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