Gradient Methods with Online Scaling

TL;DR

This paper presents a framework that adaptively scales gradients using online learning to accelerate convergence in gradient-based optimization, achieving improved complexity bounds and superlinear convergence.

Contribution

It introduces a novel online learning-based gradient scaling framework that provably accelerates convergence and improves complexity bounds over previous methods.

Findings

Achieves $O( ext{condition number} imes ext{log}(1/\varepsilon))$ complexity for strongly convex problems.

Demonstrates superlinear convergence on convex quadratics.

Shows hypergradient descent improves convergence over standard gradient descent.

Abstract

We introduce a framework to accelerate the convergence of gradient-based methods with online learning. The framework learns to scale the gradient at each iteration through an online learning algorithm and provably accelerates gradient-based methods asymptotically. In contrast with previous literature, where convergence is established based on worst-case analysis, our framework provides a strong convergence guarantee with respect to the optimal scaling matrix for the iteration trajectory. For smooth strongly convex optimization, our results provide an $O(\kappa^\star \log(1/\varepsilon)$) complexity result, where $\kappa^\star$ is the condition number achievable by the optimal preconditioner, improving on the previous $O(\sqrt{n}\kappa^\star \log(1/\varepsilon))$ result. In particular, a variant of our method achieves superlinear convergence on convex quadratics. For smooth convex optimization, we show for the first time that the widely-used hypergradient descent heuristic improves on the convergence of gradient descent.