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

arXiv: https://arxiv.org/abs/2512.19172 · HTML: https://mlsanity.com/paper/2512.19172/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 57 references that are in the map.
- First posted on arXiv 22 Dec 2025.

## Builds on (most cited first, 22 of 22)

- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [Stability of Stochastic Gradient Descent on Nonsmooth Convex Losses](https://mlsanity.com/paper/2006.06914/) (2020)
- [Sharper bounds for uniformly stable algorithms](https://mlsanity.com/paper/1910.07833/) (2019)
- [Data-Dependent Stability of Stochastic Gradient Descent](https://mlsanity.com/paper/1703.01678/) (2017)
- [Variance-Reduced Methods for Machine Learning](https://mlsanity.com/paper/2010.00892/) (2020)
- [Multi-Agent Connected Autonomous Driving using Deep Reinforcement Learning](https://mlsanity.com/paper/1911.04175/) (2019)
- [Statistical Learning Theory for Control: A Finite Sample Perspective](https://mlsanity.com/paper/2209.05423/) (2022)
- [Stochastic Variance Reduction for Variational Inequality Methods](https://mlsanity.com/paper/2102.08352/) (2021)
- [Train simultaneously, generalize better: Stability of gradient-based minimax learners](https://mlsanity.com/paper/2010.12561/) (2020)
- [Toward Better Generalization Bounds with Locally Elastic Stability](https://mlsanity.com/paper/2010.13988/) (2020)
- [Stability and Generalization of Stochastic Gradient Methods for Minimax Problems](https://mlsanity.com/paper/2105.03793/) (2021)
- [Extragradient method with variance reduction for stochastic variational inequalities](https://mlsanity.com/paper/1703.00260/) (2017)
- [The Scenario Approach for Stochastic Model Predictive Control with Bounds on Closed-Loop Constraint Violations](https://mlsanity.com/paper/1307.5640/) (2013)
- [What is a Good Metric to Study Generalization of Minimax Learners?](https://mlsanity.com/paper/2206.04502/) (2022)
- [McDiarmid-Type Inequalities for Graph-Dependent Variables and Stability Bounds](https://mlsanity.com/paper/1909.02330/) (2019)
- [Stochastic generalized Nash equilibrium seeking in merely monotone games](https://mlsanity.com/paper/2002.08318/) (2020)
- [Distributed Forward-Backward algorithms for stochastic generalized Nash equilibrium seeking](https://mlsanity.com/paper/1912.04165/) (2019)
- [Probabilistic feasibility guarantees for solution sets to uncertain variational inequalities](https://mlsanity.com/paper/2005.09420/) (2020)
- [A stochastic generalized Nash equilibrium model for platforms competition in the ride-hail market](https://mlsanity.com/paper/2203.15412/) (2022)
- [Projected-gradient algorithms for generalized equilibrium seeking in Aggregative Games are preconditioned Forward-Backward methods](https://mlsanity.com/paper/1803.10441/) (2018)
- [On the robustness of equilibria in generalized aggregative games](https://mlsanity.com/paper/2005.09408/) (2020)
- [Learning Variational Inequalities from Data: Fast Generalization Rates under Strong Monotonicity](https://mlsanity.com/paper/2410.20649/) (2024)

## Built on by (most cited first, 1 of 1)

- [Generalization as a robust performance property of learning-enabled dynamical systems](https://mlsanity.com/paper/2608.30431/) (2026)

## Abstract (arXiv)

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.