Finite-sample guarantees for data-driven operator splitting methods via martingale inequalities

RESUMO

Operator splitting methods are a fundamental class of algorithms for solving structured monotone inclusion problems arising in optimization, control, and game theory. We consider the case, common in stochastic regimes, where the forward evaluation of one of the constituent operators is either unavailable in closed form or computationally expensive to evaluate, and is therefore approximated using a finite number of noisy oracle samples. We establish distribution-free, finite-sample certificates for the quality of the output produced by data-driven Davis-Yin splitting algorithms. Unlike previous works, our analysis directly controls the residual error via martingale inequalities instead of relying on algorithmic stability arguments for a tailored surrogate loss, yielding the first a priori certificates whose statistical excess provably vanishes with the sample size. We further show that, under linear convergence of the Davis-Yin splitting algorithm, the dependence of our bounds on the iteration count improves from linear growth to exponential decay. We validate our theoretical results on a stochastic portfolio optimization problem with uncertain asset returns.