Strong Duality in Risk-Constrained Nonconvex Functional Programming

TL;DR

This paper proves strong duality for a broad class of risk-constrained nonconvex optimization problems, including neural network policies and various risk measures, with implications for multiple application domains.

Contribution

It establishes strong duality results for risk-constrained nonconvex functional programming under new assumptions, extending prior work to complex policy spaces and diverse risk measures.

Findings

Strong duality holds for decomposable and nondecomposable policy spaces.

Results apply to popular risk measures like CVaR and MAD.

The proof introduces a novel risk conjugate duality technique.

Abstract

We show that a wide class of risk-constrained nonconvex functional optimization problems exhibit strong duality, regardless of nonconvexity. We develop two novel results under distinct sets of assumptions, establishing strong duality over both decomposable policy spaces (matching and extending prior work in the risk neutral case), and nondecomposable policy spaces with structure (e.g., continuity or smoothness), including certain universal finite-dimensional (fixed depth/width) neural network parametrizations as special cases (improving established results in the risk-neutral setting as well). We consider constraints featuring convex and positively homogeneous risk measures with bounded risk envelopes, generalizing expectations. Popular risk measures supported within our setting include the conditional value-at-risk (CVaR), the (even non-monotone) mean-absolute deviation (MAD), certain distributionally robust representations and more generally all real-valued coherent risk measures on the space $L_1$. We further discuss various generalizations of our base model, extensions for risk measures supported on $L_{p>1}$, implications in the context of mean-risk tradeoff models, as well as applications in wireless systems resource allocation, and supervised constrained learning. Our core proof technique appears to be new and relies on risk conjugate duality in tandem with J. J. Uhl's weak extension of A. A. Lyapunov's convexity theorem for vector measures taking values in infinite-dimensional Banach spaces.