Composing Optimized Stepsize Schedules for Gradient Descent

TL;DR

This paper develops a comprehensive theory for composing and optimizing stepsize schedules in gradient descent, leading to new schedules that outperform previous methods and potentially achieve minimax optimality.

Contribution

It introduces a general framework for composing stepsize schedules, constructs highly optimized sequences, and extends recent advances to broader settings.

Findings

Constructed optimized stepsize schedules generalizing exponential spacing.

Achieved improved convergence rates matching or surpassing minimax schedules.

Extended dynamic gradient norm minimizing schedules to objective gap minimization.

Abstract

Recent works by Altschuler and Parrilo and the authors have shown that it is possible to accelerate the convergence of gradient descent on smooth convex functions, even without momentum, just by picking special stepsizes. In this paper, we provide a general theory for composing stepsize schedules capturing all recent advances in this area and more. We propose three notions of ``composable'' stepsize schedules with elementary associated composition operations for combining them. From these operations, in addition to recovering recent works, we construct three highly optimized sequences of stepsize schedules. We first construct optimized stepsize schedules of every length generalizing the exponentially spaced silver stepsizes. We then construct highly optimized stepsizes schedules for minimizing final objective gap or gradient norm, improving on prior rates by constants and, more importantly, matching or beating the numerically computed minimax optimal schedules. We conjecture these schedules are in fact minimax (information theoretic) optimal. Several novel tertiary results follow from our theory including recovery of the recent dynamic gradient norm minimizing short stepsizes and extending them to objective gap minimization.