control of safety-critical systems that require both high performance and rigorous robustness guarantees. In this work, we build upon the recent "locally biased smoothing" method to develop classifiers that simultaneously inherit high accuracy from standard models and high robustness from robust models. Specifically, we extend locally biased smoothing to the multi-class setting, and then overcome its performance bottleneck by generalizing the formulation to "mix" the outputs of a standard neural network and a robust neural network. We prove that when the robustness of the robust base model is certifiable, within a closed-form ell_p radius, no alteration or attack on an input can result in misclassification of the mixed classifier; the proposed model inherits the certified robustness. Moreover, we use numerical experiments on the CIFAR-10 benchmark dataset to verify that the mixed model noticeably improves the accuracy-robustness trade-off.

Understanding and analyzing markets is crucial, yet analytical equilibrium solutions remain largely infeasible. Recent breakthroughs in equilibrium computation rely on zeroth-order policy gradient estimation. These approaches commonly suffer from high variance and are computationally expensive. The use of fully differentiable simulators would enable more efficient gradient estimation. However, the discrete allocation of goods in economic simulations is a non-differentiable operation. This renders the first-order Monte Carlo gradient estimator inapplicable and the learning feedback systematically misleading. We propose a novel smoothing technique that creates a surrogate market game, in which first-order methods can be applied. We provide theoretical bounds on the resulting bias which justifies solving the smoothed game instead. These bounds also allow choosing the smoothing strength a priori such that the resulting estimate has low variance. Furthermore, we validate our approach via numerous empirical experiments. Our method theoretically and empirically outperforms zeroth-order methods in approximation quality and computational efficiency.

Text-driven localized editing of 3D objects is particularly difficult as locally mixing the original 3D object with the intended new object and style effects without distorting the object's form is not a straightforward process. To address this issue, we propose a novel NeRF-based model, Blending-NeRF, which consists of two NeRF networks: pretrained NeRF and editable NeRF. Additionally, we introduce new blending operations that allow Blending-NeRF to properly edit target regions which are localized by text. By using a pretrained vision-language aligned model, CLIP, we guide Blending-NeRF to add new objects with varying colors and densities, modify textures, and remove parts of the original object. Our extensive experiments demonstrate that Blending-NeRF produces naturally and locally edited 3D objects from various text prompts. Our project page is available at https://seokhunchoi.github.io/Blending-NeRF/

Stochastically Extended Adversarial (SEA) model is introduced by Sachs et al. [2022] as an interpolation between stochastic and adversarial online convex optimization. Under the smoothness condition, they demonstrate that the expected regret of optimistic follow-the-regularized-leader (FTRL) depends on the cumulative stochastic variance sigma_{1:T}^2 and the cumulative adversarial variation Sigma_{1:T}^2 for convex functions. They also provide a slightly weaker bound based on the maximal stochastic variance sigma_{max}^2 and the maximal adversarial variation Sigma_{max}^2 for strongly convex functions. Inspired by their work, we investigate the theoretical guarantees of optimistic online mirror descent (OMD) for the SEA model. For convex and smooth functions, we obtain the same O(sigma_{1:T^2}+Sigma_{1:T^2}) regret bound, without the convexity requirement of individual functions. For strongly convex and smooth functions, we establish an O(min{log (sigma_{1:T}^2+Sigma_{1:T}^2), (sigma_{max}^2 + Sigma_{max}^2) log T}) bound, better than their O((sigma_{max}^2 + Sigma_{max}^2) log T) bound. For exp-concave and smooth functions, we achieve a new O(dlog(sigma_{1:T}^2+Sigma_{1:T}^2)) bound. Owing to the OMD framework, we can further extend our result to obtain dynamic regret guarantees, which are more favorable in non-stationary online scenarios. The attained results allow us to recover excess risk bounds of the stochastic setting and regret bounds of the adversarial setting, and derive new guarantees for many intermediate scenarios.

In this paper, we study the existence and uniqueness of solutions to the Euler equations with initial conditions that exhibit analytic regularity near the boundary and Sobolev regularity away from it. A key contribution of this work is the introduction of the diamond-analyticity framework, which captures the spatial decay of the analyticity radius in a structured manner, improving upon uniform analyticity approaches. We employ the Leray projection and a nonstandard mollification technique to demonstrate that the quotient between the imaginary and real parts of the analyticity radius remains unrestricted, thus extending the analyticity persistence results beyond traditional constraints. Our methodology combines analytic-Sobolev estimates with an iterative scheme which is nonstandard in the Cauchy-Kowalevskaya framework, ensuring rigorous control over the evolution of the solution. These results contribute to a deeper understanding of the interplay between analyticity and boundary effects in fluid equations. They might have implications for the study of the inviscid limit of the Navier-Stokes equations and the role of complex singularities in fluid dynamics.

Implicit models separate the definition of a layer from the description of its solution process. While implicit layers allow features such as depth to adapt to new scenarios and inputs automatically, this adaptivity makes its computational expense challenging to predict. In this manuscript, we increase the "implicitness" of the DEQ by redefining the method in terms of an infinite time neural ODE, which paradoxically decreases the training cost over a standard neural ODE by 2-4x. Additionally, we address the question: is there a way to simultaneously achieve the robustness of implicit layers while allowing the reduced computational expense of an explicit layer? To solve this, we develop Skip and Skip Reg. DEQ, an implicit-explicit (IMEX) layer that simultaneously trains an explicit prediction followed by an implicit correction. We show that training this explicit predictor is free and even decreases the training time by 1.11-3.19x. Together, this manuscript shows how bridging the dichotomy of implicit and explicit deep learning can combine the advantages of both techniques.

We present a generalization of the proximal operator defined through a convex combination of convex objectives, where the coefficients are updated in a minimax fashion. We prove that this new operator is Bregman firmly nonexpansive with respect to a Bregman divergence that combines Euclidean and information geometries.

Non-convex optimization plays a key role in a growing number of machine learning applications. This motivates the identification of specialized structure that enables sharper theoretical analysis. One such identified structure is quasar-convexity, a non-convex generalization of convexity that subsumes convex functions. Existing algorithms for minimizing quasar-convex functions in the stochastic setting have either high complexity or slow convergence, which prompts us to derive a new class of stochastic methods for optimizing smooth quasar-convex functions. We demonstrate that our algorithms have fast convergence and outperform existing algorithms on several examples, including the classical problem of learning linear dynamical systems. We also present a unified analysis of our newly proposed algorithms and a previously studied deterministic algorithm.

Textured 3D morphing creates smooth and plausible interpolation sequences between two 3D objects, focusing on transitions in both shape and texture. This is important for creative applications like visual effects in filmmaking. Previous methods rely on establishing point-to-point correspondences and determining smooth deformation trajectories, which inherently restrict them to shape-only morphing on untextured, topologically aligned datasets. This restriction leads to labor-intensive preprocessing and poor generalization. To overcome these challenges, we propose a method for 3D regenerative morphing using a 3D diffusion prior. Unlike previous methods that depend on explicit correspondences and deformations, our method eliminates the additional need for obtaining correspondence and uses the 3D diffusion prior to generate morphing. Specifically, we introduce a 3D diffusion model and interpolate the source and target information at three levels: initial noise, model parameters, and condition features. We then explore an Attention Fusion strategy to generate more smooth morphing sequences. To further improve the plausibility of semantic interpolation and the generated 3D surfaces, we propose two strategies: (a) Token Reordering, where we match approximate tokens based on semantic analysis to guide implicit correspondences in the denoising process of the diffusion model, and (b) Low-Frequency Enhancement, where we enhance low-frequency signals in the tokens to improve the quality of generated surfaces. Experimental results show that our method achieves superior smoothness and plausibility in 3D morphing across diverse cross-category object pairs, offering a novel regenerative method for 3D morphing with textured representations.

Text-to-texture generation has recently attracted increasing attention, but existing methods often suffer from the problems of view inconsistencies, apparent seams, and misalignment between textures and the underlying mesh. In this paper, we propose a robust text-to-texture method for generating consistent and seamless textures that are well aligned with the mesh. Our method leverages state-of-the-art 2D diffusion models, including SDXL and multiple ControlNets, to capture structural features and intricate details in the generated textures. The method also employs a symmetrical view synthesis strategy combined with regional prompts for enhancing view consistency. Additionally, it introduces novel texture blending and soft-inpainting techniques, which significantly reduce the seam regions. Extensive experiments demonstrate that our method outperforms existing state-of-the-art methods.

We propose a penalized nonparametric approach to estimating the quantile regression process (QRP) in a nonseparable model using rectifier quadratic unit (ReQU) activated deep neural networks and introduce a novel penalty function to enforce non-crossing of quantile regression curves. We establish the non-asymptotic excess risk bounds for the estimated QRP and derive the mean integrated squared error for the estimated QRP under mild smoothness and regularity conditions. To establish these non-asymptotic risk and estimation error bounds, we also develop a new error bound for approximating C^s smooth functions with s >0 and their derivatives using ReQU activated neural networks. This is a new approximation result for ReQU networks and is of independent interest and may be useful in other problems. Our numerical experiments demonstrate that the proposed method is competitive with or outperforms two existing methods, including methods using reproducing kernels and random forests, for nonparametric quantile regression.