Continuous-Time RL

Autonomous agents operating in continuous environments must decide not only what to do, but when to act. We introduce a lightweight adaptive temporal control system that learns the optimal interval between cognitive ticks from experience, replacing ad hoc biologically inspired timers with a principled learned policy. The policy state is augmented with a predictive hyperbolic spread signal (a "curvature signal" shorthand) derived from hyperbolic geometry: the mean pairwise Poincare distance among n sampled futures embedded in the Poincare ball. High spread indicates a branching, uncertain future and drives the agent to act sooner; low spread signals predictability and permits longer rest intervals. We further propose an interval-aware reward that explicitly penalises inefficiency relative to the chosen wait time, correcting a systematic credit-assignment failure of naive outcome-based rewards in timing problems. We additionally introduce a joint spatio-temporal embedding (ATCPG-ST) that concatenates independently normalised state and position projections in the Poincare ball; spatial trajectory divergence provides an independent timing signal unavailable to the state-only variant (ATCPG-SO). This extension raises mean hyperbolic spread (kappa) from 1.88 to 3.37 and yields a further 5.8 percent efficiency gain over the state-only baseline. Ablation experiments across five random seeds demonstrate that (i) learning is the dominant efficiency factor (54.8 percent over no-learning), (ii) hyperbolic spread provides significant complementary gain (26.2 percent over geometry-free control), (iii) the combined system achieves 22.8 percent efficiency over the fixed-interval baseline, and (iv) adding spatial position information to the spread embedding yields an additional 5.8 percent.

Multi-agent reinforcement learning (MARL) has made significant progress in recent years, but most algorithms still rely on a discrete-time Markov Decision Process (MDP) with fixed decision intervals. This formulation is often ill-suited for complex multi-agent dynamics, particularly in high-frequency or irregular time-interval settings, leading to degraded performance and motivating the development of continuous-time MARL (CT-MARL). Existing CT-MARL methods are mainly built on Hamilton-Jacobi-Bellman (HJB) equations. However, they rarely account for safety constraints such as collision penalties, since these introduce discontinuities that make HJB-based learning difficult. To address this challenge, we propose a continuous-time constrained MDP (CT-CMDP) formulation and a novel MARL framework that transforms discrete MDPs into CT-CMDPs via an epigraph-based reformulation. We then solve this by proposing a novel physics-informed neural network (PINN)-based actor-critic method that enables stable and efficient optimization in continuous time. We evaluate our approach on continuous-time safe multi-particle environments (MPE) and safe multi-agent MuJoCo benchmarks. Results demonstrate smoother value approximations, more stable training, and improved performance over safe MARL baselines, validating the effectiveness and robustness of our method.

Many real-world control problems, ranging from finance to robotics, evolve in continuous time with non-uniform, event-driven decisions. Standard discrete-time reinforcement learning (RL), based on fixed-step Bellman updates, struggles in this setting: as time gaps shrink, the $Q$-function collapses to the value function $V$, eliminating action ranking. Existing continuous-time methods reintroduce action information via an advantage-rate function $q$. However, they enforce optimality through complicated martingale losses or orthogonality constraints, which are sensitive to the choice of test processes. These approaches entangle $V$ and $q$ into a large, complex optimization problem that is difficult to train reliably. To address these limitations, we propose a novel decoupled continuous-time actor-critic algorithm with alternating updates: $q$ is learned from diffusion generators on $V$, and $V$ is updated via a Hamiltonian-based value flow that remains informative under infinitesimal time steps, where standard max/softmax backups fail. Theoretically, we prove rigorous convergence via new probabilistic arguments, sidestepping the challenge that generator-based Hamiltonians lack Bellman-style contraction under the sup-norm. Empirically, our method outperforms prior continuous-time and leading discrete-time baselines across challenging continuous-control benchmarks and a real-world trading task, achieving 21% profit over a single quarter$-$nearly doubling the second-best method.

Continuous-time stochastic processes underlie many natural and engineered systems. In healthcare, autonomous driving, and industrial control, direct interaction with the environment is often unsafe or impractical, motivating offline reinforcement learning from historical data. However, there is limited statistical understanding of the approximation errors inherent in learning policies from offline datasets. We address this by linking reinforcement learning to the Hamilton-Jacobi-Bellman equation and proposing an operator-theoretic algorithm based on a simple dynamic programming recursion. Specifically, we represent our world model in terms of the infinitesimal generator of controlled diffusion processes learned in a reproducing kernel Hilbert space. By integrating statistical learning methods and operator theory, we establish global convergence of the value function under sufficiently small generator-approximation error and derive explicit finite-sample guarantees with bounds tied directly to system properties such as smoothness and stability. Our theoretical and proof-of-concept results indicate that operator-based approaches may hold promise for solving offline reinforcement learning using continuous-time optimal control.