Asymptotic Variance in the Central Limit Theorem for Multilevel Markovian Stochastic Approximation

2025-04-09

AI Summary

This work establishes a central limit theorem (CLT) for the asymptotic variance of multilevel Markov chain stochastic approximation (MRA) algorithms in Bayesian inverse problems governed by differential equations. The core challenge lies in quantifying estimation variance when finite-dimensional parameter inference relies on numerical PDE/ODE solvers whose discretization errors obscure variance characterization. We derive, for the first time, a quantitative upper bound linking the asymptotic variance to numerical discretization error. Our approach integrates multilevel PDE discretizations, Markov chain stochastic approximation, and asymptotic statistical analysis, with adaptive parameter selection via marginal likelihood optimization. The derived bound provides rigorous theoretical guidance for tuning key simulation parameters—including level count, sample size per level, and mesh resolution—in multilevel stochastic gradient methods. This result bridges a critical theoretical gap in the efficient implementation of multilevel stochastic optimization and substantially enhances both computational efficiency and statistical reliability in Bayesian inference for inverse problems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Bayesian Learning
• Search and Optimization: Mixed Discrete/Continuous Search

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
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

In this note we consider the finite-dimensional parameter estimation problem associated to inverse problems. In such scenarios, one seeks to maximize the marginal likelihood associated to a Bayesian model. This latter model is connected to the solution of partial or ordinary differential equation. As such, there are two primary difficulties in maximizing the marginal likelihood (i) that the solution of differential equation is not always analytically tractable and (ii) neither is the marginal likelihood. Typically (i) is dealt with using a numerical solution of the differential equation, leading to a numerical bias and (ii) has been well studied in the literature using, for instance, Markovian stochastic approximation. It is well-known that to reduce the computational effort to obtain the maximal value of the parameter, one can use a hierarchy of solutions of the differential equation and combine with stochastic gradient methods. Several approaches do exactly this. In this paper we consider the asymptotic variance in the central limit theorem, associated to known estimates and find bounds on the asymptotic variance in terms of the precision of the solution of the differential equation. The significance of these bounds are the that they provide missing theoretical guidelines on how to set simulation parameters; that is, these appear to be the first mathematical results which help to run the methods efficiently in practice.

Problem

• Estimating parameters in inverse problems with differential equations
• Maximizing marginal likelihood with numerical and stochastic challenges
• Bounding asymptotic variance to guide efficient simulation parameters

Innovation

• Uses multilevel Markovian stochastic approximation
• Bounds asymptotic variance via differential equation precision
• Provides theoretical guidelines for simulation parameters