Stochastic Approximation Methods for Distortion Risk Measure Optimization

• URL: http://arxiv.org/abs/2510.04563v1
• Date: Mon, 06 Oct 2025 07:59:09 GMT
• Title: Stochastic Approximation Methods for Distortion Risk Measure Optimization
• Authors: Jinyang Jiang, Bernd Heidergott, Jiaqiao Hu, Yijie Peng,
• Abstract summary: This paper proposes descent algorithms for DRM optimization based on two dual representations.<n>The DM-form employs a three-timescale algorithm to track quantiles, compute their gradients, and update decision variables.<n>The QF-form provides a simpler two-timescale approach that avoids the need for complex quantile gradient estimation.
• Score: 2.97238992700289
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Distortion Risk Measures (DRMs) capture risk preferences in decision-making and serve as general criteria for managing uncertainty. This paper proposes gradient descent algorithms for DRM optimization based on two dual representations: the Distortion-Measure (DM) form and Quantile-Function (QF) form. The DM-form employs a three-timescale algorithm to track quantiles, compute their gradients, and update decision variables, utilizing the Generalized Likelihood Ratio and kernel-based density estimation. The QF-form provides a simpler two-timescale approach that avoids the need for complex quantile gradient estimation. A hybrid form integrates both approaches, applying the DM-form for robust performance around distortion function jumps and the QF-form for efficiency in smooth regions. Proofs of strong convergence and convergence rates for the proposed algorithms are provided. In particular, the DM-form achieves an optimal rate of $O(k^{-4/7})$, while the QF-form attains a faster rate of $O(k^{-2/3})$. Numerical experiments confirm their effectiveness and demonstrate substantial improvements over baselines in robust portfolio selection tasks. The method's scalability is further illustrated through integration into deep reinforcement learning. Specifically, a DRM-based Proximal Policy Optimization algorithm is developed and applied to multi-echelon dynamic inventory management, showcasing its practical applicability.

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