arXiv subjects

Martin Hernandez

Publications and source records attributed to Martin Hernandez.

3 recordsLinked to original sources

arXiv2026

Mean-Field PhiBE: Continuous-Time Mean-Field Reinforcement Learning from Discrete-Time Data

This paper develops a model-free framework for continuous-time mean-field control when the population evolves according to unknown controlled McKean--Vlasov dynamics and only discrete-time transition data are available. Model-based mean-field control requires the continuous-time drift and diffusion coefficients, which are not directly observed from fixed-step transitions, while a direct reduction to a discrete-time Bellman equation loses the continuous-time generator structure. To bridge these two viewpoints, we introduce a Mean-Field-PhiBE (MF-PhiBE), which incorporates discrete-time transition information into a continuous-time PDE on the Wasserstein space. The MF-PhiBE replaces the unknown infinitesimal drift and covariance in the policy-evaluation equation by one-step estimators computed from data, while preserving the generator structure of the McKean-Vlasov HJB equation. We also derive a policy-gradient theorem for entropy-regularized randomized feedback policies, expressing the actor direction through an action-wise infinitesimal advantage and the score of the policy. Combining these two ingredients yields a model-free actor-critic method. We prove a first-order consistency estimate showing that the value induced by an optimal MF-PhiBE policy approximates the optimal continuous-time value as the observation time step vanishes. For entropy-regularized LQR, we establish first-order policy convergence and second-order value convergence; under suitable conditions, the population-averaged feedback means coincide exactly. Numerical experiments on an LQR benchmark and a crowd-aversion problem illustrate the proposed framework.

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arXiv2026

Policy Gradient for Continuous-Time Mean-Field Control

This paper develops a policy gradient method for entropy-regularized mean-field control in the discounted infinite-horizon setting. We consider randomized feedback policies and a coupled representative-particle/population system, in which the representative state evolves jointly with a population law governed by a McKean--Vlasov equation. The resulting value function is therefore defined on the product space $\mathbb R^d \times \mathcal P_2(\mathbb R^d)$. A key distinction from existing policy gradient methods for mean-field control is that, after computing the value function under a fixed policy, our approach does not require solving an additional equation to obtain the policy gradient. Instead, we derive an explicit policy gradient formula directly in terms of the value function. The formulation is based on an instantaneous advantage function, which quantifies the gain of taking a given action relative to the current randomized policy. We establish a G\^ateaux policy-gradient formula, which gives the first-order variation of the objective along arbitrary policy perturbations, and then derive the corresponding ascent direction under finite-dimensional policy parametrization. The resulting formula leads to a model-based actor--critic scheme. The critic is obtained by solving the associated linear stationary Hamilton--Jacobi--Bellman equation for the value function, using cylindrical functions to represent dependence on the population law. The actor is then updated according to the derived policy-gradient formula. We further analyze the well-posedness of the PDE in a polynomial-growth function class. Finally, we illustrate the proposed method through numerical experiments on an LQR model and a crowd-motion problem.

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arXiv2023

Uniform Turnpike Property and Singular Limits

Motivated by singular limits for long-time optimal control problems, we investigate a class of parameter-dependent parabolic equations. First, we prove a turnpike result, uniform with respect to the parameters within a suitable regularity class and under appropriate bounds. The main ingredient of our proof is the justification of the uniform exponential stabilization of the corresponding Riccati equations, which is derived from the uniform null control properties of the model. Then, we focus on a heat equation with rapidly oscillating coefficients. In the one-dimensional setting, we obtain a uniform turnpike property with respect to the highly oscillatory heterogeneous medium. Afterward, we establish the homogenization of the turnpike property. Finally, our results are validated by numerical experiments.

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