Online Projected Gradient Descent for Stochastic Optimization with Decision-Dependent Distributions

Abstract

This paper investigates the problem of tracking solutions of stochastic optimization problems with time-varying costs and decision-dependent distributions. In this context, the paper focuses on the online stochastic gradient descent method, and establishes its convergence to the sequence of optimizers (within a bounded error) in expectation and in high probability.In particular, high-probability convergence results are derived by modeling the gradient error as a sub-Weibull random variable. The theoretical findings are validated via numerical simulations in the context of charging optimization of a fleet of electric vehicles.

Stochastic Saddle Point Problems with Decision-Dependent Distributions

“…For these reasons, we introduce an equilibrium problem associated with (1) for which solutions will be the saddle points of the stationary problem that they induce. These can be seen as the counterparts of the so-called performatively stable points in [18,42,56] in our stochastic minimax setup (1). We first introduce the equilibrium problem in the static (or time invariant) setting.…”

“…When the projection onto the convex set X t ∩ {x ∈ R n : g(x) ≤ 0} exists in closed form or is cheap to compute, projected gradient methods have been proposed for computing optimizers of (7) [18,42,56]. When the projection is computationally heavy, typical approaches for stationary problems include primaldual methods and interior point methods [40].…”

“…An extension of the decision dependent framework was proposed in [13] as a game between the institution and an adversary that chooses a distribution dependent on the last available classifier. In [56], first moment and high probability tracking analysis are provided under a sub-Weibull gradient error for a time-varying optimization problem.…”

“…Convergence rates of Lasso estimates using Sub-Weibull covariates and additive noise assumptions are provided in [30]. High probability bounds for the tracking error of time-varying minimization problems have been established using sub-Weibull distributions [5,56].…”

“…Furthermore, one may make assumptions of convexity and concavity on φ, but this does not guarantee that the same property hold for Φ [18,42]. For these reasons, and in the spirit of [18,42,56], in the following we turn the attention to a class of solutions that are saddle points for the stationary problem that they induce.…”