We establish strong duality relations for functional two-step compositional
risk-constrained learning problems with multiple nonconvex loss functions
and/or learning constraints, regardless of nonconvexity and under a minimal set
of technical assumptions. Our results in particular imply zero duality gaps
within the class of problems under study, both extending and improving on the
state of the art in (risk-neutral) constrained learning. More specifically, we
consider risk objectives/constraints which involve real-valued convex and
positively homogeneous risk measures admitting dual representations with
bounded risk envelopes, generalizing expectations and including popular
examples, such as the conditional value-at-risk (CVaR), the mean-absolute
deviation (MAD), and more generally all real-valued coherent risk measures on
integrable losses as special cases. Our results are based on recent advances in
risk-constrained nonconvex programming in infinite dimensions, which rely on a
remarkable new application of J. J. Uhl's convexity theorem, which is an
extension of A. A. Lyapunov's convexity theorem for general, infinite
dimensional Banach spaces. By specializing to the risk-neutral setting, we
demonstrate, for the first time, that constrained classification and regression
can be treated under a unifying lens, while dispensing certain restrictive
assumptions enforced in the current literature, yielding a new state-of-the-art
strong duality framework for nonconvex constrained learning.

我们建立了对于功能性的两步组合风险受限学习问题的强对偶关系，涵盖多个非凸损失函数和 / 或学习约束的情况，不受非凸性和最小技术假设的限制。我们的结果特别意味着在所研究的问题类中存在零对偶间隙，扩展和改进了（风险中性）受限学习的最新进展。