Exact Risk Curves of signSGD in High-Dimensions: Quantifying Preconditioning and Noise-Compression Effects

In recent years, signSGD has garnered interest as both a practical optimizer as well as a simple model to understand adaptive optimizers like Adam. Though there is a general consensus that signSGD acts to precondition optimization and reshapes noise, quantitatively understanding these effects in theoretically solvable settings remains difficult. We present an analysis of signSGD in a high dimensional limit, and derive a limiting SDE and ODE to describe the risk. Using this framework we quantify four effects of signSGD: effective learning rate, noise compression, diagonal preconditioning, and gradient noise reshaping. Our analysis is consistent with experimental observations but moves beyond that by quantifying the dependence of these effects on the data and noise distributions. We conclude with a conjecture on how these results might be extended to Adam.

Stepping on the Edge: Curvature Aware Learning Rate Tuners

Curvature information -- particularly, the largest eigenvalue of the loss Hessian, known as the sharpness -- often forms the basis for learning rate tuners. However, recent work has shown that the curvature information undergoes complex dynamics during training, going from a phase of increasing sharpness to eventual stabilization. We analyze the closed-loop feedback effect between learning rate tuning and curvature. We find that classical learning rate tuners may yield greater one-step loss reduction, yet they ultimately underperform in the long term when compared to constant learning rates in the full batch regime. These models break the stabilization of the sharpness, which we explain using a simplified model of the joint dynamics of the learning rate and the curvature. To further investigate these effects, we introduce a new learning rate tuning method, Curvature Dynamics Aware Tuning (CDAT), which prioritizes long term curvature stabilization over instantaneous progress on the objective. In the full batch regime, CDAT shows behavior akin to prefixed warm-up schedules on deep learning objectives, outperforming tuned constant learning rates. In the mini batch regime, we observe that stochasticity introduces confounding effects that explain the previous success of some learning rate tuners at appropriate batch sizes. Our findings highlight the critical role of understanding the joint dynamics of the learning rate and curvature, beyond greedy minimization, to diagnose failures and design effective adaptive learning rate tuners.

A Clipped Trip: the Dynamics of SGD with Gradient Clipping in High-Dimensions

The success of modern machine learning is due in part to the adaptive optimization methods that have been developed to deal with the difficulties of training large models over complex datasets. One such method is gradient clipping: a practical procedure with limited theoretical underpinnings. In this work, we study clipping in a least squares problem under streaming SGD. We develop a theoretical analysis of the learning dynamics in the limit of large intrinsic dimension-a model and dataset dependent notion of dimensionality. In this limit we find a deterministic equation that describes the evolution of the loss. We show that with Gaussian noise clipping cannot improve SGD performance. Yet, in other noisy settings, clipping can provide benefits with tuning of the clipping threshold. In these cases, clipping biases updates in a way beneficial to training which cannot be recovered by SGD under any schedule. We conclude with a discussion about the links between high-dimensional clipping and neural network training.

High dimensional analysis reveals conservative sharpening and a stochastic edge of stability

Recent empirical and theoretical work has shown that the dynamics of the large eigenvalues of the training loss Hessian have some remarkably robust features across models and datasets in the full batch regime. There is often an early period of progressive sharpening where the large eigenvalues increase, followed by stabilization at a predictable value known as the edge of stability. Previous work showed that in the stochastic setting, the eigenvalues increase more slowly - a phenomenon we call conservative sharpening. We provide a theoretical analysis of a simple high-dimensional model which shows the origin of this slowdown. We also show that there is an alternative stochastic edge of stability which arises at small batch size that is sensitive to the trace of the Neural Tangent Kernel rather than the large Hessian eigenvalues. We conduct an experimental study which highlights the qualitative differences from the full batch phenomenology, and suggests that controlling the stochastic edge of stability can help optimization.

Gradient descent induces alignment between weights and the empirical NTK for deep non-linear networks

Understanding the mechanisms through which neural networks extract statistics from input-label pairs is one of the most important unsolved problems in supervised learning. Prior works have identified that the gram matrices of the weights in trained neural networks of general architectures are proportional to the average gradient outer product of the model, in a statement known as the Neural Feature Ansatz (NFA). However, the reason these quantities become correlated during training is poorly understood. In this work, we explain the emergence of this correlation. We identify that the NFA is equivalent to alignment between the left singular structure of the weight matrices and a significant component of the empirical neural tangent kernels associated with those weights. We establish that the NFA introduced in prior works is driven by a centered NFA that isolates this alignment. We show that the speed of NFA development can be predicted analytically at early training times in terms of simple statistics of the inputs and labels. Finally, we introduce a simple intervention to increase NFA correlation at any given layer, which dramatically improves the quality of features learned.

Neglected Hessian component explains mysteries in Sharpness regularization

Recent work has shown that methods like SAM which either explicitly or implicitly penalize second order information can improve generalization in deep learning. Seemingly similar methods like weight noise and gradient penalties often fail to provide such benefits. We show that these differences can be explained by the structure of the Hessian of the loss. First, we show that a common decomposition of the Hessian can be quantitatively interpreted as separating the feature exploitation from feature exploration. The feature exploration, which can be described by the Nonlinear Modeling Error matrix (NME), is commonly neglected in the literature since it vanishes at interpolation. Our work shows that the NME is in fact important as it can explain why gradient penalties are sensitive to the choice of activation function. Using this insight we design interventions to improve performance. We also provide evidence that challenges the long held equivalence of weight noise and gradient penalties. This equivalence relies on the assumption that the NME can be ignored, which we find does not hold for modern networks since they involve significant feature learning. We find that regularizing feature exploitation but not feature exploration yields performance similar to gradient penalties.

On the Interplay Between Stepsize Tuning and Progressive Sharpening

Recent empirical work has revealed an intriguing property of deep learning models by which the sharpness (largest eigenvalue of the Hessian) increases throughout optimization until it stabilizes around a critical value at which the optimizer operates at the edge of stability, given a fixed stepsize (Cohen et al, 2022). We investigate empirically how the sharpness evolves when using stepsize-tuners, the Armijo linesearch and Polyak stepsizes, that adapt the stepsize along the iterations to local quantities such as, implicitly, the sharpness itself. We find that the surprisingly poor performance of a classical Armijo linesearch may be well explained by its tendency to ever-increase the sharpness of the objective in the full or large batch regimes. On the other hand, we observe that Polyak stepsizes operate generally at the edge of stability or even slightly beyond, while outperforming its Armijo and constant stepsizes counterparts. We conclude with an analysis that suggests unlocking stepsize tuners requires an understanding of the joint dynamics of the step size and the sharpness.

SAM operates far from home: eigenvalue regularization as a dynamical phenomenon

The Sharpness Aware Minimization (SAM) optimization algorithm has been shown to control large eigenvalues of the loss Hessian and provide generalization benefits in a variety of settings. The original motivation for SAM was a modified loss function which penalized sharp minima; subsequent analyses have also focused on the behavior near minima. However, our work reveals that SAM provides a strong regularization of the eigenvalues throughout the learning trajectory. We show that in a simplified setting, SAM dynamically induces a stabilization related to the edge of stability (EOS) phenomenon observed in large learning rate gradient descent. Our theory predicts the largest eigenvalue as a function of the learning rate and SAM radius parameters. Finally, we show that practical models can also exhibit this EOS stabilization, and that understanding SAM must account for these dynamics far away from any minima.

Second-order regression models exhibit progressive sharpening to the edge of stability

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.

Deep equilibrium networks are sensitive to initialization statistics

Deep equilibrium networks (DEQs) are a promising way to construct models which trade off memory for compute. However, theoretical understanding of these models is still lacking compared to traditional networks, in part because of the repeated application of a single set of weights. We show that DEQs are sensitive to the higher order statistics of the matrix families from which they are initialized. In particular, initializing with orthogonal or symmetric matrices allows for greater stability in training. This gives us a practical prescription for initializations which allow for training with a broader range of initial weight scales.