Markov Decision Processes

The stochastic linear bandit, where actions are represented as vectors and rewards are linear, is a central paradigm for sequential decision making. We study a partially observed variant of this problem in which the learning agent only sees a random subset of coordinates for each action. Such partial observability arises naturally in settings like recommendation and healthcare, where full action descriptions can be expensive or even impossible to obtain. In general, this makes sublinear regret information-theoretically impossible. However, we show that this barrier can be overcome when the action vectors have low intrinsic dimension. We propose an algorithm, TOFU-POV, that estimates the latent action subspace using the masked actions, imputes current actions using an epoch-wise frozen representation, and runs OFUL in the resulting low-dimensional coordinates. Our theory shows that TOFU-POV enjoys a $\sqrt{T}$ regret that scales with the intrinsic action subspace dimension as opposed to the ambient dimension and quantifies the interaction between these quantities and the missingness, decision set size, and subspace conditioning. We also devise a rank-adaptive algorithm that does not require the knowledge of the intrinsic dimension. We complement these guarantees with a lower bound based on a novel product construction that separates usual reward-learning uncertainty from a missingness-dependent cost intrinsic to partial observation. Synthetic and real data experiments support our theory and show that TOFU-POV can substantially improve upon natural baselines in this challenging problem.

Training reinforcement-learning agents directly on physical robots makes every fall costly, since a fall can damage the platform and cannot be undone like a simulator reset; the goal is therefore to minimize falls during training rather than trade them off against return, as constrained Markov decision process (MDP) formulations do. A standard mitigation hands control to a separate recovery policy whenever the agent leaves a designer-specified safe region (a subset of state space it should stay within), but the resulting mixed-policy rollouts silently bias every on-policy update, and the importance-sampling correction that would remove this bias is ill-defined whenever the recovery policy is deterministic. We address this bias with a drop-in modification of proximal policy optimization (PPO). Its core is an unbiased policy-gradient estimator that uses the score function only at safe timesteps and never evaluates the recovery policy's density, so it stays valid even when the recovery policy is deterministic, exactly where importance sampling breaks, and it empirically dominates importance sampling even when the recovery policy is stochastic. Because the recovery policy still makes credit assignment slow near the safe-region boundary, two further components accelerate learning: a closed-form value for recovery-triggering states when dynamics and recovery are deterministic, and an imitation loss that copies recovery actions only when recovery succeeds. On a three-environment, five-seed benchmark, the resulting algorithm reduces training-time falls by factors of 233x, 48x, and 26x on HalfCheetah, Ant, and Unitree Go1 over standard PPO, while matching or exceeding PPO's final reward, and on Ant, where the recovery policy is unreliable, it is the only method that reaches 80% of the best final reward.

Computing Nash equilibrium policies in multi-agent Pursuit-Evasion games (PEG) is challenging due to the exponential growth of the joint state and action spaces with the number of agents. Existing approaches either rely on offline equilibrium approximations, which may lack adaptability during execution, or online planning methods, which suffer from large branching factors. In this work, we propose Primitive-Guided Tree Search (PGTS), a hybrid framework that integrates offline exact Nash equilibrium computation with online tree search: PGTS first solves a collection of smaller, tractable sub-games offline; at deployment, PGTS performs online tree search at each time step, using the optimal sub-game policies and value functions to guide tree expansion and estimate leaf-node values. Extensive experiments on varied graph topologies, including real-world networks, demonstrate that PGTS significantly outperforms state-of-the-art learning and heuristic baselines, while maintaining robust performance against adversaries.

In this paper, we study quantile-based distributional reinforcement learning from the perspective of statistical efficiency. We focus on distributional policy evaluation, whose goal is to characterize the return distribution, namely the distribution of discounted cumulative rewards under a given policy. To obtain a finite-dimensional representation of the return distribution, we consider the quantile fixed point $η_m$ induced by the quantile-projected distributional Bellman equation. Assuming access to a generative model, we construct an estimator $η_m^{(n)}$ based on an empirical Markov decision process. For a fixed number of quantiles $m$, we establish a non-asymptotic error bound for $η_m^{(n)}$ and $η_m$ under the supremum $W_\infty$ metric, showing that the estimation error scales as $\widetilde{O}(\sqrt{m/n})$ with respect to $m$ and $n$. This implies that the quantile-based distributional policy evaluation problem can be solved with sample efficiency, achieving the optimal parametric $\sqrt{n}$ convergence rate. We derive the asymptotic distribution of the quantile parameters $\sqrt{n}(θ_m^{(n)}-θ_m)$ and characterize the semiparametric efficiency bound, which is attained by our estimator. Beyond the fixed-dimensional setting, we investigate the asymptotic regime in which the number of quantiles diverges. We characterize the limit covariance structure and show that it matches the semiparametric efficiency bound of the nonparametric model for distributional policy evaluation, showing that quantile-based estimators remain asymptotically efficient in the infinite-dimensional limit. Finally, we establish a Berry--Esseen theorem for smooth functionals $\sqrt{n}(η_m^{(n)}(s)-η_m(s))f$, thereby providing a foundation for statistically valid inference on functionals of the quantile-projected return distribution.

This paper studies an online variant of the assistance games framework, where an informed agent and an uninformed agent repeatedly interact over $T$ timesteps to optimize a common reward function. While the informed agent (the human) observes a latent state of the world, the uninformed agent (the assistant) observes only the human's actions. We provide the first provably efficient learning algorithms for repeated assistance games. We introduce the notion of assistance regret: the gap between the cumulative utility of interactions and that of the optimal joint policies in hindsight, which map latent states to action pairs. We present decentralized algorithms for both the human and the assistant that achieve a $(1-1/e)$-approximate assistance regret rate of $\widetilde{O}(T^{3/4})$, with runtime polynomial in the size of the action and state spaces. These algorithms are general; in particular, they accommodate any no-regret algorithm for the assistant. We prove that achieving a regret approximation factor better than $(1-1/e)$ is computationally intractable. Furthermore, we demonstrate how these generic no-regret algorithms can be tailored to a pseudo-decentralized setting -- using a shared random string -- to achieve a rate of $\widetilde{O}(T^{1/2})$, optimal up to logarithmic factors.

Starting from the utilization of deep neural networks to approximate the state-action value function that led to winning one of the most challenging games, to algorithmic advancements that allowed solving problems without even explicitly stating the rules of the challenge at hand, reinforcement learning research has been the center of remarkable scientific progress for the past decade. In this paper, we focus on the key ingredients of this research progress and we analyze the canonical evaluation and design paradigms in reinforcement learning. We introduce the theoretical foundations of scaling laws in reinforcement learning and show that the asymptotic performance of reinforcement learning algorithms does not have a monotone relationship between performance rankings and data-regimes. We conduct large-scale experiments and our results demonstrate that a line of reinforcement learning research under the canonical design and evaluation paradigms resulted in incorrect conclusions. Our analysis and results provide a core analysis on scaling, capacity and complexity of deep reinforcement learning.

Reinforcement learning (RL) is increasingly grounded in tools from probability, optimization, and operator theory. This survey organizes the mathematical structures that underpin the design and analysis of modern algorithms in RL. We begin from Markov decision processes (MDPs) and the Bellman operators, emphasizing contraction mappings, monotonicity, and fixed-point theory that yield convergence guarantees and rates for value and policy iteration, and temporal-difference schemes. We then develop the optimization perspective: stochastic approximation and martingale methods, convex duality and the role of regularization linking mirror/proximal methods. Function approximation is treated through linear and non-linear settings, covering stabilization, error decomposition, and sample-complexity via concentration inequalities for dependent data and mixing processes. We further cover off-policy evaluation/learning, constrained RL and constrained MDPs (CMDPs). Throughout we unify algorithmic templates under common operator and variational lenses, highlighting both finite-sample bounds and asymptotic results. Our presentation is intended to provide a unified mathematical entry point for researchers in probability, optimization, and statistics interested in reinforcement learning.

Occupancy ratios correct distribution shift in offline reinforcement learning and are central to off-policy evaluation. Existing primal-dual and minimax methods typically estimate these ratios by enforcing occupancy-balance moments over a critic class. We propose fitted occupancy-ratio evaluation (FORE), a fitted fixed-point method that characterizes the discounted occupancy ratio through an adjoint Bellman recursion. At each iteration, FORE solves a single-level density-ratio objective on one-step-transition data, thereby projecting the adjoint Bellman image onto a log-ratio class in Kullback--Leibler (KL) divergence. Unlike analyses of fitted Q-evaluation, which typically require value-function realizability together with Bellman completeness or projected-operator stability, our central approximation condition is just realizability of the discounted occupancy ratio itself. Under this condition, the population KL-projected recursion contracts in relative entropy toward the true ratio by virtue of the adjoint Bellman operator being a KL-contraction. For the empirical recursion, we establish finite-sample regret bounds that yield convergence in KL up to log-ratio approximation error and a statistical error governed by the complexity of the ratio hypothesis class. The fitted ratio supports direct value estimation by reward reweighting, occupancy-weighted fitted Q-evaluation, and doubly robust estimation that combines the fitted ratio with a fitted Q-function. Together, these results identify discounted occupancy-ratio realizability as a sufficient condition for offline policy evaluation without any completeness assumptions.

We propose the first deep learning algorithm, the Certainty Equivalent Learning (CEL) algorithm, for solving high-dimensional discrete-time dynamic programming problems with recursive utility. Dynamic programming with recursive utility is numerically challenging because the recursive utility does not have an explicit representation and the Bellman equation contains a certainty equivalent that is difficult to evaluate. The CEL algorithm learns this certainty-equivalent value directly with neural networks and jointly approximates value functions, policy functions, and certainty-equivalent functions. The CEL algorithm is mesh-free and simulation-based, allowing high-dimensional state and control spaces, and does not rely on Euler equations, first-order conditions, or differentiability of the state transition function. The CEL algorithm also works for dynamic programming problems with expected utility as expected utility is a special case of recursive utility. We apply the CEL to discounted linear exponential quadratic Gaussian control, small-noise robust control, Epstein-Zin DSGE, and multivariate strategic asset allocation problems. Compared with closed-form and VFI-based benchmarks, the CEL delivers accurate value and policy approximations, remains effective in high-dimensional problems, achieves accuracy comparable to VFI in the small-noise robust-control case, and produces out-of-sample Bellman errors and Euler or first-order residuals that are in the range from 1.0e-4 to 1.0e-3 for most problems.

Multi-task offline safe reinforcement learning (RL) promises to learn a shared optimal safe policy from offline data across multiple tasks. This paradigm provides an effective means for the widespread application of RL in multi-task scenarios with high risk and interaction costs. However, the triple challenges of multi-tasking, safety constraints, and out-of-distribution (OOD) actions pose a significant hurdle for existing methods to ensure safety while maximizing reward returns. In this work, we propose a Conditional Diffusion model with Contextual Prompts (CDCP) to address these challenges. Concretely, we first rethink the requirements and challenges in current multi-task decision-making and control scenarios and establish the objectives of multi-task offline safe RL. Subsequently, we transform the multi-task constrained optimization problem into a conditional generation problem using the diffusion model. Based on this, we design a classifier-free guided cost-constraint strategy to provide flexible cost constraints and eliminate extrapolation errors from OOD actions via supervised learning. Additionally, we introduce a novel contextual prompting method to enhance multi-task representation accuracy and adaptability to unseen tasks. A gradient loss synchronization strategy is also introduced to eliminate gradient interference, improving training stability. Finally, extensive experiments demonstrate that the CDCP algorithm exhibits higher performance and safety in multi-task scenarios than the current state-of-the-art baseline methods. It meets different cost constraints without further training, providing a more flexible cost-constraint solution for the multi-task safe RL.

Policy learning has received substantial attention with the goal of learning policies from observational data for decision-making. A majority of work in this space has focused on developing algorithms for computing policies that minimize regret compared to the optimal policy. However, in many practical settings, there is insufficient data to obtain low regret. As a result, recent work has shifted attention to alternative objectives, most notably, studying whether it is possible to learn an improving policy that statistically significantly outperforms baseline policies. We argue that there is substantial merit in studying a broader range of policy learning problems. When there is insufficient data to learn an improving policy, there may still be useful questions that can be answered. To this end, we provide a mathematical framework for studying the relationships between policy learning problems. We formalize three problems within our framework: beyond the optimal policy problem and the improving policy problem, we also propose the policy existence problem, which aims to determine if an improving policy exists. Within our framework, we show that the policy existence problem reduces to the improving policy problem, which in turn reduces to the optimal policy problem; these reductions prove that each problem is at least as easy as the next one (in sample complexity). A key question remains: is this hardness strict? We provide partial answers. First, the gap between the optimal policy and improving policy problems is strict. For the improving policy and policy existence problems, we prove that a sublinear polynomial gap exists under natural conditions on improving policy learning algorithms. Thus, we may be able to answer questions about the existence of an improving policy even when we cannot find one. These results highlight the value in studying a broader range of policy learning problems.

Entropy regularization is widely used in continuous-time reinforcement learning (RL) to reduce sensitivity to environmental perturbations, yet its robustness benefits lack a rigorous theoretical foundation. This paper establishes the first robustness guarantees for entropy-regularized continuous-time Markov decision processes. We show that maximizing an entropy-regularized objective yields a lower bound on a worst-case robust RL problem with joint reward and transition perturbations. We analytically characterize the induced robust sets and prove that they expand monotonically with the regularization strength, justifying the empirical observation that stronger entropy improves robustness. In contrast to prior discrete-time analyses, our results remove the intractable state-distribution entropy term and provide guarantees invariant to action frequency. Experiments on queueing network control and market making confirm our theory, showing that entropy-regularized policies outperform greedy and $ε$-greedy baselines under dynamics perturbations.

Reinforcement Learning (RL) has substantially improved the reasoning ability of large language models (LLMs), but sparse outcome rewards still make token-level credit assignment difficult. Existing scalable RL methods typically assign trajectory-level rewards uniformly across tokens, while recent entropy-aware approaches either rely on coarse detached heuristics or directly optimize true entropy, which can introduce non-local gradient components misaligned with sampled-token policy updates. We propose Adaptive Credit Policy Optimization (ACPO), a token-level credit assignment framework based on a mode-local surrogate entropy. ACPO asymmetrically modulates policy updates by emphasizing uncertain decisions in successful rollouts and overconfident tokens in failed rollouts. We show that the surrogate admits deterministic entropy bounds and, under modal alignment and proximal updates, preserves the policy-gradient direction to leading order. Experiments on mathematical reasoning and coding benchmarks, including AIME 2025 and HumanEvalPro, show that ACPO consistently improves over strong RL baselines such as DAPO, GTPO, and SAPO.

While pessimism counteracts overestimation bias in offline reinforcement learning (RL), being overly conservative has been associated with hindering certain forms of generalization. However, in this paper we demonstrate that being overly pessimistic does not inherently prevent optimal generalization in contextual MDPs (CMDPs). Instead, we argue successful generalization depends not on the amount of pessimism, but whether the pessimistic structure respects the underlying symmetries of the optimal solution. We prove that a mildly pessimistic, non-symmetric value function can generalize worse than an overly pessimistic, symmetric one. In offline RL, the structure of the pessimism is determined by the structure of the dataset coverage. As such, enforcing a symmetric value function can be non-trivial, and might require techniques such as data augmentation (DA). Inspired by our theoretical results, we argue that DA can best be applied through a consistency loss during policy extraction, rather than the common practice of (regular) offline training on an augmented dataset. This is empirically validated using IQL and CQL on a rotationally symmetric reacher environment.

This paper studies additive regret in the multi-secretary problem, defined as the gap between the expected offline prophet reward and the reward of the best online policy. Prior work established $O(\log T)$ regret for bounded-density distributions with connected support and $O((\log T)^2)$ upper bounds for bounded-density distributions with support gaps. It was unknown whether the extra logarithmic factor is necessary even in the one-resource model. We prove that it is necessary. For a mixture of two separated uniform distributions at the critical capacity, the optimal regret grows at least on the order of $(\log T)^2$. Thus the existing $O((\log T)^2)$ upper bounds for bounded-density gapped instances, including those implied by network revenue management models with continuous rewards, are tight in this simplest specialization. The same framework also yields a matching lower bound for gapped distributions whose gap-facing densities vanish near the support edges; this companion result is given in the appendix. The proofs use Bellman certificates: feasible solutions to a relaxation of the exact Bellman recursion. This framework converts lower bounds into explicit certificate constructions and identifies why support gaps permit larger regret.

Value functions are an essential component in actor-critic based deep reinforcement learning (RL). Conventionally, these functions are trained as a regression task by minimising the mean squared error (MSE) relative to bootstrapped target values. Meanwhile, in distributional RL, a distribution of returns is modelled based on the distributional Bellman operator. This work investigates the Gaussian Histogram Loss (HL-Gauss), a recent approach that reframes value estimation as classification by encoding each scalar Bellman target as a Gaussian-smoothed categorical target. Despite its potential, applying histogram-based losses to RL presents inherent challenges, most notably the requirement to pre-define a fixed support interval, which is often complicated by the non-stationary and stochastic nature of target values typically found in RL tasks. In this work, we propose an approach that dynamically learns the lower and upper bounds of the support instead of assigning them beforehand. We derive an objective that jointly learns these bounds whilst learning the categorical representation of the scalar values, and we show that this objective forms an upper bound on the mean-squared Bellman error. Our theoretical analysis further shows that this bound is tighter than that of non-learned supports of HL-Gauss. Empirically, the proposed objective enables stable adaptation of the support interval and matches HL-Gauss-based actor-critic algorithms on most continuous-control tasks whilst improving on a subset, without requiring a pre-specified support interval.

We study policy optimization for online episodic tabular Markov decision processes with unknown transition kernels, aiming for best-of-both-worlds guarantees together with data-dependent regret bounds. Recent work (Dann et al., 2023; Li et al., 2026) has shown that policy optimization can adapt to both adversarial and stochastic losses with first-order, second-order, and path-length bounds, but only under known transitions, leaving open whether such data-dependent guarantees are achievable by policy optimization when the transition kernel is unknown. We resolve this by developing a new algorithm based on optimistic follow-the-regularized-leader that attains these guarantees under unknown transitions. The key ingredient is a new design of optimistic $Q$-function estimators together with a data-dependent transition bonus that controls estimator bias through the loss-prediction error. Our analysis further identifies an unavoidable transition-dependent complexity term that captures the intrinsic cost of estimating the transition kernel. As a result, we obtain first-order, second-order, and path-length bounds with the transition-dependent complexity term while simultaneously achieving gap-dependent $\mathrm{polylog}(T)$ regret in the stochastic regime.

While neural network control policies are powerful, their deployment on safety critical systems depends on ensuring that they obey strict constraints. Existing work often treats safety as a metric to optimize for, which competes with other performance objectives, if training converges at all. Instead, we introduce ShardNet, a neural network architecture that strictly enforces unions of polyhedral constraints by construction, using a differentiable projection layer parameterized by a classification network. The key insight is to embed safety into the neural network's structure, allowing performance to be optimized independently because formal safety guarantees are always given. In contrast with existing neural architectures that can only enforce simple convex constraints, ShardNet enables the first safe-by-construction synthesis of forward-invariant neural network controllers on closed-loop systems where safety constraints are expressed as nonconvex unions of polyhedras or learned value function level sets. To support this, we also introduce a technique to verify and train such value functions correctly as rectified linear unit (ReLU) networks, which has not previously been possible. On double integrator benchmarks drawn from the literature, ShardNet policies maintain 100% safety on verified sets and achieves significantly lower objective loss compared to existing formal methods. Furthermore, our value function training technique also produces safe sets more than 3 times larger than existing verification approaches.

Probabilistic model checking for Markov decision processes (MDPs) provides quantitative guarantees, but often offers limited insight into why undesired outcomes occur. Probability-raising (PR) causality addresses this by identifying states whose visitation increases the probability of reaching designated states. Existing PR-cause identification methods, however, use MDP modifications not well-suited for learning: the gap between conditional and unconditional reachability probabilities can be hard to detect from transition samples, and construction requires reachability probabilities of the MDP, which are unavailable when transition probabilities are unknown. We study unknown MDPs and propose a learning approach with probabilistic guarantees for PR-cause identification. Our key ingredient is a restart-based MDP modification that reduces PR-cause checking to two conditional reachability queries without using reachability values of the original MDP. We prove correctness, establish sample-complexity bounds, and develop an anytime learning-and-checking algorithm based on two-sided value iteration that progressively classifies states as causal, non-causal, or undecided. Experiments on two benchmarks demonstrate reliable and fast identification of PR causes.

In a changing decision problem, standard dynamic-regret analyses have often equated the cost of non-stationarity to how far loss moves. However, it is simultaneously possible for a loss sequence to travel far and retain the same optimal policy, or for a small movement in loss to force the optimal policy to change completely. Thus, the size of the movement through loss variation, transition variation, or comparator path length describe the adversary's motion, but not the cost of that motion to the control problem. For a more faithful analytic interpretation, this paper develops a normal-fan geometry for finite-horizon adversarial MDPs with fixed transitions. Occupancy measures form a polytope, and each loss vector exposes an optimal face of that polytope. Non-stationarity in rewards is therefore a path through the normal fan, where motion inside one cone leaves the optimal face unchanged, while crossing a wall may carry regret. We pose the notion of a face-crossing price, which is the minimum regret incurred by remaining on the previous optimal face under the new loss. For any learner that tracks the previous face, dynamic regret decomposes exactly into intrinsic priced face motion plus within-face selection error. The resulting theory separates consequential from harmless non-stationarity, where loss variation can be arbitrarily large at zero price, and identical one-coordinate variation can hide horizon-scale differences in regret.

We propose KL-Coupled Policy Regularization (KCPR), a policy coordination framework for Reward-Punishment Reinforcement Learning (RPRL). Based on KCPR, we derive KL-Coupled Soft Optimality (KCSO) and develop its deep realization, klDMP. Unlike existing RPRL approaches that optimize reward-seeking and punishment-related policies largely independently, KCPR enables direct interactions between companion policies by treating each as a dynamically learned prior for the other. KCSO yields coupled soft-optimal policies and KL-regularized Bellman operators, allowing reward and punishment information to jointly influence value propagation. To improve learning stability, we introduce a companion-prior softening mechanism and evaluate separate replay-buffer designs for balancing reward- and punishment-related experience. Experiments in grid-world and Gazebo robotic navigation tasks demonstrate that klDMP improves safety and learning stability while maintaining competitive task performance compared with DQN, SQL and softDMP. These results suggest that policy-level coordination provides an effective mechanism for integrating multiple behavioral objectives and may serve as a useful design principle for reinforcement learning systems with interacting motivational processes.

This paper studies the problem of regret minimization in Markovian bandits with \emph{non-observable states} and possibly \emph{constrained} decision epochs. The focus is restricted to a ``pure'' regret benchmark, that compares the performance of the learning algorithm to the best \emph{pure policy} which -- akin to optimal policies of stochastic bandits -- picks the optimal arm from start to finish without ever switching. We introduce a generalization of rested Markovian bandits, \emph{self-degrading Markovian bandits}, for which pure policies are always asymptotically optimal.We show that without prior knowledge on the underlying bandit, the regret of algorithms that switch arms rarely necessarily scales super-logarithmically for every bandit, i.e., as $ω(\log(T))$, where $T$ is the learning horizon. Despite the unreachability of the logarithmic regime, we design UCB-NOM, an optimistic algorithm inspired by UCB, of which the regret is nearly logarithmic. Lastly, we show that given prior knowledge on the Markovian bandit in the form of a bound on the bias functions of its arm, a proper instantiation of UCB-NOM achieves $O(\log(T))$ regret. We further show that this prior knowledge allows for a $O(\sqrt{T \log(T)})$ worst-case regret bound for UCB-NOM. Notably, our regret bounds do not depend on the number of states of the underlying Markov chains. Our findings suggest that the non-observability of states is a mild inconvenience in self-degrading Markovian bandits.

We study learning in prophet inequalities with i.i.d. rewards drawn from an exponential-type parametric family with an unknown parameter $θ$, a class that includes exponential, Pareto, and bounded-support power-family distributions. We first characterize the optimal full-information asymptotic competitive ratio for this family. In the unbounded-support case, the limit is ${\left(θ/({θ-c_+})\right)^{c_+/θ}}/ {Γ(1-c_+/θ)},$ while in the bounded-support case, the limit is $1$. We then propose a confidence-based dynamic-programming policy for online learning. By exploiting the explicit parametric structure, the policy achieves the same optimal asymptotic competitive ratio using only online observations, without external offline samples. We further derive distribution-specific convergence rates for canonical examples. Finally, numerical experiments on synthetic instances illustrate the performance of our algorithm.

This paper addresses model-free continuous-time mean-field control in a setting where the population dynamics evolve continuously according to an unknown McKean-Vlasov stochastic differential equation, while only discrete-time transition data are available. In the model-based formulation, policy evaluation is naturally described by a stationary Hamilton-Jacobi-Bellman equation on $\mathcal P_2(\mathbb R^d)$, but this equation involves the drift and diffusion coefficients of the controlled McKean-Vlasov dynamics, which are not identifiable when only discrete-time data are available. On the other hand, a direct reduction to a time-discrete Bellman equation avoids the non-identifiability issue but loses the differential equation structure. To bridge these two viewpoints, we introduce a Mean-Field-PhiBE (MF-PhiBE), which incorporates discrete-time transition information into a continuous-time PDE on the Wasserstein space. The MF-PhiBE replaces the unknown infinitesimal drift and covariance in the policy-evaluation equation by one-step estimators computed from data, while preserving the generator structure of the McKean-Vlasov HJB equation. We also derive a policy-gradient theorem for entropy-regularized randomized feedback policies, expressing the actor direction through an action-wise infinitesimal advantage and the score of the policy. Combining these two ingredients yields a model-free actor-critic method. We prove a first-order consistency estimate showing that the value induced by an optimal MF-PhiBE policy approximates the optimal continuous-time value with an error of order $Δt$. In the linear-quadratic case, we show our approximation achieves second-order accuracy with only one-step data. Numerical experiments on an LQR benchmark and a crowd-aversion problem illustrate the proposed framework.

Real-world decision-making often requires balancing multiple conflicting objectives, a challenge that standard Reinforcement Learning (RL) frequently addresses by aggregating rewards into a single scalar signal. While effective for simple tasks, this approach often fails to capture the full spectrum of optimal trade-offs, known as the Pareto frontier. In this paper, we introduce a novel preference-conditioned Bellman operator, motivated from the Chebyshev scalarization, designed to compute deterministic Pareto-optimal policies for Multi-Objective Markov Decision Processes (MOMDPs). We prove that this operator satisfies an enveloping property, where the estimated value functions upper-bound the true Pareto frontier, and demonstrate that it monotonically converges to a coverage set of this frontier. Furthermore, we also show how to extract deterministic policies from these converged Q-estimates. This ensures the agent can recover a policy for any given preference, capturing the entire Pareto-optimal frontier while guaranteeing each synthesized policy remains approximately Pareto-optimal. Experimental results validate that our algorithm successfully recovers complex trade-offs, providing a solution for deterministic Pareto-optimal policy synthesis.

Markov decision processes (MDPs) are a classic model of decision making under uncertainty, exhibiting both non-deterministic choice as well as probabilistic uncertainty. Traditionally, exact knowledge of the underlying probabilities is assumed. However, this often is unrealistic, e.g.\ when modelling cyber-physical systems or biological processes. Here, statistical methods provide a way towards obtaining meaningful guarantees. The classical approach is to gather samples in the MDP, use these to draw statistical conclusions about the transition probabilities, and from there obtain bounds on the true value; then, if these bounds are too broad, repeat. However, existing implementations of this approach are either subtly incorrect or sub-optimal, and quite often both. We present several \emph{confidence sequences}, which are specifically designed for such \enquote{online} settings, implement all of them in an efficient tool, and show their practical applicability. In particular, we show that they outperform classical \enquote{union-bound} style approaches, and overall our implementation requires 50x less samples on average than previous state of the art.

We study optimal-policy geometry in structured Markov decision processes. While approximate dynamic programming and reinforcement learning typically approximate high-dimensional value functions, we show that optimal policies induce simpler decision tessellations. We propose boundary-based policy approximations that learn policy regions directly. A policy-loss decomposition links performance degradation to action margins and explains why errors concentrate near indifference boundaries. Inventory control and queue admission experiments show lower policy error, smaller value gaps, faster error decay, and stability than reinforcement learning baselines.

Living systems navigate environments using noisy and incomplete sensory signals. In unicellular algae, phototaxis is often modeled as a mechanistic run--tumble process driven by stimulus--response rules. However, such descriptions overlook how organisms actively sample their environment to reduce sensory ambiguity. From a minimal cognition perspective, we reframe this navigation as a subjective, information-driven sensorimotor process. To this end, we propose a framework linking a Partially Observable Markov Decision Process (POMDP) with biochemical reaction dynamics. Environmental variables are hidden, while the cell updates a minimal internal state from each observation through a memoryless Bayesian step. These internal dynamics balance orienting toward light with exploratory reorientation and can be implemented through Chemical-Reaction-Network Ordinary Differential Equations (CRN--ODEs). Our model includes a biophysical observation process for photoreception and a chemically computable polynomial bound on information gain. Using Inverse Reinforcement Learning (IRL) on 30 experimentally recorded Chlamydomonas trajectories, we infer the behavioral objective consistent with observed phototactic motion and benchmark the resulting dynamics with standard Stochastic Simulation Algorithm (SSA) baselines. Our model reproduces the empirical alignment-to-light distribution, comparable to objective SSA baselines on this dataset. Within this framework, run--tumble alternation emerges as an information-acquisition strategy: tumbling reorients the cell to sample new sensory configurations and resolve sensor ambiguity, demonstrating how intracellular biochemical networks can support adaptive information-seeking behavior in cellular navigation.

We study PAC learning in tabular discounted Markov decision processes with exogenous i.i.d. contexts, with discount factor $γ$, finite state space $\mathcal X$, action space $\mathcal A$, and context space $\mathcal Z$. At each time step, a context is drawn independently from an unknown distribution $μ$ and revealed before the agent acts. This context may affect both rewards and transitions, while remaining uncontrolled by the agent. Depending on the regime, the learner has access either to a sampling oracle for $μ$, to a sampling oracle for the transition kernel conditioned on state-context-action tuples, or to both. Oracles can be accessed before and during policy execution. The sample complexity is measured by a couple $(n,m)$, where $n$ is the number of calls to the sampling oracles before execution and $m$ is the number of calls to the sampling oracles during execution. When rewards and transitions are known and only the context distribution $μ$ is sampled, we give a variance-reduced algorithm that solves policy evaluation (PE), best-value estimation (BVE), and best-policy extraction (BPE) with $\left(\widetilde O\left(1/((1-γ)^3\varepsilon^2)\right), 0 \right)$ sample complexity. The rate is independent of $|\mathcal Z|$ and minimax optimal up to logarithmic factors. As a corollary, we also obtain tight rates in the case of one-step perfect look-ahead, improving upon the existing guarantees. In the fully unknown regime, where both $μ$ and P must be learned, we show that PE remains $|\mathcal Z|$-free, with matching upper and lower bounds $\bigl(\widetilde O(|\mathcal X|/((1-γ)^3\varepsilon^2)),\, \widetilde O(1/((1-γ)^2\varepsilon^2))\bigr)$.

Generalist value models play a pivotal role in scaling robotic policy learning from large-scale, mixed-quality data. Mathematically, accurate value estimation demands deep temporal understanding, requiring models to both ground the current belief using historical context and plan over future outcomes. However, most existing robotic value models are built on Vision-Language Model (VLM) backbones that are pretrained primarily on static or temporally sparse visual observations, lacking the requisite temporal modeling capabilities for value estimation. Unlike VLMs, world models naturally excel at temporal modeling and future planning, making them ideal foundations for learning generalizable value functions. Driven by this insight, we marry world models with value estimation to construct a new generalist robotic value model, World Value Model (WVM), that offers accurate task progressions to assess data quality. On standard benchmarks, WVM delivers state-of-the-art (SOTA) Value-Order Correlation (VOC) results. Complementing standard evaluation suites that contains only expert data, we further introduce Suboptimal-Value-Bench, a multi-embodiment benchmark consisting of 800 suboptimal trajectories with high-fidelity, human-labeled frame annotations. Our evaluations show that WVM maintains its SOTA performance on Suboptimal-Value-Bench, establishing its robustness in handling both expert and suboptimal data. When deployed for policy learning, WVM improves manipulation performance across various policy extraction approaches in both simulated and real-world deployment, providing robust guidance for learning from mixed-quality data.

Model Predictive Control (MPC) is widely used in industrial and robotic systems for enforcing constraints and embedding domain knowledge through finite-horizon optimization-based planning. However, despite these strengths, an MPC scheme typically does not yield optimal policies for sequential decision-making problems formulated as Markov Decision Processes (MDPs). Recent combinations of MPC with Reinforcement Learning (RL) alleviate this issue by treating MPC as a parameterized model of the optimal policy of an MDP and adjusting its parameters using data. While these approaches typically consider classical MDPs, many real-world problems include future information--such as forecasts, prices, or reference trajectories--at decision time, which must be included in the MDP state for optimal decision-making. Current MPC-RL approaches do not directly account for this augmented-state structure, raising the question of how to incorporate future information into MPC to obtain an optimal policy. This work establishes the structural requirements under which a parameterized MPC can exactly represent the optimal value functions and policy of an MDP with future information. We further demonstrate that such a parameterized MPC can serve as a structured function approximator, with its parameters learned using RL. The approach is illustrated on a point-mass racing task with future reference information.

Human demonstrations for robot imitation learning often contain mistakes and corrective behaviors, such as imprecise grasps, object misalignment, unstable contact, and repeated attempts. While these segments are commonly treated as noisy or suboptimal data, they provide valuable evidence about when execution deviates from a desirable path and how task feasibility can be restored. However, existing reward and value models often rely on monotonic progress assumptions, which capture coarse task advancement but may overlook local execution errors and corrective behaviors in imperfect demonstrations. In this work, we propose ReTVL (ReTry-Supervised Value Learning), a framework for learning mistake-sensitive value functions from mixed-quality robot demonstrations by leveraging retry events as sparse supervision. ReTVL captures the local degradation-and-recovery structure around mistakes by combining global progress calibration with local pairwise preference learning induced by sparsely annotated retry keypoints. The learned value model is then used to reweight demonstration chunks for downstream behavior cloning, reducing the influence of harmful execution errors while preserving useful corrective behaviors. Experiments on real-robot manipulation tasks show that ReTVL produces more fine-grained value estimates than progress-based baselines and improves imitation learning from imperfect demonstrations.

At the core of most socio-technical systems lies a scarce resource that is allocated among agents: highway lanes, public transit, road space, water rights, energy access, grid capacity, user attention, pollution rights, etc. With further automation of the underlying allocation processes, control engineers are increasingly tasked to make decisive assumptions regarding what society wants. In practice to date, design choices are largely driven by industry norms and conventions rather than a result of conscientiously responsible and ethical design. In this paper, we look at tools available to control engineers to design systems in a more principled manner in order to match the societal mandate. We consider three control design paradigms: online feedback optimization, control of Markov decision processes, and model predictive control. Beginning with aggregating individual agents' preferences into control design objectives, subsequently ensuring and certifying the fulfillment of those specifications, we argue that the feedback nature of control systems enables appropriate allocation of the shared resources in ways hitherto unparalleled.

A data-driven method is developed for approximating value functions in deterministic optimal control problems with nonlinear control-affine dynamics. The Pontryagin Maximum Principle optimality system is solved from multiple initial conditions to generate training data consisting of values, gradients, and Hessians of the value function, where Hessian information is obtained from a matrix Riccati equation along optimal trajectories. These quantities augment a weighted least-squares regression over sparse polynomial bases on hyperbolic cross index sets, with gradients and Hessians contributing additional linear equations per sample and substantially reducing sample complexity compared to value-only regression. Feedback laws are recovered analytically from the learned value function. In high dimensions, a partial Hessian strategy controls the cost of data generation. The approach is validated on problems of increasing state dimension, where second-order data augmentation is shown to improve approximation accuracy and closed-loop performance, with up to an order-of-magnitude reduction in the number of training samples required relative to lower-order methods.

Deploying multi-agent reinforcement learning (MARL) in the real world is often limited by model mismatches between the training simulators and the true environment, which could be further amplified through strategic interactions and result in severe performance degradation upon deployment. Distributional robustness offers a principled response by optimizing policies against worst-case transition models drawn from an uncertainty set, but standard robust MARL frameworks become increasingly intractable as the number of agents grows. This paper develops an infinite-horizon, stationary mean-field game framework that incorporates distributional model uncertainty directly into the population-coupled dynamics. We establish a robust dynamic programming principle with a contractive Bellman operator and prove the existence of a stationary robust mean-field equilibrium via a fixed-point argument. We further develop the first concrete algorithm with convergence guarantees. We then connect the mean-field solution to a finite-population robust game whose ambiguity sets depend on the empirical distribution, showing that the mean-field equilibrium policy induces approximate equilibrium behavior as the population size increases. Under a contractive robust-dynamics regime, we further obtain explicit non-asymptotic error bounds. Numerical experiments further illustrate the qualitative and quantitative impact of robustness under multiple uncertainty models, validating our theoretical findings.

Vision-Language-Action (VLA) models provide a unified paradigm for robotic manipulation, yet their real-world deployment is often bottlenecked by execution efficiency. While existing efforts predominantly focus on compute-centric efficiency to reduce per-step inference latency, the intrinsic \textbf{policy efficiency} of these models remains largely unexplored. Policy efficiency is fundamentally affected by two factors, namely the effective executable length of predicted action chunks and the total physical steps required to complete a task. These two factors jointly determine the total number of forward inference calls during execution. We observe that current VLA policies struggle with planning unreliability and action redundancy, suffering from severe prediction degradation at the tail of action chunks and tending to generate unnecessarily redundant physical steps. To address this, we propose \textbf{PolicyTrim}, a reinforcement learning-based post-training framework that extends the reliable action chunk length and reduces redundant physical steps. For reliable chunk extension, we employ a dynamic exploration strategy that explicitly rewards the successful completion of longer executable lengths, progressively pushing the trustworthy prediction horizon to its empirical limit. For step efficiency, we design a redundancy-aware reward that directly favors successful task completions with fewer steps while penalizing unreproducible shortcuts, effectively eliminating redundant physical actions. Extensive experiments across three benchmarks and three VLA models demonstrate that PolicyTrim improves action chunk utilization by 3$\times$ and reduces physical execution steps by 51.4%. Ultimately, our framework delivers up to a 5.83$\times$ end-to-end deployment speedup without compromising task success rates.

While federated learning enables collaborative modelling on decentralised data, standard methods merely fit historical observations. This purely observational approach is fundamentally insufficient for interventional inference and policy evaluation, as sequential actions dynamically alter future states. We propose \textbf{Fed-CausalDiff}, a federated causal diffusion framework for do-simulation. The architecture decomposes the evolution of the latent state into a global causal score function and a local confounding score function. This design enables \emph{decoupled synchronisation} (DSS), where clients aggregate only the shared causal mechanism while retaining site-specific confounders locally to handle heterogeneity. Experiments on four datasets demonstrate that Fed-CausalDiff achieves better ATE and policy-value estimation accuracy, offering a favorable trade-off between communication cost and inference fidelity.

Online planning in continuous partially observable Markov decision processes (POMDPs) using $ω$-regular specifications requires handling continuous belief dynamics within the finite symbolic memory in order to track temporal progress. Existing methods based on either direct search in belief space or predefined discrete abstractions suffer from drawbacks, e.g., lack of symbolic memory for long-horizon logical progress or difficult to certify from noisy online beliefs. As such, obtaining reliable symbolic states online from continuous observations remains a challenge. To address this issue, we introduce the Revealed Belief Automaton (REBA), an event-driven framework that advances the research from global belief-space discretization to a fundamental new way of thinking, namely online certification of revelation events. Specifically, we propose an online revelation method that, through information-theoretic gates, can dynamically analyse and establish belief abstraction from the continuous belief space by discovering reliable anchors among noisy beliefs. We then develop an incremental topology adaptation mechanism over the certified anchors to realise the online finite Belief Automaton. By combining with the $ω$-regular specification, REBA is able to support formal parity policy synthesis without a predefined discrete abstraction, which in turn can guide the Monte Carlo Tree Search process to perform online search beyond its local horizon. In addition, we design an error decomposition analysis which can assess the effectiveness and reliability of this discrete guidance for the underlying continuous POMDP. Empirical evaluations in patrolling and navigation scenarios show that REBA matches or exceeds all evaluated baselines, with primary metric gains of +17.0% to +47.4% over state-of-the-art approaches.

Reinforcement learning (RL) has become central to LLM post-training, yet the methods that dominate current pipelines, PPO and GRPO, represent only a narrow slice of what RL offers. Understanding why these methods prevail, and what alternatives exist, requires a principled examination of the design decisions that underlie any RL algorithm. This survey organizes that examination around three stages of algorithm construction. We begin with MDP creation: how the reward function, state space, action space, termination condition, and discount factor are, or could be, defined for LLM training. We then turn to exploration, covering temperature sampling, entropy regularization, intrinsic motivation, tree search, and curriculum learning. Finally, we address learning along four classical RL dimensions: model-free versus model-based, value-based versus policy-based versus actor-critic, on-policy versus off-policy, and credit assignment, including both Monte Carlo methods, which rely on full return estimates, and bootstrapping methods, which update estimates using other learned predictions. Mapping the LLM literature onto this taxonomy reveals a strikingly non-uniform distribution of research effort. Critic-free policy gradients and Monte Carlo credit assignment are densely populated, while value-based methods, off-policy actor-critic training, and bootstrapping-based credit assignment remain largely unexplored despite well-established counterparts in classical RL. These gaps represent concrete opportunities for transferring proven RL techniques to LLM training. By making these gaps explicit alongside the methods that have proven effective, this survey offers researchers in both RL and LLMs a shared framework for understanding current practice and identifying promising directions for future work.

We consider the problem of synthesizing control policies that enforce a concentration property on the state distributions of a stochastic system. We present a formalization of this problem in terms of synthesizing strategies for maintaining an entropy-based objective in Markov Decision Processes (MDPs). We first show that even relaxed versions of this problem are complexity-theoretically hard. We then present a sound and (conditionally) relatively complete method to verify and synthesize strategies for such entropy objectives. The main challenge is the non-linear nature of such objectives, and our approach addresses this by exploiting and combining ideas from convex duality and invariant synthesis. We also investigate the role of memory and randomization in ensuring entropy objectives. Finally, we implement our ideas to evaluate our approach empirically on a few illustrative benchmarks.

A finite agent, a machine's digital twin or any bounded reasoner, infers a fixed and noisy world through finite sensors, so its coherent output is a belief: a probability density over states (the Bayes posterior). Such an agent stops short of certainty, and revising a belief carries a cost. We propose a framework for belief costs based on optimal transport, motivated by these facts. We pose two postulates. P0 (the arena): a revision cost is a scalar price on optimal transport, so beliefs live in Wasserstein space. P1 (uniform pricing): one nat of knowledge costs the same metric length everywhere, the eikonal condition. Among conceivable pricing rules we study this one. Under P0 and P1 the cost metric is optimal transport conformally reweighted by Fisher information, $\tilde g_{e,U}=2(e+U)\,g_{W_2}$, and the Fisher family is a characterization: among continuous reliefs, uniform pricing is equivalent to $U=cJ$. Two consequences follow on the conformal class. Certainty sits at infinite cost-distance once the relief dominates the Fisher information, so a well-posed inference has a cost floor diverging at certainty (necessity conjectural beyond power laws). On location-scale leaves the geometry is hyperbolic, and the Stam bound places the Gaussian as the most curved one (at $e=0$). The results are geometric, in nats, and hold up to units: a change of cost unit rescales all distances and preserves every conclusion (boundary, eikonal family, hyperbolicity, Gaussian extremum), a gauge theorem; a global change of state units at $e=0$ is an isometry; the content lies in signs, rankings and ratios. Via Landauer (one nat worth $k_BT$) the cost floor becomes an energy floor: revising toward certainty would demand unbounded energy. Physics anchors the unit and enters no theorem. Removing either postulate leaves the selection open.

Model-based and model-free reinforcement learning are traditionally viewed as separate paradigms: instead of learning a model of the transition kernel $P$, model-free agents typically estimate value functions tied to a specific policy and reward. In this paper, we challenge this dichotomy by proving that value-based agents trained on a sufficiently rich set of reward functions, e.g. using goal-conditioned RL, implicitly encode a unique and accurate world model. To extract this model in practice, we introduce \textit{$P$-learning}, an inverse analogue to $Q$-learning that samples from an agent's $Q$-values, policies and rewards to decode its internal model of the environment. We then provide sufficient conditions on the type and number of goals for which agents encode the true kernel $P$, covering both stochastic and deterministic MDPs over finite or continuous state spaces. Even when our assumptions are violated, we empirically demonstrate that agents trained on a handful of reward functions encode accurate dynamics in $\texttt{Reacher}$, $\texttt{MountainCar}$ and stochastic variants of $\texttt{FourRooms}$. Surprisingly, we find that policies trained exclusively on a \texttt{Reacher} agent's implicit world model are quasi-optimal on out-of-distribution, velocity-based goals despite position-only training -- suggesting that agents contain hidden generalisation capabilities and providing a new lens into the connection between model-based, model-free, and goal-conditioned RL.

In this article, we present a robust $Q$-learning algorithm for discrete-time mean-field control problems under Wasserstein uncertainty in the common noise law. The algorithm combines a quantization-and-projection scheme with a Wasserstein dual reformulation on the common-noise space. We establish its convergence together with finite-time iteration bounds for both synchronous and asynchronous learning schemes. Numerical experiments on systemic risk and epidemic models compare the asynchronous implementation with an idealized Bellman iteration, illustrate the robustness-performance tradeoff under common-noise misspecification, and report the observed convergence behavior of the asynchronous $Q$-learning algorithm.

This paper investigates mobile target search under imperfect perceptions caused by sensor limitations, malicious jamming, or communication noise. Searchers and targets operate in a grid-shaped area with bounded mobility, leading to a dynamic interplay between search and evasion. To capture this adversarial interaction under imperfect perceptions, we adopt the partially observable stochastic game (POSG) approach, which generalizes partially observable Markov decision processes (POMDPs) by incorporating target intelligence. To handle false alarms and missed detections caused by perceptual uncertainties, we propose a novel detectability concept to determine whether a search strategy guarantees eventual detection, and provide sufficient detectability criteria based on stochastic recurrence analysis. We further develop a server-assisted distributed algorithm that utilizes the aggregative potential game structure for searchers and a KL-divergence-based reduction for target prediction. Numerical simulations validate the effectiveness of the proposed algorithm and support the detectability analysis.

In offline Reinforcement Learning, immediate rewards in logged batch data are often unobserved due to sparse or irregular record-keeping, or censored beyond certain reward values. This issue arises in practical settings, including health care and marketing. We investigate off-policy evaluation (OPE) in finite-horizon Markov decision processes when rewards are missing not at random (MNAR), which breaks ignorability and induces selection bias even after conditioning on states and actions. To address this, we formalize a reward-dependent propensity model and use future states as shadow variables to identify the full-data conditional mean reward. We further introduce a bridge function that recovers the conditional mean reward without explicitly modeling the MNAR mechanism, and estimate it via a min-max procedure to avoid double sampling. Building upon these identification results, we propose an Fitted-Q-Evaluation-style estimator that propagates the recovered rewards while allowing target policies to depend on past missingness indicators. Finally, we establish consistency and finite-sample error bounds for our OPE estimator, and show through experiments the strong performance of our method compared to existing methods on simulated and MIMIC-III Sepsis data.

Reinforcement learning with verifiable rewards has become a central tool for improving the reasoning ability of large language models, but current methods face a trade-off between simplicity and credit assignment. Group-relative methods such as GRPO avoid training a critic, but typically assign a trajectory-level advantage to every token. Actor-critic methods provide denser learning signals, but require a learned value function with its own training instability. We introduce VIMPO, a critic-free policy optimization method that derives a policy-implied value function from the optimality conditions of KL-regularized reinforcement learning. For autoregressive generation, the resulting value recurrence can be written in terms of policy-reference log-ratios and anchored by the terminal condition that no future reward remains at the end of a trajectory. This gives a simple value loss that incorporates outcome-level verifiable rewards without training a critic. The same derivation also yields a critic-free actor advantage, allowing VIMPO to separate reward incorporation through the value loss from policy improvement through a PPO-style actor update. On mathematical RLVR benchmarks, VIMPO improves over GRPO across MATH-500, AIME 2024, AIME 2025, and OlympiadBench, with especially larger gains on competition-style evaluations. Under noisy rewards, VIMPO retains a consistent advantage over GRPO, suggesting that policy-implied value optimization can provide finer credit assignment while preserving the practical simplicity of critic-free training.

Planning under uncertainty is an essential capability for autonomous robots. The Partially Observable Markov Decision Process (POMDP) provides a powerful framework for such a capability. Although POMDP-based planning has advanced significantly, its application to real-world problems is often limited by the difficulty of obtaining faithful POMDP models. We present Vectorized Online planning wIth Learned diffusion model for POMDP Agents (VOiLA), a framework that learns task-agnostic POMDP models for online planning under uncertainty. VOiLA learns transition and observation samplers using conditional diffusion models and learns observation-likelihood models for particle-based belief updates. To enable efficient online planning, the diffusion samplers are distilled into compact feedforward generators and integrated with Vectorized Online POMDP Planner (VOPP), an online POMDP planner designed to leverage GPU parallelization. Experimental results indicate the distillation strategy reduces sampling cost by up to nearly three orders of magnitude, making learned generative POMDP models practical for online planning. Evaluation of VOiLA on three benchmark problems indicate that VOiLA achieves equal or better performance than Recurrent Soft Actor Critic while using less than 10% training data, and generalizes much better to unseen environment configurations. Physical robot evaluation indicates VOiLA uses the models learned using only simulated data and generates a policy that successfully accomplish the task in 10 of 10 runs.

Effective machine learning depends not only on how we model data, but also on what data we choose to collect. While large sequence models have revolutionized data modeling, the problem of automated data selection, or "intrinsic curiosity", remains a significant challenge. Classic approaches incentivize exploration by rewarding an agent based on its "learning progress", which measures how much a newly acquired observation improves a world model's predictive ability. However, evaluating these rewards traditionally requires expensive inner loops of gradient descent updates within each trajectory, rendering them computationally impractical at scale. In this work, we investigate whether the emergent in-context learning (ICL) capabilities of sequence models can eliminate this bottleneck by serving as immediate, update-free world models. Specifically, we evaluate whether an exploration policy can be trained to maximize learning progress, using solely the prediction errors and counterfactual context manipulations of an in-context learner. We first prove that in general Markov decision processes, this is in fact impossible in an unbiased way: the resulting intrinsic rewards either suffer from nuisance terms that bias their estimation of true learning progress, or they cannot be implemented using an in-context learner's prediction errors. Conversely, we prove a positive result for a broad subclass of non-temporal settings, encompassing active learning and Bayesian Experimental Design: here, ICL-derived rewards successfully bound and asymptotically converge to the true learning progress. We corroborate our theory with controlled experiments across continuous and symbolic environments, demonstrating that our ICL-driven framework successfully trains curious data-collection policies that explore optimally.

We present Pareto Q-Learning with Reward Machines (PQLRM), a multi-objective reinforcement learning algorithm for tasks whose reward structure is specified by a set of reward machines (RMs). PQLRM combines Pareto Q-Learning (PQL), which maintains sets of vector-valued Q-estimates to approximate the Pareto front, with enhancements from Q-Learning with Reward Machines (QRM), which exploits the factored automaton structure of the reward signal. This yields a multi-policy algorithm that remains sample-efficient under non-Markovian, RM-encoded rewards. Experimental trials show that PQLRM converges faster than a naive PQL baseline applied to the cross-product MDP and can synthesize Pareto-optimal policies that QRM cannot.

Sequential decision problems often exhibit an asymmetric evolution of information and decision flexibility: as a decision cycle unfolds, the agent receives richer information while feasible actions expire due to operational cutoffs, commitments, or resource constraints. Standard MDP formulations typically flatten this structure into stage-dependent state descriptions and action masks, thereby obscuring the nested information--action asymmetry that determines which decisions are urgent and which can be deferred. We introduce Maturing Markov Decision Processes (MMDPs), a formulation built around this information--action asymmetry. We characterize one of its key consequences through an expiring-action priority principle, which identifies the actions that must be resolved before the next stage. Motivated by this structure, we develop a structure-aware reinforcement learning framework with stage-aware policy design, expiring-action abstraction, and search-augmented learning with distillation. Experiments on a controlled multi-supplier replenishment problem, simplified cash-management environments of increasing complexity, and a production-scale simulator show that explicitly modeling this asymmetry improves learning efficiency and becomes increasingly valuable as decision problems scale.

Decision making in lithium production is challenging, whether from an investor's perspective or a strategic production standpoint. Determining which mines to open and when to open them involves not only geological and price uncertainties, but also complexities around the choice of extraction method, from direct lithium extraction to hard rock mining. Prior work explored models of this problem and different methods to optimize mining decisions; these models did not account for uncertainty in pricing, uncertainty in demand, or different mining technologies to extract lithium. Incorporating different pricing models and extraction technology into these models enables more robust strategies for determining not only when and where to open a mine, but also which method of production to pursue. We frame the problem as a partially observable Markov decision process (POMDP) and solve using belief state planning methods to get optimal decision making. In our study, we show that POMDP solvers outperform human inspired heuristics by dynamically adapting to shifting lithium price regimes (static, linear, exponential, and stochastic) through belief state planning and explicit uncertainty management. By optimally sequencing exploration, production, and technology choice, the framework achieves higher demand fulfillment and more balanced economic environmental outcomes over the projects lifetime in all different pricing and deposit scenarios.

In this paper, we study a sequential workforce management problem in a contingent labor setting with uncertainty in both worker production and labor supply. A firm seeks to maximize cumulative profit by maintaining an active team of fixed size while learning worker productivity over time. We emphasize two critical operational frictions in this problem: replacing workers is costly, and workers may not be available immediately for hiring because of, for example, prior job commitments, scheduling constraints, or onboarding procedures. Thus, hiring decisions take effect only after a random delay. We formulate this problem as a stochastic multi-play bandit with costly switching and delayed actions, and develop a learning-based hiring policy, DR-UCB (DelayedReplacement-UCB), that makes replacement and hiring decisions sequentially through learning cycles. In each cycle, the policy uses real-time production data to determine when to initiate workforce changes and which workers to replace and hire. We show that the leading-order regret of the proposed policy matches its lower bound in its dependence on the time horizon. Our numerical experiments show that DR-UCB outperforms benchmark policies.

Temporal difference (TD) learning with linear function approximation is a core method for policy evaluation. Its classical continuous-time description is an ordinary differential equation (ODE), which captures the asymptotic mean dynamics but neglects stochastic fluctuations determining the error floor. We introduce a stochastic differential equation (SDE) approximation for linear TD(0) under Markovian noise. The resulting model distinguishes the contraction dynamics governed by the projected Bellman operator from the influence of Markovian sampling. As a consequence, the model explains the constant-stepsize error floor through the interaction between Markovian long-run covariance and the contraction geometry of the projected Bellman operator.

Fairness is an important aspect of decision-making in multi-objective reinforcement learning (MORL), where policies must ensure both optimality and equity across multiple, potentially conflicting objectives. While single-policy MORL methods can learn fair policies for fixed user preferences using welfare functions such as the generalized Gini welfare function (GGF), they fail to provide the diverse set of policies necessary for dynamic or unknown user preferences. To address this limitation, we formalize the fair optimization problem in multi-policy MORL, where the goal is to learn a set of Pareto-optimal policies that ensure fairness across all possible user preferences. Our key technical contributions are threefold: (1) We show that for concave, piecewise-linear welfare functions (e.g., GGF), fair policies remain in the convex coverage set (CCS), which is an approximated Pareto front for linear scalarization. (2) We demonstrate that non-stationary policies, augmented with accrued reward histories, and stochastic policies improve fairness by dynamically adapting to historical inequities. (3) We propose three novel algorithms, which include integrating GGF with multi-policy multi-objective Q-Learning (MOQL), state-augmented multi-policy MOQL for learning non-statoinary policies, and its novel extension for learning stochastic policies. We evaluate our algorithms across various domains and compare our methods against the state-of-the-art MORL baselines. The empirical results show that our methods learn a set of fair policies that accommodate different user preferences.

Robots operating in non-stationary environments must continually adapt their policies as the dynamics drift, but onboard energy and compute budgets cap how often a full state estimation and re-planning step can be performed. This raises a question: \emph{when}, along a horizon, should a robot spend its limited budget? We formulate this problem in time-varying Markov decision processes (TVMDPs) with a known bound on the rate of transition drift. We model execution as a \emph{skip-update} scheme in which, at chosen update times, the agent estimates the transition kernel by maximum likelihood and computes a finite-horizon policy, and between updates reuses this policy under a propagated state estimate. We analyze the dynamic regret of this scheme and show how it grows during skip intervals in terms of the properties of the TVMDP and the skip lengths; the resulting bound answers the opening question via an online, regret-guided update rule that allocates the budget adaptively. We evaluate the rule in a simulated Mars-rover navigation task with time-varying slip dynamics and on a Crazyflie quadrotor in indoor obstacle fields. Adaptive allocation outperforms other budgeted baselines.

We study the operator-theoretic core of Q-learning in continuous-time stochastic control with continuous states and actions. In value-based reinforcement learning, each Q-learning or DQN update is built from a Bellman optimality target; our analysis isolates this target in a diffusion setting and studies its regularity and approximation complexity. Under uniform ellipticity and Hölder-regular coefficients, we show that a Bellman update maps bounded inputs into an anisotropic regularity class, smoothing the state variable while leaving only Lipschitz dependence on the action variable. This yields a compact family of Bellman iterates and motivates a tensor-product DeepONet architecture adapted to the mixed regularity of the problem. We then derive explicit approximation and resource bounds, together with a stiffness--complexity trade-off as the time step $δ\to 0$. The resulting theory makes a direct contribution to Q-learning theory at the level of Bellman target regularity and approximation in continuous stochastic control. At the same time, we do not claim a full convergence theorem for practical sampled Q-learning with exploration, replay, and stochastic gradient updates.

We study inverse reinforcement learning for discrete-time, infinite-horizon mean-field games (MFGs) under an average-reward criterion. Expert demonstrations are assumed to arise from a stationary mean-field equilibrium under an unknown reward, and the goal is to recover a policy explaining the observed behaviour via the maximum causal entropy principle. We formulate the inverse problem by enforcing consistency with the expert mean-field term and long-run feature expectations, treating two reward classes within a unified occupation-measure framework. For finite-dimensional linear rewards, we give a convex dual reformulation with an explicit log-partition objective, and prove smoothness and curvature properties justifying constant-step-size gradient descent. For infinite-dimensional RKHS rewards, we develop a Lagrangian relaxation whose inner-maximising policy is characterised by a soft Bellman equation. The main obstacle is the absence of a discount-factor contraction. We resolve this by introducing a minorisation-based sub-stochastic kernel that yields a strict contraction of the soft Bellman operator. We establish Fréchet differentiability and Lipschitz smoothness of the log-likelihood score, leading to a gradient ascent algorithm with convergence guarantees. Two numerical examples, a malware-spread MFG and an RKHS-based consumer-choice model, show that the recovered policies closely match expert behaviour.

While there is an extensive body of work characterizing the sample complexity of discounted cumulative-reward MDPs, finite sample analyses for average-reward MDPs have been limited, and most existing works rely on restrictive assumptions such as ergodicity or access to a generative model. In this work, we establish the first finite sample complexity guarantees from a single trajectory for weakly communicating average-reward MDPs. To this end, we study the dynamics of a single trajectory in weakly communicating MDPs and based on this analysis, we develop novel model-free methods. Notably, our value-based and policy-based methods provide finite sample complexity guarantees of $\widetilde{O}(1/\varepsilon^2)$ and $\widetilde{O}(1/\varepsilon^4)$ from a single trajectory in weakly communicating MDPs, respectively. Furthermore, we introduce the first model-free method that requires no prior knowledge of problem-dependent quantities for communicating MDPs.

Recent advances in tool-integrated language agents have significantly improved their ability to solve complex reasoning tasks. However, existing alignment methods predominantly focus on maximizing task accuracy, while overlooking auxiliary objectives such as tool-use efficiency, which are essential for practical deployment. To address this gap, we introduce ParetoPO, a two-stage multi-objective optimization framework for aligning tool-using large language models (LLMs) under competing objectives. In the first stage, ParetoPO leverages hypervolume-guided dynamic scalarization to adapt reward weights based on global Pareto frontier progress. In the second stage, it replaces scalarized learning signals with Pareto-ranking-based advantage computation, promoting nondominated trajectories through dominance-aware credit assignment. This design enables fine-grained, action-level optimization across multiple conflicting objectives. Experimental results on mathematic reasoning and multi-hop QA tasks show that ParetoPO consistently discovers policies with superior accuracy-efficiency trade-offs compared to static and heuristic baselines.

Many games rely on storytelling combined with systems that track levelling, NPC behaviour, and consequence simulation; bridging tightly-authored narrative with deeply-simulated worlds -- most acute in sandbox and open-world settings -- has been prohibitively expensive. LLM-driven worlds open a new path: a single harness can coordinate numerical state, narrative voice, storytelling pacing, and rule logic together. Realising this requires the LLM system to sustain a persistent world (who is where, what has just happened, what is currently true), which today's deployed systems do not: the narrative voice asserts state in free prose without any validated representation, so a fully autonomous game engine remains infeasible. We treat this as an architectural choice, not a limitation of language models, and report work in progress on a framework -- orchestrated reality -- that makes the world a canonical object owned by a singleton orchestration agent analogous to the tabletop-RPG Game Master (GM). We formalise an LLM-driven game world for a human player as a Parameterized-Action POMDP: state is a tree of canonical JSON entities, actions decompose as $a=(k, x_k)$ (a discrete intent kind plus structured JSON parameters), the agent observes only a narrative projection $o=O(s)$ of state, and the transition kernel $F$ is an LLM-driven Plan-Diff-Validate-Apply (PDVA) pipeline that commits schema-validated, content-hashed JSON deltas. We give the formal model, a JSON-state example, a worked single-turn example, and a catalogue of 15 illustrative incidents drawn from a real deployment showing the framework in action. Empirical validation through a planned human player study -- together with multi-NPC concurrent agency and deployment as an RL environment -- is situated as future work.