Finite-sample guarantees for data-driven forward-backward operator methods
Provides theoretical guarantees for stochastic optimization algorithms in data-driven settings, with applications to equilibrium seeking in smart grids.
The paper establishes finite-sample guarantees for data-driven forward-backward operator splitting methods, deriving probabilistic bounds on solution quality without distributional assumptions. It shows stability bounds scale with iteration count under weak conditions but become iteration-independent under stronger assumptions, validated on a smart grid control problem.
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.