A Stochastic Primal-Dual Method for Optimization with Conditional Value at Risk Constraints

We study a first-order primal-dual subgradient method to optimize risk-constrained risk-penalized optimization problems, where risk is modeled via the popular conditional value at (CVaR) measure. The algorithm processes independent and identically distributed samples from underlying uncertainty in an online fashion, produces $\eta/\sqrt{K}$-approximately feasible optimal point within $K$ iterations with constant step-size, $\eta$ increases tunable risk-parameters of CVaR. find optimized step sizes using our bounds precisely characterize computational cost aversion as revealed by growth $\eta$. Our proposed makes simple modification typical stochastic algorithm. With this mild change, analysis surprisingly obviates need for priori or complex adaptive bounding schemes dual variables assumed many prior works. also draw interesting parallels sample complexity that chance-constrained programs derived literature very different solution architecture.