Data-Driven Performance Guarantees for Classical and Learned Optimizers
This work provides performance guarantees for optimization algorithms, which is important for researchers and practitioners in fields like signal processing and control, though it is incremental as it builds on existing statistical learning theory.
The paper tackles the problem of analyzing optimization algorithm performance by introducing a data-driven approach using generalization guarantees from statistical learning theory, applied to both classical and learned optimizers, resulting in tighter bounds than worst-case guarantees for classical ones and outperforming empirical outcomes for learned ones.
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.