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Mirror descent for stochastic control problems with measure-valued controls

Author

Listed:
  • Kerimkulov, Bekzhan
  • Šiška, David
  • Szpruch, Łukasz
  • Zhang, Yufei

Abstract

This paper studies the convergence of the mirror descent algorithm for finite horizon stochastic control problems with measure-valued control processes. The control objective involves a convex regularisation function, denoted as h, with regularisation strength determined by the weight τ≥0. The setting covers regularised relaxed control problems. Under suitable conditions, we establish the relative smoothness and convexity of the control objective with respect to the Bregman divergence of h, and prove linear convergence of the algorithm for τ=0 and exponential convergence for τ>0. The results apply to common regularisers including relative entropy, χ2-divergence, and entropic Wasserstein costs. This validates recent reinforcement learning heuristics that adding regularisation accelerates the convergence of gradient methods. The proof exploits careful regularity estimates of backward stochastic differential equations in the bounded mean oscillation norm.

Suggested Citation

  • Kerimkulov, Bekzhan & Šiška, David & Szpruch, Łukasz & Zhang, Yufei, 2025. "Mirror descent for stochastic control problems with measure-valued controls," Stochastic Processes and their Applications, Elsevier, vol. 190(C).
    • Handle: RePEc:eee:spapps:v:190:y:2025:i:c:s0304414925002091
    DOI: 10.1016/j.spa.2025.104765
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    References listed on IDEAS

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    1. Haihao Lu & Robert M. Freund & Yurii Nesterov, 2018. "Relatively smooth convex optimization by first-order methods, and applications," LIDAM Reprints CORE 2965, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    3. Heinz H. Bauschke & Jérôme Bolte & Marc Teboulle, 2017. "A Descent Lemma Beyond Lipschitz Gradient Continuity: First-Order Methods Revisited and Applications," Mathematics of Operations Research, INFORMS, vol. 42(2), pages 330-348, May.
    4. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
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