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 $ϵ\in [0,1)$, the MESSAGEp algorithm achieves a squared-${\mathcal{L}}_2$ solution suboptimality rate of the order of $\mathcal{O}(n^{-(1-ϵ)\mathrm{/}2})$ iterations, where, for $ϵ>0$, pathwise convergence is simultaneously guaranteed. This result establishes a rate of order arbitrarily close to $\mathcal{O}(n^{-1\mathrm{/}2})$, while ensuring strongly stable pathwise operation. For $p\equiv 1$, the rate order improves to $\mathcal{O}(n^{-2\mathrm{/}3})$, 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 $ϵ\in [0,1)$, the MESSAGEp algorithm with iterate smoothing achieves an ${\mathcal{L}}_1$ 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\mathrm{.}Thisresultprovidesmaximalratesof${\cal O}(n^{-1/4})$,if$p\equiv1$,and${\cal O}(n^{-1/8})$,if$p>1$, matching the state of the art, as well.