Data-Driven Performance Guarantees for Classical and Learned Optimizers

2024-04-22

arXiv.org

AI Summary

This work addresses the family of parametric optimization problems and proposes the first unified, data-driven framework for analyzing the generalization performance of both classical and learned optimizers. Methodologically: (1) it introduces PAC-Bayes theory to the analysis of learned optimizers, deriving verifiable, high-probability generalization upper bounds; (2) it establishes performance bounds for classical optimizers based on empirical convergence rates; and (3) it pioneers a learning paradigm that directly minimizes the PAC-Bayes bound during training. Evaluated on signal processing, control, and meta-learning tasks, the derived bounds are significantly tighter than conventional worst-case guarantees. Moreover, the theoretical generalization guarantees for learned optimizers consistently exceed the empirical performance of their non-learned baselines—thereby unifying theoretical rigor with practical efficacy.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization

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.

Problem

• Analyzing performance of classical and learned optimization algorithms
• Providing generalization guarantees using statistical learning theory
• Developing tighter performance bounds than worst-case guarantees

Innovation

• Data-driven approach analyzes optimization algorithm performance
• Generalization guarantees for classical optimizers via sample convergence
• PAC-Bayes framework trains learned optimizers with gradient minimization