Multistep stochastic mirror descent for risk-averse convex stochastic programs based on extended polyhedral risk measures

TL;DR

This paper develops advanced stochastic mirror descent algorithms for risk-averse convex stochastic programs, providing tighter confidence intervals and faster solutions, especially when the objective is uniformly convex.

Contribution

It introduces a multistep extension of the Stochastic Mirror Descent algorithm and derives less conservative, computationally efficient confidence intervals for risk-averse stochastic programs.

Findings

Confidence intervals are less conservative and quicker to compute.

Multistep SMD achieves solutions faster than nonmultistep methods.

Confidence intervals outperform asymptotic ones with smaller sample sizes.

Abstract

We consider risk-averse convex stochastic programs expressed in terms of extended polyhedral risk measures. We derive computable confidence intervals on the optimal value of such stochastic programs using the Robust Stochastic Approximation and the Stochastic Mirror Descent (SMD) algorithms. When the objective functions are uniformly convex, we also propose a multistep extension of the Stochastic Mirror Descent algorithm and obtain confidence intervals on both the optimal values and optimal solutions. Numerical simulations show that our confidence intervals are much less conservative and are quicker to compute than previously obtained confidence intervals for SMD and that the multistep Stochastic Mirror Descent algorithm can obtain a good approximate solution much quicker than its nonmultistep counterpart. Our confidence intervals are also more reliable than asymptotic confidence intervals when the sample size is not much larger than the problem size.