Statistical Optimality of Stochastic Gradient Descent on Hard Learning Problems through Multiple Passes

TL;DR

This paper demonstrates that for complex, high-dimensional least-squares problems, multiple passes of stochastic gradient descent (SGD) are statistically optimal, contrasting with the single-pass optimality suggested by previous theory for simpler problems.

Contribution

It establishes that multiple passes of SGD are necessary and optimal for hard problems, especially in high-dimensional settings, and characterizes the optimal number of passes based on sample size.

Findings

Multiple passes of SGD improve predictive performance on hard problems.

Optimal number of passes increases with sample size in complex models.

Experimental validation on synthetic and benchmark datasets supports theoretical results.

Abstract

We consider stochastic gradient descent (SGD) for least-squares regression with potentially several passes over the data. While several passes have been widely reported to perform practically better in terms of predictive performance on unseen data, the existing theoretical analysis of SGD suggests that a single pass is statistically optimal. While this is true for low-dimensional easy problems, we show that for hard problems, multiple passes lead to statistically optimal predictions while single pass does not; we also show that in these hard models, the optimal number of passes over the data increases with sample size. In order to define the notion of hardness and show that our predictive performances are optimal, we consider potentially infinite-dimensional models and notions typically associated to kernel methods, namely, the decay of eigenvalues of the covariance matrix of the features and the complexity of the optimal predictor as measured through the covariance matrix. We illustrate our results on synthetic experiments with non-linear kernel methods and on a classical benchmark with a linear model.