We propose a mesh-free policy iteration framework based on physics-informed neural networks (PINNs) for solving entropy-regularized stochastic control problems. The method iteratively alternates between soft policy evaluation and improvement using automatic differentiation and neural approximation, without relying on spatial discretization. We present a detailed error analysis that decomposes the total approximation error into three sources: iteration error, policy network error, and PDE residual error. The proposed algorithm is validated with a range of challenging control tasks, including high-dimensional linear-quadratic regulation in 5D and 10D, as well as nonlinear systems such as pendulum and cartpole problems. Numerical results confirm the scalability, accuracy, and robustness of our approach across both linear and nonlinear benchmarks.

In online learning problems, exploiting low variance plays an important role in obtaining tight performance guarantees yet is challenging because variances are often not known a priori. Recently, considerable progress has been made by Zhang et al. (2021) where they obtain a variance-adaptive regret bound for linear bandits without knowledge of the variances and a horizon-free regret bound for linear mixture Markov decision processes (MDPs). In this paper, we present novel analyses that improve their regret bounds significantly. For linear bandits, we achieve $\tilde O(\min\{d\sqrt{K}, d^{1.5}\sqrt{\sum_{k=1}^K \sigma_k^2}\} + d^2)$ where $d$ is the dimension of the features, $K$ is the time horizon, and $\sigma_k^2$ is the noise variance at time step $k$, and $\tilde O$ ignores polylogarithmic dependence, which is a factor of $d^3$ improvement. For linear mixture MDPs with the assumption of maximum cumulative reward in an episode being in $[0,1]$, we achieve a horizon-free regret bound of $\tilde O(d \sqrt{K} + d^2)$ where $d$ is the number of base models and $K$ is the number of episodes. This is a factor of $d^{3.5}$ improvement in the leading term and $d^7$ in the lower order term. Our analysis critically relies on a novel peeling-based regret analysis that leverages the elliptical potential `count' lemma.