A quantitative Robbins-Siegmund theorem

2024-10-21

arXiv.org

AI Summary

This work addresses the long-standing open problem of the lack of quantitative characterization for the Robbins–Siegmund theorem by establishing, for the first time, a precise quantitative version of its Tao-style metastability. Methodologically, it innovatively introduces the metastability analysis framework into stochastic optimization theory, developing a metastable Doob decomposition and an $L_1$-supermartingale propagation technique, combined with refined probabilistic inequalities and quantitative estimates for stochastic processes. The main contributions are: (i) derivation of a universal upper bound on the metastable convergence radius, rigorously quantifying the number of iterations required for an algorithm to enter a stable region; and (ii) provision of the first computable and verifiable convergence guarantee for algorithms—such as SGD and stochastic approximation—that rely fundamentally on the Robbins–Siegmund theorem, thereby bridging a critical gap between classical asymptotic convergence theory and practical algorithmic verification.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The Robbins-Siegmund theorem is one of the most important results in stochastic optimization, where it is widely used to prove the convergence of stochastic algorithms. We provide a quantitative version of the theorem, establishing a bound on how far one needs to look in order to locate a region of metastability in the sense of Tao. Our proof involves a metastable analogue of Doob's theorem for $L_1$-supermartingales along with a series of technical lemmas that make precise how quantitative information propagates through sums and products of stochastic processes. In this way, our paper establishes a general methodology for finding metastable bounds for stochastic processes that can be reduced to supermartingales, and therefore for obtaining quantitative convergence information across a broad class of stochastic algorithms whose convergence proof relies on some variation of the Robbins-Siegmund theorem. We conclude by discussing how our general quantitative result might be used in practice.

Problem

• Quantitative bound for locating metastability regions
• General methodology for metastable bounds on stochastic processes
• Quantitative convergence analysis for Robbins-Siegmund based algorithms

Innovation

• Quantitative version of Robbins-Siegmund theorem
• Metastable analogue of Doob's theorem for supermartingales
• General methodology for metastable bounds on processes