Last Iterate Risk Bounds of SGD with Decaying Stepsize for Overparameterized Linear Regression

TL;DR

This paper analyzes the risk bounds of stochastic gradient descent with decaying stepsize in overparameterized linear regression, providing nearly matching bounds and demonstrating the benefits of geometric decay in the last iterate.

Contribution

It offers a problem-dependent analysis of last iterate SGD risk bounds with decaying stepsize in overparameterized linear regression, filling a gap in theoretical understanding.

Findings

Nearly matching upper and lower bounds on excess risk for geometric decay

Lower bounds for polynomial decay stepsize

Geometric decay stepsize offers advantages in risk bounds

Abstract

Stochastic gradient descent (SGD) has been shown to generalize well in many deep learning applications. In practice, one often runs SGD with a geometrically decaying stepsize, i.e., a constant initial stepsize followed by multiple geometric stepsize decay, and uses the last iterate as the output. This kind of SGD is known to be nearly minimax optimal for classical finite-dimensional linear regression problems (Ge et al., 2019). However, a sharp analysis for the last iterate of SGD in the overparameterized setting is still open. In this paper, we provide a problem-dependent analysis on the last iterate risk bounds of SGD with decaying stepsize, for (overparameterized) linear regression problems. In particular, for last iterate SGD with (tail) geometrically decaying stepsize, we prove nearly matching upper and lower bounds on the excess risk. Moreover, we provide an excess risk lower bound for last iterate SGD with polynomially decaying stepsize and demonstrate the advantage of geometrically decaying stepsize in an instance-wise manner, which complements the minimax rate comparison made in prior works.