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

TL;DR

This paper provides finite-sample probabilistic guarantees for data-driven forward-backward operator splitting methods, addressing the challenge of noisy operator evaluations in stochastic settings without assuming specific data distributions.

Contribution

It introduces stability-based probabilistic bounds for FB schemes with noisy operators, applicable under weaker convergence conditions and validated on a smart grid control problem.

Findings

Stability bounds grow proportionally with iterations under weaker assumptions.

Stronger assumptions lead to iteration-independent stability guarantees.

Theoretical bounds are validated through a smart grid control case study.

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.