Understanding the robustness difference between stochastic gradient descent and adaptive gradient methods

TL;DR

This paper investigates why stochastic gradient descent (SGD) yields more robust models than adaptive methods like Adam, revealing that SGD-trained models have smaller Lipschitz constants and are less sensitive to irrelevant frequencies in data.

Contribution

The study provides empirical and theoretical insights into the robustness differences between SGD and adaptive methods, highlighting the role of weight norms and Lipschitz constants.

Findings

SGD-trained models exhibit greater robustness to input perturbations.

Models trained with adaptive methods are sensitive to irrelevant frequencies.

Smaller weight norms correlate with improved robustness in linear models.

Abstract

Stochastic gradient descent (SGD) and adaptive gradient methods, such as Adam and RMSProp, have been widely used in training deep neural networks. We empirically show that while the difference between the standard generalization performance of models trained using these methods is small, those trained using SGD exhibit far greater robustness under input perturbations. Notably, our investigation demonstrates the presence of irrelevant frequencies in natural datasets, where alterations do not affect models' generalization performance. However, models trained with adaptive methods show sensitivity to these changes, suggesting that their use of irrelevant frequencies can lead to solutions sensitive to perturbations. To better understand this difference, we study the learning dynamics of gradient descent (GD) and sign gradient descent (signGD) on a synthetic dataset that mirrors natural signals. With a three-dimensional input space, the models optimized with GD and signGD have standard risks close to zero but vary in their adversarial risks. Our result shows that linear models' robustness to $\ell_2$-norm bounded changes is inversely proportional to the model parameters' weight norm: a smaller weight norm implies better robustness. In the context of deep learning, our experiments show that SGD-trained neural networks have smaller Lipschitz constants, explaining the better robustness to input perturbations than those trained with adaptive gradient methods.