Painless Stochastic Gradient: Interpolation, Line-Search, and Convergence Rates

TL;DR

This paper introduces line-search techniques for stochastic gradient descent (SGD) that automatically determine optimal step-sizes, achieving fast convergence rates in interpolation settings for convex, non-convex, and saddle-point problems, with practical benefits demonstrated on classification tasks.

Contribution

It proposes stochastic Armijo line-search methods for SGD that adaptively set step-sizes, ensuring convergence rates comparable to full-batch gradient descent under interpolation conditions.

Findings

SGD with Armijo line-search attains deterministic convergence rates for convex functions.

The methods are robust to hyper-parameter choices and improve practical convergence.

Experiments show faster convergence and better generalization on classification tasks.

Abstract

Recent works have shown that stochastic gradient descent (SGD) achieves the fast convergence rates of full-batch gradient descent for over-parameterized models satisfying certain interpolation conditions. However, the step-size used in these works depends on unknown quantities and SGD's practical performance heavily relies on the choice of this step-size. We propose to use line-search techniques to automatically set the step-size when training models that can interpolate the data. In the interpolation setting, we prove that SGD with a stochastic variant of the classic Armijo line-search attains the deterministic convergence rates for both convex and strongly-convex functions. Under additional assumptions, SGD with Armijo line-search is shown to achieve fast convergence for non-convex functions. Furthermore, we show that stochastic extra-gradient with a Lipschitz line-search attains linear convergence for an important class of non-convex functions and saddle-point problems satisfying interpolation. To improve the proposed methods' practical performance, we give heuristics to use larger step-sizes and acceleration. We compare the proposed algorithms against numerous optimization methods on standard classification tasks using both kernel methods and deep networks. The proposed methods result in competitive performance across all models and datasets, while being robust to the precise choices of hyper-parameters. For multi-class classification using deep networks, SGD with Armijo line-search results in both faster convergence and better generalization.