Optimal and Adaptive Monteiro-Svaiter Acceleration

TL;DR

This paper introduces a modified Monteiro-Svaiter acceleration framework that reduces computational costs and adapts to problem parameters, achieving optimal complexity for convex optimization with Lipschitz derivatives.

Contribution

It presents a new variant of MS acceleration that eliminates expensive implicit equations and offers an adaptive subproblem solver suitable for first- and second-order methods.

Findings

Improves complexity bounds for convex optimization with Lipschitz pth derivatives.

Outperforms previous second-order momentum methods on logistic regression.

First-order adaptive solver performs comparably to L-BFGS.

Abstract

We develop a variant of the Monteiro-Svaiter (MS) acceleration framework that removes the need to solve an expensive implicit equation at every iteration. Consequently, for any $p\ge 2$ we improve the complexity of convex optimization with Lipschitz $p$th derivative by a logarithmic factor, matching a lower bound. We also introduce an MS subproblem solver that requires no knowledge of problem parameters, and implement it as either a second- or first-order method by solving linear systems or applying MinRes, respectively. On logistic regression our method outperforms previous second-order momentum methods, but under-performs Newton's method; simply iterating our first-order adaptive subproblem solver performs comparably to L-BFGS.