Bayesian stochastic gradient descent for stochastic optimization with streaming input data

We consider stochastic optimization under distributional uncertainty, where the unknown distributional parameter is estimated from streaming data that arrive sequentially over time. Moreover, data may depend on the decision at the time when they are generated. For both decision -independent and decision -dependent uncertainties, we propose an approach to jointly estimate the distributional parameter via Bayesian posterior distribution and update the decision by applying stochastic gradient descent (SGD) on the Bayesian average of the objective function. Our approach converges asymptotically over time and achieves the convergence rates of classical SGD in the decision -independent case. We demonstrate the empirical performance of our approach on both synthetic test problems and a classical newsvendor problem.