Convergence and Regret of the Policy Gradient for Multi-Armed Bandits in Diffusion Environment

This paper studies the policy gradient update for a multi-arm bandit problem in diffusion environment that is described by a stochastic differential equation (SDE) under the continuous-time reinforcement learning framework by Wang et al. (2020), Jia and Zhou (2022b). With the logit parameterization for the stochastic policy, we show that it converges almost surely to the optimal arm under an arbitrary constant learning rate. Furthermore, we derive the non-asymptotic regret upper bound when the constant learning rate is below a time-invariant threshold; and the regret bound has order $O(\log T)$. We improve the analysis in Lattimore (2026a) for the same SDE by constructing a novel Lyapunov function and demonstrate the transparency of analyzing policy gradient using the tools in SDEs. In addition, the same Lyapunov function is also helpful in analyzing the discrete-time policy gradient algorithm.

A Zeroth-Order Deep Learning Method for Fully Nonlinear Parabolic Partial Differential Equations with Unknown Coefficients

High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging. Existing deep learning solvers often rely on repeated automatic differentiation to evaluate differential operators, which can cause instability and amplify derivative errors in high dimensions, while probabilistic methods based on stochastic representations require explicit knowledge of the data-generating dynamics and therefore do not apply to black-box environments. We introduce two types of simulators as data-generating mechanisms, and take a ``representing-then-learning" approach that learns the solutions and their derivatives under settings where the underlying PDE operators are accessible only through simulations and pointwise evaluations. Our representation of derivatives relies on the zeroth-order derivative (ZOD) estimators derived from perturbed Monte Carlo trajectories. This fully model-free approach generates targets for the gradient and Hessian networks using only function evaluations. We provide a statistical learning analysis of the proposed approach, including a bias--variance tradeoff for ZODs. Assuming a standard contraction property of the underlying operator, we establish a non-asymptotic error bound that decomposes the total error into discretization error, approximation error, statistical error, and ZOD bias. Crucially, we derive the sample complexity of the learned representations in (weighted) Sobolev space, characterizing the error up to second-order derivatives. Numerical experiments illustrate the competitive performance of the method in moderate and high dimensions.

Accuracy of Discretely Sampled Stochastic Policies in Continuous-time Reinforcement Learning

Stochastic policies are widely used in continuous-time reinforcement learning algorithms. However, executing a stochastic policy and evaluating its performance in a continuous-time environment remain open challenges. This work introduces and rigorously analyzes a policy execution framework that samples actions from a stochastic policy at discrete time points and implements them as piecewise constant controls. We prove that as the sampling mesh size tends to zero, the controlled state process converges weakly to the dynamics with coefficients aggregated according to the stochastic policy. We explicitly quantify the convergence rate based on the regularity of the coefficients and establish an optimal first-order convergence rate for sufficiently regular coefficients. Additionally, we show that the same convergence rates hold with high probability concerning the sampling noise, and further establish a $1/2$-order almost sure convergence when the volatility is not controlled. Building on these results, we analyze the bias and variance of various policy evaluation and policy gradient estimators based on discrete-time observations. Our results provide theoretical justification for the exploratory stochastic control framework in [H. Wang, T. Zariphopoulou, and X.Y. Zhou, J. Mach. Learn. Res., 21 (2020), pp. 1-34].