Projected Stochastic Gradient Descent with Decision Dependent Distributions: Extended Version

Abstract

Online feedback optimization (OFO) leverages real-time output measurements to optimize the operation of networked systems without requiring full knowledge of system dynamics or disturbances. We develop an OFO approach for constrained stochastic optimization problems in which the distribution of the system's random parameters shifts in response to the control actions. We propose a projected primal-dual algorithm where the true dual constraint sets are replaced by surrogate sets. Our main result is an upper bound on the mean-square tracking error, which decomposes into four interpretable terms reflecting: (i) the stochasticity of the problem, (ii) output measurement errors, (iii) time-variability of the problem, and (iv) the mismatch between surrogate and true dual constraint sets. The theory is illustrated in a numerical experiment for power grids with price-responsive assets.