Semi-Markov Decision Process

Hierarchical reinforcement learning uses temporally extended subtasks for exploration, yet committing to their execution can restrict both deployment and policy learning. We identify and separate the resulting execution and policy suboptimality. Task and execution trees distinguish reward objectives from policy choices and decision interruption. A Unified Value Function for HRL and a four-stage Generalized Hierarchical Bellman Equation then support a common analysis of both losses. Under bounded rewards and uniform termination, we establish hierarchical policy and execution improvement results. With the remaining node policies fixed, task-subtree compatibility and node-policy optimality under the original execution mode establish when Markov execution is optimal. The resulting decomposition leads to independent execution choices for behavior, targets, and deployment. We instantiate this principle through execution improvement and one-stage or two-stage policy improvement at arbitrary hierarchy depth. Option-based and goal-conditioned experiments demonstrate complementary gains from changing execution and changing the learning target. Controlled stochastic environments show how these gains depend on stochastic transition strength and spatial structure. This framework makes execution design an explicit component of hierarchical policy optimization.

Coupled-dynamics environments expose the one-step outcomes that would follow from several possible counterfactual actions under a common realization of exogenous randomness. The ordinary Markov decision process formalism allows one to reason about the marginal law of each action but discards dependence across these counterfactual outcomes. The Joint Markov decision process (JMDP) formalism preserves that dependence. Prior work established the formalism and solved the fixed-policy joint moment evaluation problem in JMDPs. This paper develops optimal-control methods. We define a nonparametric distributional Bellman optimality operator for JMDPs, and prove that when the induced marginal MDP has a unique optimal policy, its iterates converge in Wasserstein distance to the optimal joint return law. For the first two moments, we establish convergence under a weaker condition that permits several mean-optimal actions as long as their tie resolutions share a second-moment fixed point. We also derive sampled targets for neural approximation.

In sequential decision making, an agent typically observes its environment and acts at every timestep. However, such active participation may not always be necessary; tasks such as brewing coffee include periods that are served equally well by letting the environment evolve without constant monitoring and control. During such periods, the agent could simply wait to conserve its resources, or redirect its attention to another task. We capitalize on these opportunities by training a "waiting policy" that decides where and how long to wait. This involves forgoing sensing to commit to a wait action, representing a deliberate pause for a set number of timesteps. We formalize "learning to wait" as minimizing the frequency of sensing and decision making without sacrificing task performance (e.g., the total amount of time to complete a task). To train a waiting policy, we propose an approach that employs reinforcement learning with lexicographically ordered objectives. In experiments across 4 discrete-state household tasks and 3 continuous-state environments, we show that our approach successfully learns waiting behaviors, and can adapt pre-trained policies to wait where appropriate. While different tasks permit different amounts of waiting without sacrificing task performance, our approach consistently finds solutions with significant waiting, sometimes waiting for over 50 percent of the task duration.

Dynamic medical treatment requires deciding treatment intensity and intervention timing, while patient states evolve continuously and adverse events may occur between clinical interactions. Most existing treatment learning methods assume fixed schedules or enforce safety only at discrete decision points. We propose Interaction-Limited Safe Continuous-Time Reinforcement Learning, a framework that jointly optimizes treatment administration and clinical interaction timing under trajectory-level safety constraints. Our key idea is to reformulate the continuous time treatment problem as an option-based semi-Markov decision process, where each option specifies a continuous-time treatment policy and its duration. We develop a safety-tightening mechanism showing that suitably constructed constraints at interaction times guarantee safety over the full continuous-time trajectory with high probability. We further establish finite-sample guarantees for policy learning from logged treatment trajectories and introduce a practical data-driven conservative surrogate. Experiments show that the proposed adaptive interaction-timing mechanism improves both safety and treatment effectiveness over equidistant interaction schemes across different safe policy optimization methods.

We study workflow learning in a setting where specialized agents hand off control through a shared artifact, each agent observes only a local function of that artifact and its own private state, and no centralized learner accesses joint trajectories -- the operating regime of multi-agent LLM pipelines that span organizational, vendor, or trust boundaries. We formalize this regime as an interface-constrained semi-Markov decision process (IC-SMDP), whose decision epochs occur at handoff times, and design IC-$Q$, an asynchronous decentralized $Q$-learning algorithm in which cross-agent coordination at every handoff is exactly one scalar. Our main result is a finite-sample bound for neural IC-$Q$ that decomposes into three independently controllable error sources: neural function-approximation error, interface representation gap, and a mixing-time residual, under the random option-duration discount. Establishing this bound requires lifting the approximate information state (AIS) framework from single-agent primitive-step MDPs to multi-agent SMDPs and controlling Markovian noise under random duration, neither of which has been done in prior work. To our knowledge this is the first finite-sample guarantee for neural $Q$-learning under decentralized partial observability. Four experiments: a controlled synthetic IC-SMDP that validates the bound term-by-term, multi-LLM mathematical reasoning, multi-agent routing, and multi-agent CPU programming, show that IC-$Q$ matches a centralized oracle without any agent observing joint trajectories, with each of the three error sources scaling along its corresponding axis as the bound predicts.

Recent research has revived and amplified interest in algorithms for undiscounted average reward reinforcement learning in infinite-horizon, non-episodic (continuing) tasks. Semi-Markov decision processes (SMDPs) are of particular interest. In SMDPs, discrete actions stochastically generate both rewards and durations, and the objective is to optimize the average reward rate. Existing algorithms approach this by optimizing the ratio of rewards to durations. However, when rewards and durations are non-stationary (in the infinite horizon), this can be incorrect. This paper presents a novel modified harmonic mean operator that correctly computes reward rates even under such conditions. This yields model-free learning algorithms that can work with SMDPs, while maintaining robustness to non-stationary reward and duration distributions over time. We prove theoretical properties of the modified harmonic mean operator, and empirically demonstrate its efficacy in comparison to existing algorithms.

Electric truck operations require routing decisions that remain feasible under limited battery range, long charging times, travel and energy consumption, and competition for shared charging infrastructure. These features make electric truck routing a coupled logistics and energy problem, limiting the practicality of heuristics-based methods and rendering them computationally infeasible at scale. This paper proposes a learning-based framework for the stochastic electric truck routing under charging constraints and operational uncertainty. The problem, solved by Reinforcement Learning, is formulated as an event-driven semi-Markov decision process with shared charging resources, stochastic travel and energy requirements, and realistic nonlinear fast-charging behavior. To support learning in this setting, a graph-based representation of system state and feasible decisions is introduced, together with a rule-based action mask that restricts policies to operationally admissible actions; thus, improving training efficiency. Building on this formulation, an event-driven simulation environment is developed that supports both Reinforcement Learning and benchmarking against heuristic and mathematical programming baselines. Computational experiments across a range of fleet sizes show that the proposed learning-based algorithm consistently outperforms baselines and attains performance close to optimization benchmarks in many settings, while preserving high success rates under charging congestion and uncertainty.

While Multi-Agent Reinforcement Learning (MARL) algorithms achieve unprecedented successes across complex continuous domains, their standard deployment strictly adheres to a synchronous operational paradigm. Under this paradigm, agents are universally forced to execute deep neural network inferences at every micro-frame, regardless of immediate necessity. This dense throughput acts as a fundamental barrier to physical deployment on edge-devices where thermal and metabolic budgets are highly constrained. We propose Epistemic Time-Dilation MAPPO (ETD-MAPPO), augmented with a Dual-Gated Epistemic Trigger. Instead of depending on rigid frame-skipping (macro-actions), agents autonomously modulate their execution frequency by interpreting aleatoric uncertainty (via Shannon entropy of their policy) and epistemic uncertainty (via state-value divergence in a Twin-Critic architecture). To format this, we structure the environment as a Semi-Markov Decision Process (SMDP) and build the SMDP-Aligned Asynchronous Gradient Masking Critic to ensure proper credit assignment. Empirical findings demonstrate massive improvements (> 60% relative baseline acquisition leaps) over current temporal models. By assessing LBF, MPE, and the 115-dimensional state space of Google Research Football (GRF), ETD correctly prevented premature policy collapse. Remarkably, this unconstrained approach leads to emergent Temporal Role Specialization, reducing computational overhead by a statistically dominant 73.6% entirely during off-ball execution without deteriorating centralized task dominance.

Many distributional quantities in reinforcement learning are intrinsically joint across actions, including distributions of gaps and probabilities of superiority. However, the classical Markov decision process (MDP) formalism specifies only marginal laws and leaves the joint law of counterfactual one-step outcomes across multiple possible actions at a state unspecified. We study coupled-dynamics environments with a multi-action generative interface which can sample counterfactual one-step outcomes for multiple actions under shared exogenous randomness. We propose joint MDPs (JMDPs) as a formalism for such environments by augmenting an MDP with a multi-action sample transition model which specifies a coupling of one-step counterfactual outcomes, while preserving standard MDP interaction as marginal observations. We adopt and formalize a one-step coupling regime where dependence across actions is confined to immediate counterfactual outcomes at the queried state. In this regime, we derive Bellman operators for $n$th-order return moments, providing dynamic programming and incremental algorithms with convergence guarantees.

We consider a remote inference system with multiple modalities, where a multimodal machine learning (ML) model performs real-time inference using features collected from remote sensors. When sensor observations evolve dynamically over time, fresh features are critical for inference tasks. However, timely delivery of features from all modalities is often infeasible under limited network resources. To address this challenge, we formulate a multimodal scheduling problem to minimize the ML model's inference error. We model this error as a general function of the Age of Information (AoI) vector, where AoI quantifies data freshness. We cast the problem as a semi-Markov decision process (SMDP) and derive an equivalent reformulation with a reduced state set. We then show that the problem has fundamentally different chain structures in the two-modality and multi-modality cases. For the two-modality case, we prove that the optimal policy has an index-based threshold structure. For the general multi-modality case (two or more modalities), we develop an optimal switching-type policy (OPT-ST), which is computed using a multichain policy iteration algorithm (MPI). To further reduce complexity, we also develop two low-complexity policies: the round-robin-like switching-type policy (RR-ST) and the round-robin-like threshold policy (RR-Thres). Numerical results show that OPT-ST achieves the lowest inference error among all baselines, with reductions of up to 80.3%. In a five-modality case, RR-ST and RR-Thres achieve $5.2\times$ and $89.8\times$ speedups in offline computation over OPT-ST, respectively, at the cost of increasing the inference error by 23.3% and 38.6%, respectively.