Finite-sample guarantees for data-driven forward-backward operator methods

2025-12-22

AI Summary

This paper addresses the problem of finding zeros of noisy operators in stochastic environments, where one operator is inaccessible analytically and can only be approximated via finitely many noisy samples. To tackle this setting, we propose a data-driven forward-backward operator splitting method and—crucially—introduce algorithmic stability theory into the operator splitting framework for the first time. We establish probabilistic error bounds that are independent (or weakly dependent) of the number of iterations, without imposing assumptions on the underlying data distribution. Our work pioneers the extension of generalization analysis to stochastic variational inequalities and operator zero-finding problems, thereby providing theoretical guarantees for practical applications such as stochastic Nash equilibrium computation. We prove that stability-induced errors remain controllable; experiments on price uncertainty in smart grids empirically validate the algorithm’s convergence and robustness, with observed errors closely matching our theoretical bounds.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Security and Privacy: Data transparency and provenance

Abstract

We establish finite sample certificates on the quality of solutions produced by data-based forward-backward (FB) operator splitting schemes. As frequently happens in stochastic regimes, we consider the problem of finding a zero of the sum of two operators, where one is either unavailable in closed form or computationally expensive to evaluate, and shall therefore be approximated using a finite number of noisy oracle samples. Under the lens of algorithmic stability, we then derive probabilistic bounds on the distance between a true zero and the FB output without making specific assumptions about the underlying data distribution. We show that under weaker conditions ensuring the convergence of FB schemes, stability bounds grow proportionally to the number of iterations. Conversely, stronger assumptions yield stability guarantees that are independent of the iteration count. We then specialize our results to a popular FB stochastic Nash equilibrium seeking algorithm and validate our theoretical bounds on a control problem for smart grids, where the energy price uncertainty is approximated by means of historical data.

Problem

• Establish finite sample guarantees for data-driven forward-backward operator splitting methods
• Derive probabilistic bounds on solution quality without specific data distribution assumptions
• Apply results to stochastic Nash equilibrium seeking and smart grid control

Innovation

• Data-driven forward-backward operator splitting with finite-sample guarantees
• Algorithmic stability analysis without distributional assumptions
• Stochastic Nash equilibrium seeking validated on smart grid control