Simran Kaur

Simran Kaur, Jeremy Cohen, Zachary C. Lipton

The mechanisms by which certain training interventions, such as increasing learning rates and applying batch normalization, improve the generalization of deep networks remains a mystery. Prior works have speculated that "flatter" solutions generalize better than "sharper" solutions to unseen data, motivating several metrics for measuring flatness (particularly $λ_{max}$, the largest eigenvalue of the Hessian of the loss); and algorithms, such as Sharpness-Aware Minimization (SAM) [1], that directly optimize for flatness. Other works question the link between $λ_{max}$ and generalization. In this paper, we present findings that call $λ_{max}$'s influence on generalization further into question. We show that: (1) while larger learning rates reduce $λ_{max}$ for all batch sizes, generalization benefits sometimes vanish at larger batch sizes; (2) by scaling batch size and learning rate simultaneously, we can change $λ_{max}$ without affecting generalization; (3) while SAM produces smaller $λ_{max}$ for all batch sizes, generalization benefits (also) vanish with larger batch sizes; (4) for dropout, excessively high dropout probabilities can degrade generalization, even as they promote smaller $λ_{max}$; and (5) while batch-normalization does not consistently produce smaller $λ_{max}$, it nevertheless confers generalization benefits. While our experiments affirm the generalization benefits of large learning rates and SAM for minibatch SGD, the GD-SGD discrepancy demonstrates limits to $λ_{max}$'s ability to explain generalization in neural networks.

Zachary Novack, Simran Kaur, Tanya Marwah et al.

A number of competing hypotheses have been proposed to explain why small-batch Stochastic Gradient Descent (SGD)leads to improved generalization over the full-batch regime, with recent work crediting the implicit regularization of various quantities throughout training. However, to date, empirical evidence assessing the explanatory power of these hypotheses is lacking. In this paper, we conduct an extensive empirical evaluation, focusing on the ability of various theorized mechanisms to close the small-to-large batch generalization gap. Additionally, we characterize how the quantities that SGD has been claimed to (implicitly) regularize change over the course of training. By using micro-batches, i.e. disjoint smaller subsets of each mini-batch, we empirically show that explicitly penalizing the gradient norm or the Fisher Information Matrix trace, averaged over micro-batches, in the large-batch regime recovers small-batch SGD generalization, whereas Jacobian-based regularizations fail to do so. This generalization performance is shown to often be correlated with how well the regularized model's gradient norms resemble those of small-batch SGD. We additionally show that this behavior breaks down as the micro-batch size approaches the batch size. Finally, we note that in this line of inquiry, positive experimental findings on CIFAR10 are often reversed on other datasets like CIFAR100, highlighting the need to test hypotheses on a wider collection of datasets.