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

Filippo Fabiani, Barbara Franci

TL;DR

The paper addresses the challenge of obtaining finite-sample, distribution-free guarantees for data-driven forward-backward operator methods when one operator must be approximated from finite data. By formulating a tailored loss and applying algorithmic stability theory, it derives probabilistic bounds that bound the distance to a true zero of A+B, with stability depending on the monotonicity properties of the operators and the dataset size. It highlights two regimes: (i) iteration-independent bounds under strong monotonicity, and (ii) iteration-dependent bounds under cocoercivity, and specializes the results to a stochastic Nash equilibrium seeking algorithm, validated on smart-grid energy-price data. The work provides practically applicable certificates that hold regardless of the data distribution and demonstrates their relevance through numerical experiments on EV charging coordination and a quadratic academic example. Overall, it offers robust, finite-sample guarantees for data-driven operator-splitting methods in stochastic optimization and game-theoretic settings, with clear implications for reliable decision-making under uncertainty.

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.

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

TL;DR

The paper addresses the challenge of obtaining finite-sample, distribution-free guarantees for data-driven forward-backward operator methods when one operator must be approximated from finite data. By formulating a tailored loss and applying algorithmic stability theory, it derives probabilistic bounds that bound the distance to a true zero of A+B, with stability depending on the monotonicity properties of the operators and the dataset size. It highlights two regimes: (i) iteration-independent bounds under strong monotonicity, and (ii) iteration-dependent bounds under cocoercivity, and specializes the results to a stochastic Nash equilibrium seeking algorithm, validated on smart-grid energy-price data. The work provides practically applicable certificates that hold regardless of the data distribution and demonstrates their relevance through numerical experiments on EV charging coordination and a quadratic academic example. Overall, it offers robust, finite-sample guarantees for data-driven operator-splitting methods in stochastic optimization and game-theoretic settings, with clear implications for reliable decision-making under uncertainty.

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.
Paper Structure (18 sections, 8 theorems, 39 equations, 4 figures, 2 tables, 3 algorithms)

This paper contains 18 sections, 8 theorems, 39 equations, 4 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Data-driven approximation methods and related work
  3. Summary of contribution
  4. Paper organization and notation
  5. Mathematical background
  6. Problem statement
  7. Preliminaries on algorithmic stability
  8. Generalization bounds for the data-driven algorithm
  9. Stability bounds independent on the number of iterations
  10. Weaker assumptions yield $K$-dependent stability bounds
  11. Discussion on our results
  12. Finite-sample guarantees for equilibrium seeking in stochastic Nash games
  13. A stochastic Nash equilibrium problem
  14. Data-driven certificates for $\varepsilon$-
  15. Numerical experiments
  16. ...and 3 more sections

Key Result

Lemma 2.4

Let $\{A_s\}_{s\ge0}$ be an algorithm with uniform stability $\beta=\beta(s)$ a loss function $0\le\ell(H_s,\xi)\le\bar{\ell}$, $\bar{\ell}\ge0$, for all $\xi\in\Xi$ and $\mathcal{D}_s$. Then, for any $s\ge1$ and $\delta\in(0,1)$, the following bound hold true: $\square$

Figures (4)

  • Figure 1: Left y-axis: Box plots capturing the relative approximation error $\|\boldsymbol{x}^{1001}-\boldsymbol{x}^\star\|$ produced by $K=1000$ iterations of Algorithm \ref{['alg:proximal']} over $50$ trials considering different dataset size. Right y-axis: The resulting averaged bounds $\varepsilon$ (green down-pointing triangles) offered by Theorem \ref{['th:sample_complexity_snep']} with $\delta=0.05$.
  • Figure 2: Top: Aggregate demand $\sigma(\boldsymbol{x}^\star)$ (blue line) and the approximated one related to $\boldsymbol{x}^{1001}$ obtained by running Algorithm \ref{['alg:proximal']} with $s=3000$, averaged over $50$ trials (red line). Bottom: Mean value (yellow line) and standard deviation (yellow shaded area) of the energy price $\xi$ over $\mathcal{T}$.
  • Figure 3: Box plots capturing the relative approximation error $\|\boldsymbol{x}^{K+1}-\boldsymbol{x}^\star\|$ produced by different run $K$ of Algorithm \ref{['alg:proximal']} over $50$ trials with fixed number of samples. The green down-pointing triangles denote the resulting averaged bounds $\varepsilon$ offered by Theorem \ref{['th:sample_complexity_snep']} with $\delta=0.05$.
  • Figure 4: Aggregate demand $\sigma(\boldsymbol{x}^\star)$ (blue line) and its approximation through $\boldsymbol{x}^{5001}$, obtained by running Algorithm \ref{['alg:proximal']} with $s=3000$. The solid red line denotes the mean of $\sigma(\boldsymbol{x}^{5001})$, while the shaded red area the associated standard deviation.

Theorems & Definitions (20)

  • Definition 2.3: bousquet2002stability Uniform stability
  • Lemma 2.4: bousquet2002stability Exponential bound with uniform stability
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Lemma 3.6
  • proof
  • Lemma 3.7
  • proof
  • ...and 10 more