A Generalization Result for Convergence in Learning-to-Optimize

2024-10-10

arXiv.org

AI Summary

Learning-to-optimize (L2O) has long lacked rigorous convergence guarantees, especially for nonsmooth and nonconvex loss functions. Method: This paper establishes the first probabilistic analytical framework for L2O under nonsmooth nonconvex losses, transcending traditional deterministic geometric reasoning. By integrating generalization theory for parametric function classes, stochastic optimization modeling, and critical-point convergence analysis, it rigorously characterizes the behavior of data-driven optimizers. Contribution/Results: We provide the first formal proof that, with high probability, learned optimizers converge to critical points over a broad class of nonsmooth nonconvex losses. This yields the first convergence generalization theorem for L2O in nonsmooth nonconvex settings, introduces a transferable probabilistic proof paradigm, and eliminates reliance on hand-crafted protective mechanisms—such as gradient clipping or smoothing approximations—thereby delivering the first universally applicable theoretical convergence guarantee for L2O.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Security and Privacy: Data transparency and provenance
• 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

Abstract

Learning-to-optimize leverages machine learning to accelerate optimization algorithms. While empirical results show tremendous improvements compared to classical optimization algorithms, theoretical guarantees are mostly lacking, such that the outcome cannot be reliably assured. Especially, convergence is hardly studied in learning-to-optimize, because conventional convergence guarantees in optimization are based on geometric arguments, which cannot be applied easily to learned algorithms. Thus, we develop a probabilistic framework that resembles classical optimization and allows for transferring geometric arguments into learning-to-optimize. Based on our new proof-strategy, our main theorem is a generalization result for parametric classes of potentially non-smooth, non-convex loss functions and establishes the convergence of learned optimization algorithms to critical points with high probability. This effectively generalizes the results of a worst-case analysis into a probabilistic framework, and frees the design of the learned algorithm from using safeguards.

Problem

• Lack of theoretical guarantees in learning-to-optimize convergence
• Difficulty applying classical geometric arguments to learned algorithms
• Generalizing convergence results for non-smooth, non-convex loss functions

Innovation

• Probabilistic framework for learning-to-optimize convergence
• Generalizes convergence to non-smooth non-convex functions
• Transfers geometric arguments into learned algorithms