Recursive Optimization of Convex Risk Measures: Mean-Semideviation Models

We develop recursive, data-driven, stochastic subgradient methods for optimizing a new, versatile, and application-driven class of convex risk measures, termed here as mean-semideviations, strictly generalizing the well-known and popular mean-upper-semideviation. We introduce the MESSAGEp algorithm, which is an efficient compositional subgradient procedure for iteratively solving convex mean-semideviation risk-averse problems to optimality. We analyze the asymptotic behavior of the MESSAGEp algorithm under a flexible and structure-exploiting set of problem assumptions. In particular: 1) Under appropriate stepsize rules, we establish pathwise convergence of the MESSAGEp algorithm in a strong technical sense, confirming its asymptotic consistency. 2) Assuming a strongly convex cost, we show that, for fixed semideviation order $p>1$ and for $\epsilon\in\left[0,1\right)$, the MESSAGEp algorithm achieves a squared-${\cal L}_{2}$ solution suboptimality rate of the order of ${\cal O}(n^{-\left(1-\epsilon\right)/2})$ iterations, where, for $\epsilon>0$, pathwise convergence is simultaneously guaranteed. This result establishes a rate of order arbitrarily close to ${\cal O}(n^{-1/2})$, while ensuring strongly stable pathwise operation. For $p\equiv1$, the rate order improves to ${\cal O}(n^{-2/3})$, which also suffices for pathwise convergence, and matches previous results. 3) Likewise, in the general case of a convex cost, we show that, for any $\epsilon\in\left[0,1\right)$, the MESSAGEp algorithm with iterate smoothing achieves an ${\cal L}_{1}$ objective suboptimality rate of the order of ${\cal O}(n^{-\left(1-\epsilon\right)/\left(4\bf{1}_{\left\{ p>1\right\} }+4\right)})$ iterations. This result provides maximal rates of ${\cal O}(n^{-1/4})$, if $p\equiv1$, and ${\cal O}(n^{-1/8})$, if $p>1$, matching the state of the art, as well.