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In this paper we study a mean field control problem in which particles are absorbed when they reach the boundary of a smooth domain. The value of the N-particle problem is described by a hierarchy of Hamilton-Jacobi equations which are…

Analysis of PDEs · Mathematics 2026-05-14 Pierre Cardaliaguet , Joe Jackson , Panagiotis E. Souganidis

We consider reinforcement learning (RL) in continuous time and study the problem of achieving the best trade-off between exploration of a black box environment and exploitation of current knowledge. We propose an entropy-regularized reward…

Optimization and Control · Mathematics 2019-02-14 Haoran Wang , Thaleia Zariphopoulou , Xunyu Zhou

Fitted $Q$-iteration (FQI) and soft FQI are widely used value-based methods for offline reinforcement learning, but their standard stability guarantees often depend on Bellman completeness, a strong closure condition that can fail under…

Machine Learning · Statistics 2026-05-11 Lars van der Laan , Nathan Kallus

We study the policy evaluation problem in multi-agent reinforcement learning, modeled by a Markov decision process. In this problem, the agents operate in a common environment under a fixed control policy, working together to discover the…

Optimization and Control · Mathematics 2020-01-13 Thinh T. Doan , Siva Theja Maguluri , Justin Romberg

A key problem in reinforcement learning for control with general function approximators (such as deep neural networks and other nonlinear functions) is that, for many algorithms employed in practice, updates to the policy or $Q$-function…

Machine Learning · Computer Science 2016-03-01 Joshua Achiam

Q-learning with neural network function approximation (neural Q-learning for short) is among the most prevalent deep reinforcement learning algorithms. Despite its empirical success, the non-asymptotic convergence rate of neural Q-learning…

Machine Learning · Computer Science 2020-03-05 Pan Xu , Quanquan Gu

Mean field games (MFG) and mean field control problems (MFC) are frameworks to study Nash equilibria or social optima in games with a continuum of agents. These problems can be used to approximate competitive or cooperative games with a…

Optimization and Control · Mathematics 2021-06-28 Andrea Angiuli , Jean-Pierre Fouque , Mathieu Lauriere

We propose a mathematical framework to explain implicit regularization from early stopping during the training of overparametrized neural networks. In the mean-field limit, the parameter distribution evolves according to a gradient flow on…

Optimization and Control · Mathematics 2026-03-24 Beatrice Acciaio , Jakob Heiss , Gudmund Pammer , Qinxin Yan

We analyze the consequences that the so-called turnpike property has on the long-time behavior of the value function corresponding to a finite-dimensional linear-quadratic optimal control problem with general terminal cost and constrained…

Analysis of PDEs · Mathematics 2021-11-23 Carlos Esteve , Hicham Kouhkouh , Dario Pighin , Enrique Zuazua

Following Kolokoltsov's work [1], we present an extension of mean-field control theory in quantum framework. In particular such an extension is done naturally by considering the Belavkin quantum filtering and control theory in a mean-field…

Optimization and Control · Mathematics 2023-06-27 Sofiane Chalal , Nina H. Amini , Gaoyue Guo

We exploit the separation of the filtering and control aspects of quantum feedback control to consider the optimal control as a classical stochastic problem on the space of quantum states. We derive the corresponding Hamilton-Jacobi-Bellman…

Quantum Physics · Physics 2007-05-23 J. Gough , V. P. Belavkin , O. G. Smolyanov

Soft Q-learning is a variation of Q-learning designed to solve entropy regularized Markov decision problems where an agent aims to maximize the entropy regularized value function. Despite its empirical success, there have been limited…

Machine Learning · Computer Science 2024-09-06 Narim Jeong , Donghwan Lee

We replace a Hamiltonian with a modular Hamiltonian in the spectral form factor and the level spacing distribution function. This study establishes a connection between quantities within Quantum Entanglement and Quantum Chaos. To have a…

High Energy Physics - Theory · Physics 2022-11-15 Chen-Te Ma , Chih-Hung Wu

The optimal \(H_{\infty}\) control problem over an infinite time horizon, which incorporates a performance function with a discount factor \(e^{-\alpha t}\) (\(\alpha > 0\)), is important in various fields. Solving this optimal…

Optimization and Control · Mathematics 2024-10-04 Guoyuan Chen , Yi Wang , Qinglong Zhou

We introduce \textbf{QuantFPFlow}, a reinforcement learning framework that integrates quantum amplitude estimation into the Fokker--Planck~(FP) formulation of stochastic policy optimisation. Classical continuous-space RL agents must…

Machine Learning · Computer Science 2026-05-19 Abraham Itzhak Weinberg

This work proposes a novel numerical scheme for solving the high-dimensional Hamilton-Jacobi-Bellman equation with a functional hierarchical tensor ansatz. We consider the setting of stochastic control, whereby one applies control to a…

Numerical Analysis · Mathematics 2025-07-01 Xun Tang , Nan Sheng , Lexing Ying

We develop the theory of linear-quadratic (LQ) mean field games (MFGs) in Hilbert spaces with common noise modeled by an infinite-dimensional Wiener process that affects the dynamics of all agents. In the presence of common noise, the…

Optimization and Control · Mathematics 2026-05-28 Hanchao Liu , Dena Firoozi

Online off-policy reinforcement learning (RL) is shaped by two coupled choices: the policy class and the update rule. Gaussian policies are fast and have tractable entropy, but struggle with multimodal action distributions. Generative…

Machine Learning · Computer Science 2026-05-22 Zeyuan Wang , Da Li , Yulin Chen , Yuehu Gong , Yanming Guo , Ye Shi , Liang Bai , Tianyuan Yu , Yanwei Fu

Independent learners are agents that employ single-agent algorithms in multi-agent systems, intentionally ignoring the effect of other strategic agents. This paper studies mean-field games from a decentralized learning perspective, with two…

Computer Science and Game Theory · Computer Science 2025-02-04 Bora Yongacoglu , Gürdal Arslan , Serdar Yüksel

We develop an exhaustive study of Markov decision process (MDP) under mean field interaction both on states and actions in the presence of common noise, and when optimization is performed over open-loop controls on infinite horizon. Such…

Optimization and Control · Mathematics 2021-09-10 Médéric Motte , Huyên Pham