Data-Driven Performance Guarantees for Classical and Learned Optimizers

TL;DR

This paper presents a novel data-driven framework that applies statistical learning theory to provide generalization guarantees for both classical and learned optimization algorithms across various problem domains.

Contribution

It introduces a unified approach using convergence bounds and PAC-Bayes theory to analyze and guarantee optimizer performance, including training learned optimizers via PAC-Bayes bounds.

Findings

Classical optimizer bounds are significantly tighter than worst-case guarantees.

Learned optimizer bounds outperform empirical results of non-learned optimizers.

Framework applies to diverse fields like signal processing, control, and meta-learning.

Abstract

We introduce a data-driven approach to analyze the performance of continuous optimization algorithms using generalization guarantees from statistical learning theory. We study classical and learned optimizers to solve families of parametric optimization problems. We build generalization guarantees for classical optimizers, using a sample convergence bound, and for learned optimizers, using the Probably Approximately Correct (PAC)-Bayes framework. To train learned optimizers, we use a gradient-based algorithm to directly minimize the PAC-Bayes upper bound. Numerical experiments in signal processing, control, and meta-learning showcase the ability of our framework to provide strong generalization guarantees for both classical and learned optimizers given a fixed budget of iterations. For classical optimizers, our bounds are much tighter than those that worst-case guarantees provide. For learned optimizers, our bounds outperform the empirical outcomes observed in their non-learned counterparts.