Implicit Regularization in Feedback Alignment Learning Mechanisms for
  Neural Networks
Feedback Alignment (FA) methods are biologically inspired local learning
rules for training neural networks with reduced communication between layers.
While FA has potential applications in distributed and privacy-aware ML,
limitations in multi-class classification and lack of theoretical understanding
of the alignment mechanism have constrained its impact. This study introduces a
unified framework elucidating the operational principles behind alignment in
FA. Our key contributions include: (1) a novel conservation law linking changes
in synaptic weights to implicit regularization that maintains alignment with
the gradient, with support from experiments, (2) sufficient conditions for
convergence based on the concept of alignment dominance, and (3) empirical
analysis showing better alignment can enhance FA performance on complex
multi-class tasks. Overall, these theoretical and practical advancements
improve interpretability of bio-plausible learning rules and provide groundwork
for developing enhanced FA algorithms.
Zachary Robertson, Oluwasanmi Koyejo
# Layer-Wise Feedback Alignment is Conserved in Deep Neural Networks

###### Abstract

In the quest to enhance the efficiency and bio-plausibility of training deep neural networks, Feedback Alignment (FA), which replaces the backward pass weights with random matrices in the training process, has emerged as an alternative to traditional backpropagation. While the appeal of FA lies in its circumvention of computational challenges and its plausible biological alignment, the theoretical understanding of this learning rule remains partial. This paper uncovers a set of conservation laws underpinning the learning dynamics of FA, revealing intriguing parallels between FA and Gradient Descent (GD). Our analysis reveals that FA harbors implicit biases akin to those exhibited by GD, challenging the prevailing narrative that these learning algorithms are fundamentally different. Moreover, we demonstrate that these conservation laws elucidate sufficient conditions for layer-wise alignment with feedback matrices in ReLU networks. We further show that this implies over-parameterized two-layer linear networks trained with FA converge to minimum-norm solutions. The implications of our findings offer avenues for developing more efficient and biologically plausible alternatives to backpropagation through an understanding of the principles governing learning dynamics in deep networks.

Machine Learning, ICML

## 1 Introduction

Backpropagation, a widely successful learning rule for artificial neural networks, has been instrumental in advancing deep learning. Nevertheless, it presents significant challenges, including the intricate backward pass mechanism, which complicates training parallelism due to communication bottlenecks. In addition, it lacks biological plausibility, further limiting its practical utility.

As an alternative, Feedback Alignment (FA) offers an attractive learning rule that alleviates the computational complexities and bio-plausibility issues associated with backpropagation (Lillicrap et al., 2016). FA replaces the backward pass with random feedback matrices, promising a more straightforward approach to training neural networks.

Despite its advantages, the understanding of FA's underlying principles, particularly in the context of deep neural networks, remains elusive. This gap in knowledge motivates our current investigation, which seeks to unravel the inherent laws that govern the learning dynamics under FA.

Our work makes several significant contributions. Firstly, we establish a set of conservation laws for the learning dynamics under FA, elucidating the implicit bias exhibited by this learning rule--a bias distinct from that of backpropagation. Secondly, these conservation laws enable us to identify a sufficient condition for layer-wise alignment with feedback matrices. Lastly, we provide evidence that two-layer linear networks trained with FA converge to a global optimum.

We believe that our results will have broad implications for future research and practical applications. By quantifying the properties of alternative learning rules like FA, our analysis provides valuable insights that can inform both theoretical advancements and the design of more efficient and biologically plausible alternatives to backpropagation.

## 2 Related Work

The pioneering work by Stork (1989) questioned the biological plausibility of backpropagation, leading to the exploration of alternative learning rules. Feedback Alignment (FA) emerged as a promising candidate in this regard. Lillicrap et al. (2016) first introduced FA as an effective learning rule for deep neural networks. This approach was further investigated by Nokland (2016), who demonstrated that random synaptic feedback weights could support error backpropagation for deep learning.

The dynamics of learning with FA have also been the subject of several studies (Refinetti et al., 2021; Song et al., 2021; Launay et al., 2020; Lechner, 2020; Bordelon and Pehlevan, 2022). For instance, Refinetti et al. (2021) highlighted the dynamics of aligning before memorizing in learning with FA, while a parallel line of work by Launay et al. (2020)
showcased that FA can be successfully scaled to modern deep learning tasks and architectures. The work of Lechner (2020) also shows alignment of the learning rule updates with the gradient for single-output networks trained with a variant of feedback alignment, but there are counter-examples for the multi-output setting. There have also been works investigating bio-plausible learning rules applied to networks with large width (Song et al., 2021; Bordelon and Pehlevan, 2022). Our analysis can be seen as a generalization of the approach taken by Lillicrap et al. (2016) to investigate convergence of FA. In particular, our Theorem 5.1 is inspired by a relation satisfied by the parameter updates used in their main convergence result for linear feedback alignment.

Finally, our study is related to the implicit bias of models trained with gradient descent (Vallet et al., 1989; Duin, 2000; Du et al., 2018; Soudry et al., 2018; Belkin et al., 2019). There is a longer history of investigating the phenomena of gradient descent picking solutions that generalize well in linear models (Vallet et al., 1989; Duin, 2000). However, more recently, there has been work looking into similar phenomena involving deep neural networks (Du et al., 2018; Soudry et al., 2018; Belkin et al., 2019). In particular, a related conservation law of neural networks was studied by Du et al. (2018), where they examined the self-balancing property of layers in deep homogeneous models. While their work provides valuable insights into the role of conservation laws in learning dynamics, it does not directly address the implications on alignment as a result of these laws.

Our work diverges from this existing body of literature by proposing a set of conservation laws specifically tailored to the learning dynamics under FA. A unique feature of our study is that these conservation laws yield immediate implications on alignment as a corollary, offering a more comprehensive understanding of the FA's underlying principles. To the best of our knowledge, this is the first result establishing layer-wise alignment for a non-linear network trained with feedback alignment, paving the way for future research in layer-wise alignment in more general settings such as wide neural networks or for other learning rules.

## 3 Basic Notation

To fully comprehend the dynamics underpinning Feedback Alignment (FA), we first need to establish the appropriate mathematical notation and formalism. This section will clarify the key definitions and operations used throughout this study. We will denote scalars and vectors by lower-case letters (e.g., \(x,y,z\)) and matrices by uppercase letters (e.g., \(A,B,C\)). The symbol \(\odot\) represents the Hadamard (element-wise) product of two matrices or vectors of the same dimensions. Importantly, we denote the trace of a square matrix \(A\) by \(Tr(A)\). Finally, we note that the trace operator can be used to compute the inner product between two matrices \(A\) and \(B\) of the same dimension. We denote this inner product as \(\langle A,B\rangle=Tr(A^{T}B)\), where \(A^{T}\) is the transpose of \(A\).

## 4 Feedback Alignment: A Closer Look

Consider a neural network \(f\) parameterized with \(L\) layers, where each layer is denoted as \(l\), with \(l\in 1,2,\ldots,L\). Each layer is associated with an activation function \(\phi\) and a weight matrix \(W_{l}\in\mathbb{R}^{n_{l+1}\times n_{l}}\). Here, \(n_{l}\) refers to the number of neurons in layer \(l\).

For a given input \(x\in\mathbb{R}^{n_{0}}\), the pre-activation \(h^{l}\) and the activation \(a^{l}\) of layer \(l\) are computed recursively as follows:

\[h_{l}=W_{l}a_{l-1},\;a_{l}=\phi(h_{l})\]

where \(\phi\) denotes the non-linear activation function, \(a_{l-1}\) is the activation of the previous layer, and \(a_{0}=x\).

In FA, the feedback weights, denoted as \(B_{l}\in\mathbb{R}^{n_{l}\times n_{l+1}}\), are fixed random matrices independent of \(W_{l}\). The error \(\delta_{l}\) at layer \(l\) is calculated as:

\[\delta_{l}=\phi^{\prime}(h_{l})\odot B_{l+1}\delta_{l+1},\;\delta_{L}=\nabla_{a _{L}}\mathcal{L}(f)\]

where \(\mathcal{L}(f)\) is the loss function applied to the network, and \(\nabla_{a_{L}}\mathcal{L}\) is the gradient of \(\mathcal{L}\) with respect to the final layer activation \(a_{L}\).

The weight update under the FA rule is then given by:

\[\Delta W_{l}=-\eta\cdot(a_{l})^{\top}\delta_{l+1}^{\top}\]

where \(\eta\) is the learning rate.

We note that unlike in backpropagation, the feedback matrices \(B_{l}\) are not tied to the feedforward weights \(W_{l}\). In that setting, we would have time-dependent feedback matrices \(B_{l}(t)=W_{l}(t)\). This introduces an alignment challenge since there is no guarantee the feedback matrices align with the learned weights.

## 5 Theoretical Analysis

Building upon the FA formalism detailed in the previous section, we aim to present our main results and provide insight on their distinctive characteristics and how they differ from the traditional backpropagation (BP) in the context of learning rules.

It has been previously observed that the matrices learned through feedback alignment tend to align with their respective feedback matrices (Lillicrap et al., 2016; Nokland, 2016;
Launay et al., 2020). Our first main result concerns a conservation law for the FA learning dynamics that can explain the layer-wise alignment phenomena. Specifically, we introduce a conserved quantity, which remains invariant throughout the training process. This invariance holds under some general conditions related to the activation function and the loss function.

**Theorem 5.1**.: _Suppose that we apply feedback alignment to a scalar output ReLU network with differentiable loss function. Then the flow of the layer weights under feedback alignment for all \(t\in\mathbb{R}_{\geq 0}\) maintains,_

\[\frac{1}{2}\|W_{i}(t)\|_{F}^{2}-\langle W_{i+1}(t),B_{i+1}\rangle\]

\[=\frac{1}{2}\|W_{i}(0)\|_{F}^{2}-\langle W_{i+1}(0),B_{i+1}\rangle\]

Proof Sketch.: The major technical part of the proof is to show that we have:

\[\langle\dot{W}_{i},W_{i}\rangle=\langle\dot{W}_{i+1},B_{i+1}\rangle.\]

The trace map is linear so after an integration by parts we have that,

\[\Rightarrow\int_{0}^{t}\text{Tr}(\dot{W}_{i}W_{i}^{T})ds=\text{ Tr}\left[\int_{0}^{t}\dot{W}_{i}W_{i}^{T}\right]ds\]

\[=\frac{1}{2}\|W_{i}(t)\|_{F}^{2}-\frac{1}{2}\|W_{i}(0)\|_{F}^{2}\]

Finally,

\[\int_{0}^{t}\text{Tr}(\dot{W}_{i+1}B_{i+1}^{T})ds=\text{Tr}\left[\int_{0}^{t} \dot{W}_{i+1}B_{i+1}^{T}ds\right]\]

\[=\langle W_{i+1}(t),B_{i+1}\rangle-\langle W_{i+1}(0),B_{i+1}\rangle\]

The result follows. 

This conservation law has several intriguing implications. One such implication is the existence of an implicit bias in FA that is analogous to, but distinct from, the bias in gradient descent. This bias effectively governs the learning trajectory of the FA rule, directing it towards certain types of solutions over others.

The conservation law also offers an insight into the alignment challenge in FA. As a corollary, if we initialize \(W_{i+1}(0)=B_{i+1}\) such that \(\|W_{i}(0)\|\leq\|W_{i+1}(0)\|\) then the conservation law simplifies to,

\[\langle W_{i+1}(t),B_{i+1}\rangle\geq\|W_{i+1}(0)\|_{F}^{2}-\frac{1}{2}\|W_{i }(0)\|_{F}^{2}\geq 0\]

Thus, there are general initialization schemes that guarantee layer-wise alignment.

Lastly, we focus on the case of two-layer linear networks trained with FA. By exploiting the conservation law, we show that these networks are capable of converging to a global optimum.

**Theorem 5.2**.: _Assume that we are to fit data \(y\) with squared-loss and an over-parameterized two-layer network \(f_{w_{t}}(X)=Xw_{t}=XW_{1}(t)W_{2}(t)\) with data \(X\) such that rows are linearly-independent. Assume we may pick \(w_{0}\in\text{span}(X^{T})\) such that we have positive alignment for all time. If we run (direct) feedback alignment flow then we have the following,_

\[\lim_{t\rightarrow\infty}e^{rt}\cdot\|y-Xw_{t}\|_{2}\to 0\]

_for some \(r>0\). Moreover, \(w_{\infty}=W_{1}(\infty)W_{2}(\infty)\) is the minimum-norm solution._

Proof Sketch.: We prove an auxiliary lemma that shows that the network weights stay in the span of the input data matrix

Figure 1: We plot the ratio \(\frac{\langle W_{2}(t),B_{2}\rangle-\frac{1}{2}\|W_{1}(t)\|_{F}^{2}}{\langle W _{2}(0),B_{2}\rangle-\frac{1}{2}\|W_{1}(0)\|_{F}^{2}}\) during training to verify Theorem 5.1 for two-layer linear and ReLU networks.

Figure 2: We plot the loss during training for two-layer linear and ReLU networks.
\(X^{T}\). To show this we derive the update rules in continuous time, under the assumption that the weights are initialized in the span of the data. A simple option is to initialize with \(W_{1}=0\). Following this, we show that the evolution of the error is decreasing geometrically over time implying that the algorithm will converge. Finally, we demonstrate that under these conditions, the solution found by the flow will be related to the Moore-Penrose psuedoinverse which minimizes the norm of the solution. 

This result implies that feedback-alignment applied to linear networks in the over-parameterized regime enjoy similar generalization properties to those of networks trained with gradient descent. These types of results have been popular as a way to explain why deep neural networks are able to generalize (Belkin et al., 2019).

These theoretical findings indicate that FA shares some properties with the more traditional gradient descent approach to learning. In particular, we also establish a connection between initialization, layer-wise alignment, and convergence with our results. This could set the stage for further exploration of alternative learning rules that can overcome the practical challenges associated with BP, whilst retaining its effectiveness.

## 6 Experiments

In our effort to comprehend the fundamental principles governing Feedback Alignment (FA), we devised a series of experiments to validate our theoretical assertions. We specifically aimed to observe the FA learning dynamics, understanding its implicit bias and ultimately, its convergence to the global optimum in over-parameterized two-layer linear networks.

Our network design involved multi-layer configurations, with the task of finding the global minimizers for the squared-loss function. For the generation of training data, we relied on a multivariate normal distribution, where the true response variable was a noiseless linear transformation of the input.

A crucial element of our experiments was implementing FA to update the network weights. Theorem 5.1 predicts that the quantity \(\frac{1}{2}\|W_{i}(t)\|_{F}^{2}-\langle W_{i+1}(t),B_{i+1}\rangle\) should remain invariant under the FA learning flow. Tracking this quantity provided empirical validation of this theoretical result. Furthermore, comparing the FA-learned weights to the Moore-Penrose pseudoinverse solution allowed us to observe the convergence of FA to the global optimum.

We carried out these procedures for two network architectures - a two-layer linear network and a two-layer ReLU network. To assure high-probability positive layer-wise alignment and convergence, we initialized the first layer weights using a distribution \(\mathcal{N}(0,1/10)\), and the second layer weights with a distribution \(\mathcal{N}(0,1)\). We initialized the feedback matrices to match the initialization of the second layer weights.

### Discussion of Results

The experimental results, as displayed in Figures 1 and 2, corroborate the conservation law postulated by Theorem 5.1. The tracked quantity \(\frac{1}{2}\|W_{i}(t)\|_{F}^{2}-\langle W_{i+1}(t),B_{i+1}\rangle\) remained nearly constant across iterations which provides empirical support to our theoretical results. This manifestation of the conservation law substantiates the intriguing parallels between FA and Gradient Descent (GD), specifically with regards to their implicit biases. In the context of the two-layer linear network, the convergence of FA-learned weights to the global optimal solution was observed. We verified this convergence is identical to the minimum norm solution up to three significant digits. This underscores the implicit bias of feedback alignment towards solutions that generalize well.

In particular, we find the (nearly) exact conservation of layer-wise in non-linear networks updating with the feedback alignment learning rule to be compelling. While we do not show layer-wise alignment is sufficient for convergence in the non-linear setting, it seems plausible that such a condition would become useful in the large-width setting which has been successfully analyzed for gradient descent. Overall, the implications of these shared similarities warrant further investigation, potentially fostering the development of more powerful theoretical techniques capable of distinguishing these two learning rules. We think that research into establishing (or failing) implicit bias results for bio-plausible learning rules could elucidate higher-level principles behind good learning rules.

## 7 Conclusion

In conclusion, our findings challenge the prevailing narrative that FA and GD are fundamentally different learning algorithms. We demonstrate that, under certain conditions, FA can mirror the behavior of GD, offering a computationally efficient and biologically plausible alternative. Our main motivation is to pave the way for developing more efficient deep learning algorithms, better approximating the learning dynamics in biological systems. Overall, our results connecting layer-wise alignment with convergence in linear-models suggest that layer-wise alignment may also be a useful tool for analyzing learning dynamics in wide neural networks. Future work aims to extend these results to nonlinear models, facilitating the creation of more biologically plausible models.
## Acknowledgements

We'd like to thank Tengyu Ma for stimulating conversation about the project and inspiring the focus on investigating implicit bias presented.
