Big-Step-Little-Step: Efficient Gradient Methods for Objectives with Multiple Scales

TL;DR

This paper introduces new gradient methods tailored for multiscale, ill-conditioned optimization problems, achieving near-optimal complexity that significantly outperforms traditional accelerated gradient methods, especially for decomposable functions.

Contribution

The paper presents a novel recursive 'Big-Step-Little-Step' approach for efficiently solving multiscale convex optimization problems with unknown decompositions, improving complexity bounds.

Findings

Nearly optimal complexity bounds for multiscale problems

Efficient algorithms for stochastic quadratic variants

Improved understanding of convex optimization complexity

Abstract

We provide new gradient-based methods for efficiently solving a broad class of ill-conditioned optimization problems. We consider the problem of minimizing a function $f : \mathbb{R}^d \rightarrow \mathbb{R}$ which is implicitly decomposable as the sum of $m$ unknown non-interacting smooth, strongly convex functions and provide a method which solves this problem with a number of gradient evaluations that scales (up to logarithmic factors) as the product of the square-root of the condition numbers of the components. This complexity bound (which we prove is nearly optimal) can improve almost exponentially on that of accelerated gradient methods, which grow as the square root of the condition number of $f$. Additionally, we provide efficient methods for solving stochastic, quadratic variants of this multiscale optimization problem. Rather than learn the decomposition of $f$ (which would be prohibitively expensive), our methods apply a clean recursive "Big-Step-Little-Step" interleaving of standard methods. The resulting algorithms use $\tilde{\mathcal{O}}(d m)$ space, are numerically stable, and open the door to a more fine-grained understanding of the complexity of convex optimization beyond condition number.