On the asymptotic behaviour of stochastic processes, with applications to supermartingale convergence, Dvoretzky's approximation theorem, and stochastic quasi-Fej'er monotonicity

2025-04-17

AI Summary

研究随机过程的渐近行为，提出一种新方法，通过松弛超鞅条件提供定量收敛速率，应用于随机逼近中的多个定理，并评估其适用性。

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Applications
• Machine Learning: Other Foundations of Machine Learning

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: Models for Web evolution
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience

Abstract

We prove a novel and general result on the asymptotic behavior of stochastic processes which conform to a certain relaxed supermartingale condition. Our result provides quantitative information in the form of an explicit and effective construction of a rate of convergence for this process, both in mean and almost surely, that is moreover highly uniform in the sense that it only depends on very few data of the surrounding objects involved in the iteration. We then apply this result to derive new quantitative versions of well-known concepts and theorems from stochastic approximation, in particular providing effective rates for a variant of the Robbins-Siegmund theorem, Dvoretzky's convergence theorem, as well as the convergence of stochastic quasi-Fej'er monotone sequences, the latter of which formulated in a novel and highly general metric context. We utilize the classic and widely studied Robbins-Monro procedure as a template to evaluate our quantitative results and their applicability in greater detail. We conclude by illustrating the breadth of potential further applications with a brief discussion on a variety of other well-known iterative procedures from stochastic approximation, covering a range of different applied scenarios to which our methods can be immediately applied. Throughout, we isolate and discuss special cases of our results which even allow for the construction of fast, and in particular linear, rates.

Problem

• Study asymptotic behavior of stochastic processes under relaxed supermartingale conditions
• Derive quantitative convergence rates for stochastic approximation theorems
• Apply results to Robbins-Monro procedure and other iterative methods

Innovation

• General asymptotic behavior of stochastic processes
• Quantitative convergence rates construction
• Applications to stochastic approximation theorems