Boundary-aware Reinforcement Learning for Hypercube State Spaces via Deterministic Policy Gradient

We present a novel theoretical framework for deep reinforcement learning (RL) in continuous environments by modeling the problem as a continuous-time stochastic process, drawing on insights from stochastic control. Building on previous work, we introduce a viable model of actor-critic algorithm that incorporates both exploration and stochastic transitions. For single-hidden-layer neural networks, we show that the state of the environment can be formulated as a two time scale process: the environment time and the gradient time. Within this formulation, we characterize how the time-dependent random variables that represent the environment's state and estimate of the cumulative discounted return evolve over gradient steps in the infinite width limit of two-layer networks. Using the theory of stochastic differential equations, we derive, for the first time in continuous RL, an equation describing the infinitesimal change in the state distribution at each gradient step, under a vanishingly small learning rate. Overall, our work provides a novel nonparametric formulation for studying overparametrized neural actor-critic algorithms. We empirically corroborate our theoretical result using a toy continuous control task.

The framework of robust Markov decision processes (RMDPs) allows the design of reinforcement learning agents that satisfy performance guarantees under worst-case transition dynamics. Traditional RMDPs consider discrete-time dynamics and recently, sample-efficient policy gradient algorithms have been considered in this context. This paper investigates policy gradient algorithms within a continuous-time RMDP framework. Policy gradients and adversarial gradients are derived using pathwise and adjoint-based formulas for stochastic and ordinary differential equations. We propose double-loop optimisers to obtain linear convergence in the oracle-based setting and an $\tilde{\mathcal{O}}(\frac{1}{ε^2})$ sample complexity in the sample-based setting in an analysis which also derives novel tools for the framework of undiscounted total cost MDPs. Additionally, we propose mean-field optimisers as distributional optimisers with an $\tilde{\mathcal{O}}(\frac{1}{K})$ oracle-based convergence rate and an $\tilde{\mathcal{O}}(\frac{N^2}ε)$ sample complexity under $N$-particle approximation. The effectiveness of continuous-time policy gradient algorithms is confirmed for both optimisers on continuous-time RMDPs with neural ordinary differential equation dynamics.

We analyze a neural semi-discrete method for high-dimensional first-order Hamilton-Jacobi-Bellman (HJB) equations with known or learned dynamics. Centered differences and an artificial viscosity $Nh=O(h)$ define a monotone operator evaluated through $2d+1$ shifted network queries; policy iteration solves the resulting Bellman equation without a tensor grid. At fixed $h$, the sharp componentwise condition $\max_i|f_i|\le2N$ turns every frozen-policy operator into a nearest-neighbor Markov-chain generator with a policy-independent total jump rate. Uniformization gives whole-space well-posedness for measurable feedbacks, an explicit Poisson-tail bound on the numerical domain of dependence, and boundary-free localization. The representation also yields a posteriori policy-evaluation bounds that account for residual and learned-model errors. A greedy-gap analysis controls inexact policy iteration at fixed $h$; a separate consistency estimate connects the semi-discrete equation to the continuous HJB equation. Experiments reproduce the extremal tail, show rates consistent with $O(\sqrt h)$ and nearly $h$-independent exact-policy-iteration decay, and assess empirical estimator effectivity. A nonsmooth example shows that the continuous residual can miss a non-viscosity solution, whereas the shifted residual detects the defect. Further tests provide a structured interval-verified certificate calibration, an early-budget benefit of policy freezing for bang-bang control, and learned-dynamics diagnostics. A structured nonlinear problem with active compact-control constraints is tested against a manufactured semi-discrete reference through $d=1024$.