The Optimization Landscape of SGD Across the Feature Learning Strength

We consider neural networks (NNs) where the final layer is down-scaled by a fixed hyperparameter $\gamma$. Recent work has identified $\gamma$ as controlling the strength of feature learning. As $\gamma$ increases, network evolution changes from ``lazy'' kernel dynamics to ``rich'' feature-learning dynamics, with a host of associated benefits including improved performance on common tasks. In this work, we conduct a thorough empirical investigation of the effect of scaling $\gamma$ across a variety of models and datasets in the online training setting. We first examine the interaction of $\gamma$ with the learning rate $\eta$, identifying several scaling regimes in the $\gamma$-$\eta$ plane which we explain theoretically using a simple model. We find that the optimal learning rate $\eta^*$ scales non-trivially with $\gamma$. In particular, $\eta^* \propto \gamma^2$ when $\gamma \ll 1$ and $\eta^* \propto \gamma^{2/L}$ when $\gamma \gg 1$ for a feed-forward network of depth $L$. Using this optimal learning rate scaling, we proceed with an empirical study of the under-explored ``ultra-rich'' $\gamma \gg 1$ regime. We find that networks in this regime display characteristic loss curves, starting with a long plateau followed by a drop-off, sometimes followed by one or more additional staircase steps. We find networks of different large $\gamma$ values optimize along similar trajectories up to a reparameterization of time. We further find that optimal online performance is often found at large $\gamma$ and could be missed if this hyperparameter is not tuned. Our findings indicate that analytical study of the large-$\gamma$ limit may yield useful insights into the dynamics of representation learning in performant models.

Why do Learning Rates Transfer? Reconciling Optimization and Scaling Limits for Deep Learning

Recently, there has been growing evidence that if the width and depth of a neural network are scaled toward the so-called rich feature learning limit ($\mu$P and its depth extension), then some hyperparameters - such as the learning rate - exhibit transfer from small to very large models, thus reducing the cost of hyperparameter tuning. From an optimization perspective, this phenomenon is puzzling, as it implies that the loss landscape is remarkably consistent across very different model sizes. In this work, we find empirical evidence that learning rate transfer can be attributed to the fact that under $\mu$P and its depth extension, the largest eigenvalue of the training loss Hessian (i.e. the sharpness) is largely independent of the width and depth of the network for a sustained period of training time. On the other hand, we show that under the neural tangent kernel (NTK) regime, the sharpness exhibits very different dynamics at different scales, thus preventing learning rate transfer. But what causes these differences in the sharpness dynamics? Through a connection between the spectra of the Hessian and the NTK matrix, we argue that the cause lies in the presence (for $\mu$P) or progressive absence (for the NTK regime) of feature learning, which results in a different evolution of the NTK, and thus of the sharpness. We corroborate our claims with a substantial suite of experiments, covering a wide range of datasets and architectures: from ResNets and Vision Transformers trained on benchmark vision datasets to Transformers-based language models trained on WikiText

Towards Training Without Depth Limits: Batch Normalization Without Gradient Explosion

Normalization layers are one of the key building blocks for deep neural networks. Several theoretical studies have shown that batch normalization improves the signal propagation, by avoiding the representations from becoming collinear across the layers. However, results on mean-field theory of batch normalization also conclude that this benefit comes at the expense of exploding gradients in depth. Motivated by these two aspects of batch normalization, in this study we pose the following question: "Can a batch-normalized network keep the optimal signal propagation properties, but avoid exploding gradients?" We answer this question in the affirmative by giving a particular construction of an Multi-Layer Perceptron (MLP) with linear activations and batch-normalization that provably has bounded gradients at any depth. Based on Weingarten calculus, we develop a rigorous and non-asymptotic theory for this constructed MLP that gives a precise characterization of forward signal propagation, while proving that gradients remain bounded for linearly independent input samples, which holds in most practical settings. Inspired by our theory, we also design an activation shaping scheme that empirically achieves the same properties for certain non-linear activations.