Understanding the Generalization of Adam in Learning Neural Networks with Proper Regularization

TL;DR

This paper explains why Adam often generalizes worse than gradient descent in deep learning, showing that in nonconvex settings, Adam and GD can find different solutions with different test errors, unlike in convex cases.

Contribution

The paper provides a theoretical analysis demonstrating that Adam's inferior generalization is due to nonconvex landscape effects, contrasting with convex scenarios where solutions coincide.

Findings

Adam and GD converge to different solutions in nonconvex neural network training.

In convex settings with regularization, Adam and GD find the same solution.

The generalization gap is fundamentally linked to the nonconvexity of deep learning landscapes.

Abstract

Adaptive gradient methods such as Adam have gained increasing popularity in deep learning optimization. However, it has been observed that compared with (stochastic) gradient descent, Adam can converge to a different solution with a significantly worse test error in many deep learning applications such as image classification, even with a fine-tuned regularization. In this paper, we provide a theoretical explanation for this phenomenon: we show that in the nonconvex setting of learning over-parameterized two-layer convolutional neural networks starting from the same random initialization, for a class of data distributions (inspired from image data), Adam and gradient descent (GD) can converge to different global solutions of the training objective with provably different generalization errors, even with weight decay regularization. In contrast, we show that if the training objective is convex, and the weight decay regularization is employed, any optimization algorithms including Adam and GD will converge to the same solution if the training is successful. This suggests that the inferior generalization performance of Adam is fundamentally tied to the nonconvex landscape of deep learning optimization.