Constrained and composite optimization via adaptive sampling methods

The motivation for this paper stems from the desire to develop an adaptive sampling method for solving constrained optimization problems, in which the objective function is stochastic and the constraints are deterministic. The method proposed in this paper is a proximal gradient method that can also be applied to the composite optimization problem min f(x) + h(x), where f is stochastic and h is convex (but not necessarily differentiable). Adaptive sampling methods employ a mechanism for gradually improving the quality of the gradient approximation so as to keep computational cost to a minimum. The mechanism commonly employed in unconstrained optimization is no longer reliable in the constrained or composite optimization settings, because it is based on pointwise decisions that cannot correctly predict the quality of the proximal gradient step. The method proposed in this paper measures the result of a complete step to determine if the gradient approximation is accurate enough; otherwise, a more accurate gradient is generated and a new step is computed. Convergence results are established both for strongly convex and general convex f. Numerical experiments are presented to illustrate the practical behavior of the method.