Markov Decision Processes

Deep Q-Networks (DQNs) learn value functions through bootstrapped temporal-difference updates, where future returns are approximated using a greedy maximization over next-state action values. While effective, this aggregation rule is inherently sensitive to estimation noise: when Q-values are uncertain, the maximization operator deterministically favors the largest estimate, regardless of its reliability, leading to amplified errors through bootstrapping. In this work, we propose the \textbf{S}uccessor Rollout \textbf{A}ggregation \textbf{D}eep \textbf{Q}-Network (SADQ), a simple modification to Q-learning that regularizes how the TD target is formed. SADQ uses one-step rollout predictions from a learned dynamics model to guide the comparison among candidate next-state actions, introducing additional structure into the aggregation step without altering the underlying learning framework. The resulting mixed Bellman update attenuates unreliable maxima while preserving the standard fixed point under diminishing model error. We provide theoretical analysis showing that SADQ reduces bootstrap-induced overestimation in a pointwise manner. Empirically, SADQ consistently improves training stability across classical control tasks, real-world vector-based environments, and Atari benchmarks when compared to strong DQN variants.

Preference-based reinforcement learning (PbRL) for general stochastic MDPs often requires training a reward model. Existing reward-model-free methods are either restricted to bandits or deterministic MDPs, such as DPO or P3O, or use zeroth-order, gradient-free optimization, which in general exhibits a slower convergence rate than gradient-based algorithms. Furthermore, existing reward-model-free preference-based RL algorithms almost exclusively use trajectory-level feedback, which can require significant effort from a human evaluator when trajectories are long. On the other hand, segments are much shorter, so they are easier to compare and evaluate. In this paper, we introduce a novel reward-model-free, critic-free, and gradient-based PbRL algorithm compatible with segment preferences named Segment Pairwise Proximal Policy Optimization (SP3O). SP3O utilizes segment-level preference feedback to construct an accurate policy value difference estimator via off-policy importance sampling, and then uses the estimator to compute the policy gradient via a PPO-type loss function. We provide a theoretical basis for the algorithm and analyze the tradeoff in choosing the segment length. We also evaluate it experimentally against other PbRL/RLHF algorithms in robotic control and LLM finetuning settings to show its improved performance, especially in long-horizon tasks.

Counterfactual inference approaches for sequential decision-making typically assume deterministic causal models, where all randomness stems from latent variables. However, Markov Decision Processes (MDPs) are inherently stochastic. We address this by formalising counterfactual policy optimisation under probabilistic nondeterministic causal models, which properly separates latent confounding from irreducible stochasticity, and here propose a first practical optimisation problem for identifying robust counterfactual policies under a sensitivity analysis framework. We validate our approach on a sepsis treatment simulator, where diabetes status acts as a hidden global confounder.

Reinforcement learning is a subfield of machine learning that studies how an agent interacts with an environment in order to extract as large a reward as possible. A standard approach to study such interaction is through Markov Decision Processes (MDPs) and the task of choosing an optimal policy --- a function that tells the agent which action to take. In this work, we study two types of MDPs --- finite-horizon and infinite-horizon discounted --- and propose new quantum algorithms for computing approximate optimal policies. Our quantum algorithms are based on a new combination of standard value iteration and quantum subroutines like quantum mean estimation and quantum maximum finding, overall enhanced with techniques from sample-optimal classical algorithms. Our resulting query complexities improve upon previous works, thus approaching already established quantum lower bounds.

Effective model-based reinforcement learning in stochastic environments requires planning that accounts for predictive uncertainty. Propagating full state distributions analytically offers a principled way to do this, but has traditionally required restrictive policy or reward structures to remain tractable. Consequently, modern deep reinforcement learning has largely retreated to either stochastic sampling, which introduces significant target variance, or deterministic point estimates that ignore predictive covariance entirely. We investigate whether distribution-aware planning is possible without these constraints. Using a quadratic action-value parameterization, we first reduce the Bellman backup to an expectation over the state-value function alone; the key idea is then a compatibility principle between the predictive transition distribution and the value function class, under which this expectation is analytic in the distribution's moments. We instantiate this principle with a Gaussian transition model paired with a radial-basis value function, yielding a closed-form backup that propagates both predictive mean and covariance. Empirically, our approach reduces target variance and yields well-calibrated predictive uncertainty under stochastic observations in continuous control, providing a principled framework for planning with learned distribution models.

Sequential decision-making in real-world applications often involves uncertainty about the environment's model. Uncertain Markov decision processes (UMDPs) represent the possible environments as a set of MDPs with shared states and actions but potentially different transition probabilities and rewards. Optimizing a single policy across all possible MDPs may sacrifice performance, while preparing an individually optimized policy for every MDP may violate operational, regulatory, or interpretability constraints on the number of policies that can be prepared and deployed. We consider settings in which model uncertainty is resolved shortly before execution, allowing the most suitable policy to be selected from a limited set prepared in advance. We introduce $k$-adaptable policy synthesis, which optimizes such a set of $k$ policies under a minimax-regret objective. We prove that the problem is NP-hard and develop KAPS, an exact nested branch-and-bound algorithm with problem-specific bounds and heuristics. KAPS jointly optimizes which MDPs share a policy and the policies themselves. Experiments across various UMDP benchmarks show that the largest reduction in regret consistently occurs when increasing from one to two policies. In the single-policy setting, KAPS is competitive with existing methods in solution quality and proves optimality substantially more often.

In noisy social dilemmas, intended actions are stochastically corrupted before execution, so an observed defection may reflect hostile intent or action error. Standard Markov Decision Process (MDP) formulations treat executed actions as states, structurally precluding this distinction and causing systematic over-retaliation. We introduce a Partially Observable MDP (POMDP) formulation encoding opponent intentions as latent states and executed actions as noisy observations, solved within the active inference (AIF) framework with a cost function that decomposes into epistemic and pragmatic components that jointly address inferring current intent and learning how intent evolves. In the Iterated Prisoner's Dilemma with symmetric noise, we derive a critical noise threshold governing cooperation collapse, connecting it to a fixed-point condition on learned priors. Experiments reveal that the value of intention inference is context-dependent: the POMDP provides consistent advantages against conditionally cooperative opponents, but mutual intention inference under sufficient noise produces correlated belief-driven collapse. The advantage is specific to games where intent attribution is decision-relevant.

In this paper, we consider the parameter synthesis and optimization problem for parametric Markov decision processes (pMDPs), the extension of classical MDPs where exact probability values are replaced by parametric expressions. Computing the rational function $f_{\lsf}$ that maps parameter valuations to the satisfaction value of a PRCTL property $\lsf$ is a computationally expensive task, particularly for pMDPs where the optimal policy may vary across the parameter space. We adopt the \emph{scenario approach} to efficiently synthesize a probably approximately correct (PAC) approximation $\ApproxFunOfProperty{f}$ of $f_{\lsf}$: by sampling parameter configurations and solving a linear program, we obtain a polynomial approximation whose error margin $\margin$ is guaranteed, with prescribed confidence, for all but an $\errorRate$-fraction of the parameter domain under the sampling distribution. We further show how this PAC framework can be combined with statistical model checking (SMC), enabling the analysis of black-box parametric models. Building on the PAC approximation, we integrate the DIRECT (DIviding RECTangles) algorithm for derivative-free global optimization over the parameter space. We establish conditional optimality-gap guarantees: under explicit Lipschitz and PAC-good-set assumptions, the difference between the true optimum $f_{\lsf}(\parameters^{*})$ and the value found by DIRECT is bounded by a partition-diameter term and, in the PAC case, an additional approximation-error term. An empirical evaluation on 2997 benchmarks focuses on the new DIRECT-based optimization component. The results show that DIRECT variants solve fewer instances than the scenario optimizer, but on their common successful instances they often return slightly better objective values and usually run faster, while remaining close to the scenario values within the PAC margin.

Multi-step returns accelerate reward propagation in off-policy reinforcement learning, but couple the evaluation of each decision to the suboptimal logged actions that follow it, inducing a pessimistic bias that grows with the horizon. We propose Expectile $n$-step Q-learning (ENQ), which replaces the symmetric $n$-step temporal-difference (TD) loss with an asymmetric expectile loss on the action-value error, with expectile level $τ$ as the only method-specific hyperparameter added beyond $n$-step TD. We prove that the ENQ operator is a $γ^{n}$-contraction. Under deterministic dynamics, at $τ=1$, its bias vanishes at the optimal action-value function $Q^*$ on covered in-support pairs, and the corresponding fixed point satisfies the separation-$n$ instance and its multiples of the lower-bound inequality used by Long-Horizon Q-learning (LQL). Under stochastic dynamics, the operator bias admits two-sided bounds with horizon-independent noise constants. Using a single expectile level $τ=0.8$ and a fixed backup horizon across 27 manipulation and navigation task instances, ENQ is competitive with LQL on aggregate, achieves higher measured training-step throughput in our profiling study, and benefits more from a ten-critic ensemble in a controlled scaling experiment.

Programmable-payment protocols such as x402 enable per-request micropayments, but they do not determine which payable service an autonomous agent should buy under a finite wallet. We formulate this buyer-side problem as agent-native payment decision-making: contextual provider selection under wallet pressure, chosen-only paid feedback, and changing market conditions. We propose 402Pilot, a protocol-agnostic buyer-side decision layer between autonomous agents and payment execution that implements purchasing policies for selecting among payable providers. We instantiate it with PA-DCT, a payment-aware discounted contextual Thompson-sampling policy that adapts purchasing decisions under wallet pressure while learning from post-payment feedback. To evaluate buyer-side payment policies, we introduce 402Pilot-Bench, a frozen-replay benchmark spanning 823 tasks, five heterogeneous provider pipelines, and three market regimes, each evaluated over 30 paired seeds. PA-DCT achieves the strongest fixed-wallet adaptive trade-off among non-oracle policies: it maintains competitive service quality while spending only 39 to 43 percent of the wallet and reallocates spending as market conditions change. It attains the best non-oracle PA-gap/T under the price shock and the best mean and worst-case ranks across the nine scenario-metric combinations of quality, ROI, and PA-gap/T. Comparisons with learning baselines and component ablations further support the effectiveness and design of the proposed decision policy. These results suggest that programmable payment must be complemented by buyer-side decision-making capable of learning service value and adapting purchasing decisions accordingly.

For a trading desk, residual climate hedging valuation adjustment (HVA) is the climate cost left after its inherited hedge and any admissible overlay have been taken into account; it therefore cannot be inferred from a stand-alone stress loss. We obtain this residual by comparing paired climate-on and baseline worlds and reoptimizing the overlay for each hedge universe, which also turns hedge-instrument discovery into a valuation problem: an instrument is useful to the extent that it lowers the optimized residual cost. The linear-Gaussian case has an exact finite-horizon Riccati solution; Climate-Dyna starts from that hedge and learns the remaining nonlinear correction from paired world-model rollouts, with an independent gate deciding whether to deploy the update. In a public-data-calibrated semi-synthetic EU ETS study, crediting the inherited hedge lowers the mean climate charge from 1.517 to 0.906, and the learned overlay lowers it to 0.831 against a 0.821 exact floor; residual Dyna cuts regret by 93% relative to replay with one quarter as many trajectories, while adaptation from only 25 target transitions retains 60.7% of the exact-assisted gain.

Modern network policy control maps intent to sequential placement-control decisions. Bellman-style policy optimization primarily asks which action to optimize, while constraints are commonly handled through penalty, barrier, or Lagrangian mechanisms. We observe that before a value function can certify the best deployment, intermediate signals may already identify many candidates that should be excluded from further optimization. This motivates a complementary direction: \emph{Learning Not to Optimize}. Before a value function is accurate enough to select the best placement-control decision, intermediate signals may already show that candidates are equivalent under state--intent relabeling (quotienting), lead to a uniformly worse future state (dominance), or violate executable network laws (residual screening). \LNOQRD{} uses these computed or learned signals as a shadow process to reshape the domain on which primal policy optimization is performed, thereby reducing the action space. We prove lossless quotienting and dominance under explicit equivariance and monotonicity conditions, bound frontier size and ranking cost, and quantify losses from approximate certificates and primal estimates. Experiments show that \LNOQRD{} reduces small-instance candidates by $75.9\%$ while retaining $90.8\%$ near-oracle coverage and, on large instances, achieves the highest utility and intent satisfaction, the lowest hard-law violation and post-generation latency, and a $73.0\%$ average reduction among candidate-based baselines.

Constrained Markov Decision Processes (CMDPs) provide a natural framework for reinforcement learning in safety-critical applications, where agents maximize long-term reward while satisfying long-term constraints. Although primal-dual actor-critic methods with linear critics are well understood, extending order-optimal convergence guarantees to neural critics in average-reward CMDPs has remained open. The main challenge is a fundamental bias-cost trade-off in neural critic estimation: under Neural Tangent Kernel (NTK) analysis, reducing critic bias substantially increases critic optimization cost, preventing order-optimal convergence in the primal-dual framework. We resolve this bottleneck by introducing a hierarchical Multilevel Monte Carlo (MLMC) neural critic that performs debiasing simultaneously across trajectory sampling and critic optimization. The resulting estimator attains the bias of a long critic optimization run with only logarithmic expected sample cost. Building on this estimator, we develop a primal-dual Natural Actor-Critic algorithm that achieves both an optimality gap and a constraint violation of order $\tilde{O}(T^{-1/2})$. This establishes the first order-optimal convergence guarantees for infinite-horizon average-reward CMDPs with general policy parameterization and neural critics, while eliminating the need to know the underlying mixing time. Our results are novel even in the unconstrained setting.

Many algorithms spend an internal resource before returning a decision and are evaluated only by the quality of that terminal output. We formalize such procedures as terminal computation-allocation problems: costly computations produce observations, update beliefs about a latent environment, and matter only through terminal decision loss. Bellman equations characterize optimal allocation under fixed budgets, priced computation, and exact certification. We then relate value of computation (VOC) to information. Mutual information equals myopic VOC under log loss, whereas under simple regret VOC is a knowledge-gradient quantity; moreover, information gain can rank computations arbitrarily poorly, although it gives a one-sided upper bound on VOC. Bandit pulls, tree simulations, and node expansions illustrate the same model under different computation topologies. Finally, under an explicit frontier-resolution and heuristic-error model, maximizing approximate VOC recovers weighted A*, with A* and greedy best-first search as limiting cases. The theory identifies a shared decision problem without asserting that one acquisition rule is universally optimal.

Assistance games formalize human-robot collaboration under asymmetric information: the human knows the goal, while the robot must infer it from observation and interaction in order to assist effectively. In general, computing optimal assistance game strategies online is intractable, since exact solutions require planning in a POMDP. We identify a class of assistance games in which pragmatic-pedagogic reasoning resolves goal uncertainty in a single time step, rendering the full-horizon game exactly solvable by a tractable best-response procedure. Within this class, we show that mainstream inverse optimal control exhibits an inference ceiling that hinders alignment, while pragmatic-pedagogic reasoning overcomes this barrier by immediately disambiguating goals through actions that look equivalent under task execution alone. Finally, we validate our theoretical results and proposed method on a simple collaborative block-building example.

An agent acting under partial observability must retain a recursively updateable statistic of history that restores the Markov property, but the smallest such statistic is generally unknown. We characterize this minimal Markov sufficient statistic for holonomy-cover decision processes, a structured POMDP class in which the visible dynamics are Markov and every realized visible transition applies a fixed permutation to a hidden mode. In particular, we construct the stable quotient, the coarsest observation-wise abstraction preserving one-step rewards and quotient successors, and prove that the pair of the current observation and stable class forms an exact finite Markov state. When the current class is correctly initialized, exact class tracking requires exactly the minimal memory symbols, in the sense that under reachability and pairwise decision separation at a maximizing observation, no arbitrary finite-memory controller can use fewer. Under resettable diagnostics, nearest-prototype class inference has exponentially decaying error, and a calibrate-then-restart reduction transfers finite-MDP guarantees to the recovered state. The results enable \emph{Holonomy Memory Reinforcement Learning}. It represents memory by the current stable class, updates it through ordered edge transports, identifies local class coordinates when diagnostics are available, and applies a standard finite-MDP RL backbone after synchronization. Experiments recover an exact compression from raw states to quotient states and achieve perfect paired-order accuracy with three decision-time memory states, matching the quotient oracle and outperforming the non-oracle baselines.

Markov Decision Processes (MDPs) are widely used as decision-making models, commonly specified over factored state spaces through state variables and their valuations. The exponential blowup in the number of states renders many reasoning tasks in MDPs challenging. Abstractions are promising techniques to reduce MDPs and thus mitigate scalability issues. In this work, we introduce a notion of causality on factored MDPs and a novel property-driven causal abstraction technique that retains many characteristics of the original MDP model. For this, we rely on causal relations over state variable predicates and identify those states that share the same reasons for fulfilling or violating a given abstraction property. We theoretically and empirically compare various causal MDP abstractions using different model types such as MDPs, interval MDPs, or stochastic games. Our evaluation demonstrates the potential of our approach: For several standard benchmarks, we obtain small abstractions that allow us to compute near-optimal policies for the original MDP. Furthermore, our causal abstractions often generalize to related large-scale MDP models.

Current spacecraft collision-avoidance operations rely on intermittent ground-station contacts, requiring operators to plan with delayed and asynchronously updated information. Consequently, maneuvers must be planned with only intermittent information sharing between operators, raising the question of how much coordination is needed to achieve collision-avoidance performance comparable to centralized planning. Although decision-theoretic approaches such as partially observable Markov decision processes (POMDPs) capture the sequential and uncertain nature of collision avoidance, existing multiagent extensions typically assume either continuous information sharing or communication models that do not reflect operational ground-station constraints. To explicitly model this intermittent information availability, we formulate the spacecraft-to-spacecraft collision avoidance problem as a semi-decentralized POMDP (SDec-POMDP), where we govern information propagation directly by realistic ground-station visibility windows. Joint maneuver policies are computed using approximate Recursive Small-Step Semi-Decentralized A* (RS-SDA*), following the state-of-the-art A*-based lineage for decentralized multiagent planning. Across a representative suite of conjunction scenarios, semi-decentralized planning recovers near-centralized maneuver quality while requiring 28.5% fewer synchronization events than continuous coordination. Comparisons with representative rule-based operator heuristics further show that communication-aware planning more consistently achieves the desired operational miss-distance band while minimizing unnecessary trajectory deviation. Together, these results establish a practical planning framework for autonomous collision avoidance under realistic intermittent communication, bridging the gap between idealized centralized coordination and fully decentralized planning execution.

Over the past decade, decision trees have been used to represent controllers (a.k.a. policies) in an explainable way, with dtControl2 as a current state-of-the-art tool. However, for systems that are large or have many corner cases, even such representations tend to be too complex and not human-comprehensible. Unfortunately, reducing the size of the decision tree is not straightforward, as missing just a single crucial case might result in an incorrect controller. We tackle this issue in the setting of Markov decision processes, extending dtControl2 by "$\varepsilon$" functionality: Given an allowed imprecision $\varepsilon \geq 0$, we construct a smaller decision tree, distilling the essence of the controller, while still guaranteeing its $\varepsilon$-optimality. This enables us to provide tunably simpler explanations, omitting a controllable amount of detail. Our tool constructs decision trees that are orders of magnitude smaller than the state of the art.

This paper investigates a hybrid reinforcement learning setting in tabular Markov Decision Processes (MDPs), where an agent aims to learn an optimal policy by combining online interactions with a target environment and offline data from a source environment. A central challenge is that offline data may be collected from outdated environments with shifted transition dynamics, making naive integration of historical data ineffective. To address this, we propose a unified algorithmic framework featuring two algorithms: MIN-UCB-VI for regret minimization and MAX-LCB-VI for best policy identification. Both algorithms leverage fine-grained bias information to more effectively exploit offline data under general transition shifts. We provide theoretical guarantees for our framework, including both instance-dependent and independent upper bounds on regret and sub-optimality gap. Furthermore, we establish matching lower bounds to demonstrate the optimality of our approach and validate our theoretical findings through extensive experiments.

We revisit contextual optimization from the perspective of policy class design. A desirable policy class should be expressive enough to learn rich context-decision relationships, should enforce hard feasibility constraints rather than soft penalty terms, and should remain smooth enough for gradient-based training on downstream decision losses. Existing approaches usually emphasize only part of these requirements. We propose Legendre-regularized policies, which parameterize decisions as solutions of regularized optimization problems over the original feasible region. This construction yields policies that are feasible by construction and differentiable with respect to learned latent parameters. We prove that the associated optimizer map is single-valued, maps onto the relative interior of the feasible set, admits an explicit Jacobian, is Lipschitz continuous, and can be made arbitrarily smooth. We also establish a universal approximation result showing that the proposed class can approximate any continuous feasible policy on compact context sets. The framework unifies explicitly regularized optimizers and implicit perturbation-based smooth optimizers. Experiments on contextual newsvendor and resource allocation problems show that our approach improves prescriptive performance relative to the benchmark methods.

We study a restless multi-armed bandit (RMAB) problem for a stochastic deadline scheduling application. RMAB problems are solved using the Whittle index policy. The goal in RMAB is to maximize the expected cumulative discounted reward maximization. The Whittle index policy maximizes reward, but is not fair among two classes. In this paper, we introduce fairness criteria and study an outcome-fair model for RMAB which allows fairness for jobs and users structurally disadvantaged demographic classes. We formulate an outcome fair stochastic deadline scheduling problem as RMAB, and we develop the outcome fair Whittle index policy. We define a virtual queue mechanism that dynamically enforces long-term completion rate guaranties across demographic groups. We analyze a standard Whittle index policy and the outcome-fair index policy. We demonstrate the performance of our algorithms with numerical examples. We compare policies---Whittle index policy (no fairness), input-fairness Whittle index policy, outcome fair Whittle index policy. We observe that the outcome-fair Whittle index policy provides better fairness among classes compared to other policies. We demonstrate a trade off between fairness and profit. This decreases as the server capacity increases.

Developing efficient function-approximation methods for policy evaluation is a fundamental challenge in risk-aware reinforcement learning. Existing approaches either focus on restrictive classes of risk measures or rely on access to a simulator, limiting their applicability in fully online settings. In this work, we propose computationally efficient online learning algorithms for policy evaluation in Markov decision processes (MDPs) with dynamic utility-based shortfall risk (UBSR) measures under linear function approximation. Specifically, we introduce the UBSR-TD algorithm, establish conditions under which it converges almost surely, and develop several variants designed to accelerate convergence. Our formulation shows that existing policy evaluation algorithms for risk-neutral MDPs can be readily adapted to dynamic UBSR settings by incorporating a loss function into the temporal-difference error. Numerical experiments support our theoretical findings, and an application to a perishable inventory management problem with shelf-life uncertainty demonstrates the practical effectiveness of the proposed methods.

Natural Policy Gradient (NPG) is a well-established Reinforcement Learning algorithm that underlies widely used methods such as Trust Region Policy Optimization and Proximal Policy Optimization, both of which have demonstrated strong empirical success. In this paper, we study exact NPG in finite-horizon Markov Decision Processes with known dynamics and horizon-dependent transition kernels. We provide the first finite-time convergence guarantees for this algorithm in this setting, for which we consider both constant and increasing step size regimes. With a constant step size $η_t=η$, we prove that NPG converges sublinearly with a rate of $\mathcal{O}(H^{2}/t)$ after $t$ iterations, where $H$ is the horizon length. We also extend this constant step size analysis to linear MDPs in an exact population-projection oracle under a full support projection distribution, recovering the same sublinear rate as in the tabular setting. Furthermore, with increasing step sizes, we prove that this algorithm achieves a linear convergence rate of $\mathcal{O}\left(\left(1-\frac{1}{\vartheta_ρ}\right)^t\right)$ for a problem-dependent constant $\vartheta_ρ> 1$, and the horizon-only robust schedule of the form $η_t=η_0(H/(H-1))^t$ where $η_0>0$ and $H \geq 2$, attains this same geometric rate.

Remaining useful life (RUL) prediction and failure-mode classification are central tasks in predictive maintenance. Many data-driven pipelines use fixed-window supervised learning with complete terminal labels; such routes do not naturally encode the temporal recursion linking successive degradation-state predictions when observations are partial or unit identities are unavailable. We formulate prognostics as vector General Value Function (GVF) prediction on an absorbing degradation process, treating RUL and failure-mode probabilities as temporally consistent targets rather than independent window-level labels, and estimate them with a multi-step temporal-difference estimator, TD($n,λ$). Supporting theory identifies the Bellman fixed point of the vector GVFs, characterizes the linear projected-TD limit and its relation to complete-return Monte Carlo regression under realizability, and explains when bootstrapped TD targets are less variable than Monte Carlo returns. On an event-triggered multimode simulation and NASA C-MAPSS label-scarce stitch data, TD improves RUL and failure-mode prediction relative to a supervised same-backbone Monte Carlo control, especially under scarce complete labels. Practically, fragmented, identity-free degradation records can contribute local Bellman transitions instead of being discarded until complete run-to-failure labels are available.

We introduce trajectory-regularized stochastic optimal control (TRSOC), which augments standard stochastic optimal control (SOC) with a Kullback--Leibler (KL) divergence between controlled and reference trajectory distributions. Using Girsanov's theorem, the trajectory KL reduces to a quadratic drift mismatch penalty, yielding a modified running cost that preserves the dynamic programming (DP) structure. We derive the corresponding Hamilton--Jacobi--Bellman (HJB) equation and characterize the optimal policy. In the linear-quadratic (LQ) setting, the formulation admits a closed-form solution with an augmented control cost. Experiments show that the regularization parameter induces a trade-off between performance-driven and reference-preserving behavior, including cases with reference dynamics learned from offline data.

Path planning under partial observability remains challenging because an agent must make long-horizon navigation decisions from only locally bounded observations. Nevertheless, historical trajectories contain reusable experience-guided directional preferences. Classical planners, however, typically solve each instance from scratch and lack an explicit mechanism to exploit such transferable decision knowledge, often leading to redundant node expansions and locally myopic search behaviors. Motivated by this limitation, this paper proposes ImiPath, a prior-guided learning framework that distills reusable spatiotemporal decision priors from demonstration trajectories and uses them as experience-informed directional guidance to bias planners toward reliable and promising search directions under partial observability. Specifically, ImiPath first constructs a local spatiotemporal observation representation, which encodes the spatial information of the local environment and the temporal information of historical trajectories. The SpatioTemporal-Attention Policy Network (STAPNet) then transforms this representation into dicision priors. These priors are further incorporated into heterogeneous planners as directional guidance, biasing the search toward locally promising regions. Extensive experiments demonstrate that ImiPath achieves competitive path quality and improves search efficiency by reducing redundant node expansions under local observability. Additional physical experiments on a magnetic microrobot platform further validate the adaptability and practical deployment potential of the proposed framework.

We investigate a decentralized reinforcement learning problem involving multiple agents that interact with the same Markov Decision Process (MDP). The agents can exchange information over a network to collectively learn the optimal state-action value function. For this setting, we introduce a novel epoch-based distributed $Q$-learning algorithm called VRDQ, where within each epoch, agents locally estimate the Bellman optimality operator and diffuse information using a consensus-based protocol. For both static and time-varying networks, we establish high-probability finite-time convergence rates for VRDQ that enjoy linear speedups from collaboration. Crucially, we prove that such speedups in sample-complexity require only $\tilde{O}(1)$ communication, substantially improving upon the communication costs in prior work.

Dense per-step supervision is the standard remedy for sparse-reward long-horizon LLM agents: reward the policy for predicting its next observation, which looks provably safe under potential-based shaping. Published prediction-reward and auxiliary-loss variants report both successes and instabilities; we supply the controlled account: 74 preregistered arms dissect one fixed prediction signal under GRPO across ALFWorld, WebShop, a synthetic POMDP, and Qwen3-1.7B/4B/8B, varying only the delivery mechanism. (1) Every run sustaining this difference-form reward under untouched std normalization (no filtering, dynamic-sampling, or decoupling mitigations) collapses: eleven runs across scales, coefficients, group sizes, and groupings (the floor-bound synthetic environment stalls instead); ALFWorld runs end in an absorbing state (prediction accuracy -> 1.0, success -> 0): the optimizer builds the "dark room". The algebra is one line: in all-fail groups z-scoring cancels the shaping coefficient; removing only std normalization restores baseline parity. (2) A signal's danger is set by its within-group variance trajectory, plus hackability as a second axis; it retrodicts every reward-channel collapse and survives preregistered prospective tests. (3) The same signal as a teacher-forced auxiliary loss is harmless on ALFWorld at 4B, but the gain is not the signal's: content-free placebos as a class match or beat gold at both matched seeds (s0: 78.8 vs 68.6; s42: 67.9 vs 57.9); the auxiliary update is the regularizer. (4) At 8B the recipe turns bistable: gold full-weight locks two of three seeds; every content-free or reduced-weight arm stays healthy. No ALFWorld or WebShop reward-channel variant measurably beats its matched-normalization baseline and no gold signal measurably outperforms its content-free placebo: the delivery channel, not the content, decides; which channel is safe is regime-dependent.

In reinforcement learning, critics typically estimate absolute state values $V(s)$, estimating how good a particular situation is in isolation. However, it turns out that only differences in value are relevant for control. Motivated by this, we propose Relative Value Learning (RV), a framework that learns value differences directly via an antisymmetric function $Δ(s_i, s_j) = V(s_i) - V(s_j)$. We introduce a pairwise Bellman operator and prove it is a $γ$-contraction with a unique fixed point equal to the true value differences, derive well-posed $1$-step, $n$-step and $λ$-return targets and reconstruct generalized advantage estimation from pairwise differences to obtain an unbiased policy-gradient estimator (R-GAE). Beyond theoretical results, we integrate RV with PPO and achieve competitive performance on the Atari benchmark (49 ALE games) compared to standard PPO, indicating that relative value estimation is an effective alternative to absolute critics.

Motivated by reinforcement learning in harsh environments, we consider the problem of learning an optimal policy subject to adversarially corrupted feedback. Specifically, at each time-step, an adversary can perturb both the reward and state observations of the learner following the Huber contamination model. To defend against such data corruption, we propose BR-Async-Q: a novel, epoch-based, robust Q-learning algorithm built upon two key ideas: (i) partitioning the online data stream into batches to reduce variance, and (ii) constructing robust estimates of the Bellman optimality operator using such batched data. We prove a high-probability $\ell_\infty$ error bound for BR-Async-Q that matches that for vanilla Q-learning, up to a small additive term that scales with the fraction of corrupted samples. To our knowledge, this provides the first robustness guarantee for asynchronous Q-learning subject to both reward and state corruption. Furthermore, when only rewards are corrupted, the dependence of our algorithm's bound on the corruption fraction is minimax optimal.

Active Inference (AIF) frames adaptive behavior as the minimization of expected free energy (EFE), combining epistemic and pragmatic objectives within a single variational principle. We frame AIF as policy optimization and show that, for closed-loop control policies, EFE minimization can be formulated as a convex Markov decision process (MDP). This perspective reveals that policy-dependent reward prediction errors transmit natural gradients of the expected free energy backwards in time rather than up a hierarchy. Finally, we show that coupling world-model learning with policy optimization gives active inference the structure of performative reinforcement learning. Together this places EFE minimization within modern reinforcement learning and optimization theory and opens a route toward principled algorithms for active inference.

We study finite-horizon MDP planning under \emph{root-based} (resolute) risk objectives that apply a rank-dependent functional to the distribution of total returns. Such objectives are non-linear in the return distribution and generally break Bellman optimality, so direct optimization by scenario-tree enumeration is intractable. We propose \textbf{ERQDP}, an enumeration-free and sampling-free method that solves a rank--quantile surrogate via exact DP (Dynamic Programming), evaluates candidate policies exactly by DP over return Probability Mass Functions (PMFs) on a discretized return grid (with an explicit rounding bound), and refines the surrogate in an anytime loop that reports an explicit upper--lower gap (certificate) for the target objective up to discretization budgets. Across tested benchmarks, ERQDP returns certified solutions or explicit residual gaps, enables fast risk-parameter sweeps with substantial runtime gains, and supports both risk-averse and risk-seeking behaviors.

We study horizon-free regret minimization for finite-horizon time-homogeneous tabular Markov decision processes with $S$ states, $A$ actions, horizon $H$, and per-trajectory total reward bounded by $1$. We propose a new algorithm and prove a regret upper bound $$\tilde O(\sqrt{SAK}+S^8A^3)$$ with failure probability $δ$, where $K$ is the number of episodes and $\tilde O(\cdot)$ hides $\mathsf{poly}\log(S,A,K,1/δ)$. Thus, the regret is $H$-free and asymptotically optimal, matching the contextual-bandit lower bound $Ω(\sqrt{SAK})$ up to logarithmic factors. This completely removes the $\log H$ dependence from the previous $\tilde O(\sqrt{SAK\log H}+S^2A\log H)$ guarantee of Zhang et al. (2021), and drastically improves the prior best horizon-free regret $\tilde O(\sqrt{S^9A^3K})$ of Zhang et al. (2022) asymptotically. The main technical difficulty is that the optimal value functions $\{V_h^*\}_{h=1}^H$ are time-inhomogeneous even though the transition kernel is time-homogeneous. A direct union bound over all value functions typically incurs an additional $\min\{\log H,S\}$ factor. We avoid this factor by (i) exploiting the monotonicity of $V_h^*$ in $h$ and (ii) non-trivially projecting the value functions onto an $S$-dimensional grid. Our analysis relies on three additional ingredients. First, we introduce a horizon-truncation argument that enables reward-based exploration and removes the cost of a separate reward-free exploration phase. Second, we design a cutting bonus that preserves both optimism and the monotonicity needed for planning. Third, we prove a new bound on total deviation for time-homogeneous MDPs, which controls the clipped variance terms in the cutting bonus with adjustable polynomial dependence on $S$ and without any dependence on $H$. Together, these tools yield an asymptotically optimal horizon-free regret guarantee.

Learning local policies for continuous networked systems requires accounting for the effects of decisions beyond each agent's observation neighborhood. Spatial decay limits these effects, but a finite critic must also control representation and estimation errors throughout policy optimization. We analyze the Continuous Distributed Coupled Policy Gradient (CDCPG) algorithm using local random Fourier features and least-squares temporal-difference critics. For features that retain the boundary inputs required by the local dynamics, we derive an action-value representation with separate spatial and finite-feature residuals. A global integrated transition-approximation bound and a projected Bellman argument control population prediction error without an inverse-conditioning multiplier. We then quantify the dependence of critic estimation on feature excitation and dimension, and construct simultaneous lower confidence bounds for temporal-difference conditioning along the executed iterates. Combining critic error with localized reward aggregation bounds the expected squared projected-gradient mapping by an optimization term and an explicit residual separating spatial approximation, finite features, and omitted distant rewards. For fixed neighborhoods and feature dimension, the shared-oracle sample count is inverse-squared in the excess squared-stationarity accuracy, up to logarithmic factors. The guarantee assumes known local dynamics and rewards, independent discounted-occupancy samples, and stated excitation, decay, and smoothness conditions, and is conditional on favorable feature draws. Numerical studies illustrate related implementations on a linear-coupled-quadratic benchmark.

Target interception in crowded environments requires reaching a moving objective while navigating among multiple uncertain human agents. Since human navigation intent is not directly observable, the robot must reason over multiple possible future interaction outcomes. We formulate interception in crowds as a partially observable Markov decision process and solve it online using tree search under a fixed computational budget. In this setting, the action-space structure directly shapes the search tree and how computational effort is allocated. We perform a controlled comparison between a sequential path-speed planner, which first plans a spatial path and then modulates speed along it, and a unified planner that jointly branches over steering and speed within tree search. Across simulations with up to 200 humans, both approaches perform similarly at low crowd density but diverge sharply as density increases. At the highest crowd density, the sequential planner has a safe-interception rate 31 percentage points lower and requires 44% more time than the unified steering-speed planner, revealing a structural limitation of spatial restriction. Project webpage: https://tic-planning.github.io/

What gives the Bellman equation its form? We show that the recursive properties of optimal value functions follow from three conditions: that the dynamics decomposes through sufficient statistics, that the return decomposes recursively, and that the aggregation of uncertainty is compatible with both. When all three conditions hold on a common state, the Bellman equation arises from their mutual consistency; when one fails, tractability can often be recovered by augmenting the state or by deforming return or dynamics. The same conditions are shown to give rise to three dualities: one between probability and return, one between return and aggregation, and one between aggregation and probability. Our framework reveals these dualities as arising from a single construction, unifying methods developed separately across reinforcement learning, control, and decision theory.

Distributional soft policy iteration (DSPI) provides an important framework for combining distributional reinforcement learning (DRL) with maximum-entropy control, in which the policy evaluation step is governed by a distributional soft Bellman operator acting on entropy-regularised returns. Theoretical analysis of such an evaluation step requires a probability metric under which Bellman updates can be controlled, typically by showing that the operator contracts the distance between any two candidate return-distribution estimates. In this paper, we focus on the Cramér geometry, a cumulative distribution function (CDF)-based metric with an $L^2$ structure, and study whether the fixed-policy distributional soft Bellman operator has this contraction property and hence a unique fixed point under this metric. Working directly on an admissible CDF field domain, we formulate the CDF-level distributional soft Bellman operator, prove that it is a $\sqrtγ$-contraction, and obtain the corresponding unique fixed point together with convergent iterative policy evaluation. The CDF formulation also shows that this finite-Cramér-domain property follows from a uniform first-moment condition on the combined one-step reward entropy shift, rather than from separate uniform boundedness assumptions on the reward and entropy terms. We then transport the same evaluation problem to the spectral domain by conjugation, obtaining an equivalent Hilbert-space representation of the same decision process. Taken together, these results identify the Cramér-geometric Bellman fixed point associated with the policy-evaluation step of DSPI, providing a reference point for studying approximate critics, evaluation error, and critic-loss design in DSPI-style algorithms.

Reinforcement Learning is a cornerstone technique for modern large reasoning models. Usually, for difficult tasks such as code generation and theorem proving, the agent is evaluated by generating $K$ responses rather than sampling a single response, and performance is then measured using a retry-aware metric such as $\max$@$k$. Despite their practical importance, the theoretical foundations of learning under such criteria remain limited. In this work, we provide a theoretical study of the $\max$@$k$ learning problem in finite-horizon reinforcement learning. We show that optimizing the $\max$@$k$ objectives is fundamentally different from standard expected-return maximization. In particular, we prove that Markovian policies are in general insufficient, identify a compact state augmentation that restores optimality, and explicitly characterize the performance gap that can arise between history-dependent and non-history-dependent policies. Moreover, we show that learning $\max$@$k$-optimal policies is statistically harder than standard reinforcement learning and provide an efficient algorithm that achieves the optimal sample complexity rate.

The softmax policy $π(a \mid s) \propto \exp(βQ(s,a))$ is the default model of stochastic choice in reinforcement learning (RL). Various justifications based on robustness, exploration, and optimization have been offered in the RL literature, but none uniquely derives the softmax form from first principles. This leaves a basic tension unresolved: the entropy bonus in the soft Bellman equation violates the Independence axiom that underwrites the Markov decision process (MDP) reward structure. We dissolve this tension by distinguishing two kinds of randomness: chance and choice. By restricting von Neumann-Morgenstern (VNM) Independence to environmental lotteries over base prospects, we show that imposing independence of irrelevant alternatives (IIA) and monotonicity on the policy and value functions at choice nodes uniquely determines the Boltzmann policy, the entropy-regularized representation, and the soft Bellman equation. The choice between the soft and hard Bellman equations thus reduces to a design decision: whether the agent values its own ability to choose. We develop RL-specific consequences, including return monotonicity and convergence under generalized discounting, and synthesize the independent lines from economics and information theory that arrive at the same structure, offering a normative assessment of when IIA is appropriate for agent design.

We model cryptographic auditing of off-chain data as a Constrained MDP (CMDP) under partial observability: the storage node's hidden type and corruption state make the problem a POMDP, while a miss-rate ceiling rho imposes an explicit security constraint. We propose DRQN-CMDP, a Deep Recurrent Q-Network whose GRU layer maintains a belief over the latent node type, paired with Lagrangian dual ascent that adapts the miss-rate penalty lambda automatically. A pairing-free homomorphic-MAC primitive supplies O(1) on-chain verification cost. Across 13 methods--four DQN variants, PPO, A2C, PPO-Lagrangian, a stateful Bayesian heuristic, three fixed-rule baselines, and an oracle-informed heuristic--DRQN-CMDP achieves a favourable balance: 83% lower gas than fixed high-frequency auditing, single-digit miss rate (7.5%), and moderate detection latency--a combination no other method matches across all three objectives simultaneously.

In this work we study the Best Policy Identification (BPI) problem in online, tabular Reinforcement Learning. This is an active sequential hypothesis testing problem in which the learner's objective is to identify an optimal policy in a Markov Decision Process (MDP) with high confidence, while minimizing the expected sample complexity to do so. We consider an online setting with deterministic rewards, where the agent must strategically navigate through the MDP in order to effectively explore. Previous works in the literature have provided asymptotically optimal methods for BPI, such as the Navigate and Stop (NaS) algorithm and its variants, however existing analysis remains asymptotic. In this work, we fill that gap by providing the first non-asymptotic sample complexity guarantees for NaS, showing that its sample complexity depends not only on the characteristic time, but also on the connectivity of the underlying MDP, the curvature of the optimal characteristic time, and other instance-dependent quantities. We identify these additional attributes and make explicit their contributions to the overall sample complexity.

Vision-language models normally execute the same complete vision encoder for every question, even when OCR, counting, object, attribute, and spatial queries may not require identical computation. We study whether fixed-budget combinations of vision blocks can be skipped without fine-tuning. A shared K-block route skips one searched set of exactly K blocks for every question, while a capability-specific K-block policy selects one same-size route using a known capability label. We introduce a source-balanced evolutionary search and compare it with independent ranking, contiguous removal, and random routes at matched budgets. Experiments use Qwen2.5-VL-3B-Instruct, SmolVLM2-2.2B-Instruct, and an 876-example image-disjoint selection split. Search transfers across architectures: on SmolVLM2, the searched shared four-block route beats independent construction by 4.91 percentage points. Capability specialization is less stable. On Qwen, the six-block capability policy beats the shared route by 2.17 points, driven by a 7.10-point OCR gain. On sealed IIIT5K, however, the SmolVLM2 OCR-specific route trails its shared route by 13.6 points. Combinatorial search reliably improves route construction, but capability labels do not define universally transferable vision pathways.

This paper addresses key technical challenges in current large language model (LLM) agent applications, including long-horizon planning, sparse reward attribution, and dynamic environmental interaction, by designing and optimizing an intelligent agent workflow. The proposed architecture is based on the synthesis of core AI paradigms: Visual, Language, Generative, Graph, Multimodal, Reinforcement, and Agent Intelligence. Unlike conventional baseline models that rely on static prompting and lack robust perception-action loops, our approach introduces a Partially Observable Markov Decision Process (POMDP) routing mechanism. This mechanism is augmented with an internal, self-correcting reward model that evaluates decision trajectories before execution. By integrating multimodal inputs and advanced reinforcement learning principles (such as proximal policy optimization and value function approximation), the agent maintains long-term structural memory and dynamically adapts its reasoning pathways to mitigate error accumulation. Empirical experiments on the ALFWorld embodied simulation environment and the WebShop online navigation benchmark demonstrate a 24.5% absolute improvement in task success rate and trajectory efficiency over mainstream baselines like the standard ReAct framework. Comprehensive ablation studies confirm the significant contribution of the reward-driven critique module in suppressing hallucination rates. This research bridges theoretical foundations of reinforcement learning and graph-based memory with autonomous agent workflows. Ultimately, the resulting architecture offers a practical, scalable reference framework for developing artificial intelligence technologies in complex, multi-step autonomous systems. Code is available at https://github.com/01Amez/RLAW_Implementation.

An agent acting under partial observability must decide when to gather information and which observations are worth their cost. Standard POMDPs value information only through its eventual effect on reward. The $ρ$-POMDP framework instead rewards uncertainty reduction directly, through a belief-dependent utility $ρ$, but in practice both the choice of $ρ$ and the weight placed on it are tuned by hand for every task. We show that active inference removes this tuning entirely. Minimizing Expected Free Energy (EFE) is exactly equivalent to solving a $ρ$-POMDP whose utility is expected information gain, and the exploration weight is fixed at $w=1$ because the variational bound expresses pragmatic and epistemic value in the same units (nats). We prove this equivalence for observe-then-commit POMDPs and extend it to factored observation POMDPs, a broader class that covers interleaved observe-act problems such as non-destructive testing and mobile sensing, where gathering information leaves the hidden state unchanged. Experiments support the theory. Across environments ranging from the classic Tiger problem to RockSample and a new Structural Inspection benchmark with over 65,000 states, the untuned weight matches or outperforms reward-only planning at the same horizon, avoids the over-exploration of bonuses tuned per task, and sits near the reward-maximizing knee of the success-reward Pareto frontier. The practical payoff is an exploration objective that works out of the box. In applications such as fault detection and medical screening, where every test has a price and every missed fault has a cost, EFE supplies a belief-dependent utility that is derived rather than tuned.

Reinforcement learning (RL) is primarily known as a computational method for optimizing control tasks, but it is increasingly used to explain biological behavior. While RL successfully captures key aspects of biology, a major gap remains: between-agent behavioral variability. Consistent individual differences naturally permeate biological populations, yet RL models typically present only the single best individual or the population average. Addressing this gap requires moving beyond current practices to generate behavioral diversity using biologically plausible mechanisms. Here, we examine approaches from various subfields of RL and outline potential paths forward to close the gap between biology and simulation.

Current benchmarks for language models primarily evaluate execution on fully specified tasks. However, real user tasks are often ambiguous. Users arrive with incomplete, exploratory, or even inconsistent goals, requiring the assistant to first determine the intended task before carrying it out. We study this problem as task alignment: the ability to align with a user on their intended task. We introduce a general framework for converting specified tasks into underspecified interactions, formalized as a POMDP in which the model must infer a latent task from partial and evolving user intent. We validate our user simulator post hoc with a human user study. Across shopping, coding, and professional work settings, we find that while models often perform well once the task is specified, models still struggle with task alignment: current models act prematurely, interact ineffectively, and fail to resolve ambiguous requests. Models on average recover the user's intended task only 22-32% of the time under ambiguity. In a human study in the same setting, humans reach 48%, outperforming all evaluated models. We show that post-training with supervised fine-tuning and reinforcement learning improves task alignment, but models still lag behind humans in resolving uncertainty through interaction. Together, our results suggest that current models still lack key interaction abilities required for reliable agency.

Reinforcement learning has proven effective for improving reasoning in large language models, but extending it to Masked Diffusion Language Models (MDLMs) remains challenging due to the intractability of the log-likelihood estimation. Existing approaches approximate this log-likelihood by modeling only the token predictions, ignoring the order in which positions are unmasked during generation. We observe that MDLM generation involves two decisions at each step: what tokens to place at each masked position and which positions to remask. We formalize this as a two-stage action MDP, showing that the policy gradient naturally decomposes into a token term and a masking term. Combining optimization of both terms leads to state-of-the-art outcomes on mathematical reasoning and coding benchmarks, with scores of 87.1% on GSM8K and 53.4% on MBPP.

Machine-learned predictions can speed up offline NP-hard optimization, but asking a predictor what to do amounts to asking it to solve the problem, and committing an unchecked prediction forfeits every worst-case guarantee. CASP (Certificate-Augmented Solution Pruning) instead asks which parts of the search space may be ignored, and accepts each answer only after a sound polynomial-time verifier has checked it, so correctness never depends on prediction quality. We develop the learning theory of this design. The verifier makes the induced loss class uniformly bounded, so certificate parameters are learnable from $\tilde O(\varepsilon^{-2}\log K)$ samples ($K$ the maximum instance size), whereas the unverified commitment class admits no distribution-free rate and, under cost spread $R$, none below $Ω(R/\varepsilon^2)$. Filtering noisy predictions by verifiable confidence dominates the standard min-combiner, with a margin we compute in closed form, and the prediction stays useful even given the LP, because it breaks ties on degenerate optimal faces, where every symmetric LP policy, meaning one whose commitments depend on the instance only through the verifiable confidence values, provably stalls. Experiments on five problems test the theory's quantitative predictions. With trained predictors, unverified pruning loses up to $26$ of the optimum under distribution shift, while the verified deployment of the same predictions loses nothing.

Policy learning methods are increasingly used to inform treatment allocation under budget constraints. Most proposed methods assume complete treatment data, yet applications frequently suffer from missingness that can bias estimates and lead to suboptimal policies. We address this gap by extending efficient estimators for average treatment effect (ATE) estimation to policy value and conditional average treatment effect (CATE) estimation under missing at random (MAR) and missing completely conditionally at random (MCCAR) treatment data. Through asymptotic efficiency analysis, we prove that the MAR estimator, which leverages partially-observed units, is both valid and more efficient than the MCCAR estimator when MCCAR assumptions hold. This result provides formal justification for preferring MAR-based estimation in policy learning under both missing data settings. Our comprehensive experiments using synthetic and semi-synthetic datasets confirm that correctly specifying the missingness mechanism is crucial: misspecified estimators remain biased regardless of sample size, while our estimators achieve near-oracle performance when assumptions are satisfied. Our work provides practitioners with theoretically grounded, empirically validated tools for robust policy learning in the presence of missing treatment data.

Algorithmic collusion among pricing algorithms has raised concerns about sustained supra-competitive prices and their implications for social welfare. Existing work has largely focused on the probability that reinforcement-learning algorithms converge to cooperative strategies, typically under the assumption that exploration vanishes over time. Motivated by the observation that algorithms deployed in practice are likely to continue exploring in order to remain adaptive to changing environments, we study learning dynamics under constant exploration. In this setting, the relevant question is no longer whether an algorithm converges to a particular strategy profile, but rather what fraction of time the algorithms spend playing cooperative strategies. Even in the benchmark case of the repeated Prisoner's Dilemma with one-period memory, this yields high-dimensional stochastic learning dynamics, for which a complete analytic treatment is intractable. We show that cooperative strategies can be dominant in this time-averaged sense and derive a boundary predicting when such dominance arises, based on the expected dynamics of the Q-learning process. Extensive simulations show that this boundary is a strong predictor for non-defection-dominated behaviour under epsilon-greedy Q-learning.

In this paper, we study Reinforcement Learning in Parametrized Action Markov Decision Processes (PAMDP), where each decision consists of a symbolic action and numerical parameters. In such settings Reinforcement Learning algorithms typically determine parameters with one-shot estimators, which makes their training sample inefficient. Though in most PAMDP environments explicit but incomplete knowledge (e.g., rules, safety constraints, or expert heuristics) is available, it is rarely directly used to increase the sample-efficiency of training Reinforcement Learning agents. We step into this gap and propose our novel Neuro-Symbolic Knowledge- and Gradient-Guided Reinforcement Learning (KGRL) algorithm. KGRL uses domain knowledge in a Datalog knowledge base to derive the set of applicable actions and feasible parameters for a given state. This allows it to prune non-applicable actions from the decision-space and constrain the parameter spaces of the remaining actions. We then use a gradient-based parameter refinement loop to estimate the optimal parameters during training and deployment of the agent. By recording activated rules along the trajectory, KGRL additionally provides local procedural explanations on the pruning of actions and constraining of parameters. Overall, KGRL guides the agent's exploration and deployment toward feasible and constraint-aware decisions, while increasing sample efficiency during training. KGRL outperforms state-of-the-art RL baselines for PAMDPs in both, sample efficiency and episodic return.

Reinforcement learning (RL) is traditionally concerned with learning a control policy for a fixed environment. In many engineering systems, however, the environment itself is alterable: physical or operational parameters can be tuned to shape the transition dynamics and costs experienced by the agent. This motivates jointly optimizing both the policy and the environment design parameters. To this end, we establish an Environment Parameter Gradient Theorem -- a formal expression for the gradient of the value function with respect to environment parameters. The key theoretical device is a generalized action-value function $Q_{π,ξ}(s,a,ζ)$, which comprises two copies of the environment parameters: $ζ$ governs the cost and transition dynamics at the current state--action pair, while $ξ$ governs the future rollouts. This decoupling yields a tractable closed-form gradient expression and is essential to the theorem's derivation. Building on this result, we develop a model-free algorithm that simultaneously learns the optimal policy and the environment parameters. We demonstrate the efficacy of our framework on a UAV network design problem, where the optimal UAV placement (environment parameters) and communication routes (governed by the policy) are learned jointly to minimize the total communication cost in the network.

This paper investigates the control of discrete-time linear time-invariant (LTI) systems subject to incomplete and corrupted measurements. Specifically, we focus on designing a Linear Quadratic Gaussian (LQG) controller without relying on explicit state estimation. By leveraging minimum variance duality, our approach allows the current control input to be represented as a linear function of available measurements and previously applied inputs, successfully reducing the task to a tractable deterministic optimization problem. We provide theoretical justification for this framework and demonstrate its practical effectiveness through numerical experiments.

Finite Reliability Representations (FRR) certify when a cell-constant policy is sufficient for reliable decision-making in a partially observed system with a known physical noise floor. In practice, however, sensing and execution noise can be latent and context-dependent. This paper develops a certificate-aware active disambiguation framework for an unknown physical noise parameter theta = (sigma_y, sigma_u), with the sensor-only case obtained by fixing sigma_u. We define the Value of Noise Information (VoNI) as the expected excess FRR certificate gap caused by using a reliability cover calibrated to the current estimate rather than to the realized noise parameter. We bound VoNI using action-value model mismatch and FRR radius inflation, showing that noise estimation has low decision value in sub-crossover regimes where the FRR certificate is insensitive to theta, but becomes valuable when posterior uncertainty can invalidate the current cover. A bi-level decision maker uses a posterior over theta, obtained from innovation statistics, execution residuals, or another online estimator, and triggers diagnostic probing only when uncertainty threatens the FRR certificate. We also interpret VoNI as a tractable, certificate-aware approximation to a high-level finite POMDP for latent sensing-execution regime disambiguation. Under stationary, identifiable, and persistently exciting regimes, we establish posterior consistency and convergence of the induced policy loss to the FRR approximation floor. Closed-loop UGV simulations with EKF-based innovation residuals show earlier detection of abrupt sensing-noise jumps, lower drift-tracking error, and substantially fewer probing actions than posterior-entropy exploration over 50 Monte Carlo trials.

We study the problem of optimal continual fine-tuning for a pre-trained Foundation Model deployed at a resource-limited device. At each time slot, a new batch of training data arrives, and the controller is faced with two options: either use the data to fine-tune the model and incur a compute cost, or do not fine-tune the model and discard the data. After the decision, the performance of the current model is measured in terms of an application-specific performance metric such as classification accuracy. Our objective is to learn an optimal policy that determines \emph{when to fine-tune the model} on a single task (e.g., sentiment analysis), under a finite compute budget. We formulate this online decision-making problem as a constrained Markov Decision Process, where the system state captures three essential aspects: (\textit{i}) model's performance, (\textit{ii}) computational budget, and (\textit{iii}) data distribution relevance to historic data encountered up to that point. The transition to the next state is stochastic and therefore, we propose a reinforcement learning-based method to solve this problem, namely the \emph{actor-critic} algorithm. We also consider the special case where the performance of fine-tuning for a given model can be predicted or estimated prior to decision; in this case the problem becomes a Dynamic Programming one. Experiments with a large pre-trained model on a widely-used text classification dataset demonstrate that our method consistently outperforms fine-tuning approaches with the same compute budget by more than $4\%$ in terms of accuracy and achieves $97\%$ of full-parameter fine-tuning accuracy while requiring only $25\%$ of the fine-tuning steps.

Multi-object search in unknown household environments requires planning under extensive uncertainty - from unknown object locations to cluttered spaces with unobserved obstacles. POMDPs offer a principled framework for such problems but remain intractable in large domains. We propose Inter-POMDP, a novel interleaved POMDP planning algorithm that decomposes this challenge into two interacting levels: a high-level POUCT planner reasons over object distributions using LLM-informed histogram beliefs, while a low-level motion planner models navigation uncertainty with obstacle-aware particle beliefs as domain knowledge to guide high-level POUCT. This interleaved design balances planning quality and efficiency despite the large search space across unknown multi-room environments. Both simulation and real-world experiments show that our Inter-POMDP algorithm reduces collision counts by up to 63%, navigation steps by up to 35%, and detection counts by up to 32% compared with baseline methods. Full videos are https://sites.google.com/view/inter-pomdp

Equitable renewable-energy planning is a sequential decision problem, but the decision variables available to a public planner differ sharply between mature and emerging economies. In the former the government largely builds generation, while in the latter it steers private investment through incentives and quotas. We formulate socially-equitable renewable-energy budget allocation as a Markov Decision Process (MDP) and, using a single problem-agnostic solver interface, compare the same policies across the two settings: eight U.S. cities (a mature economy) and West Java, Indonesia (an emerging economy). The results show that across both settings, a receding-horizon value-iteration policy dominates. In the U.S., it reaches 66% renewable penetration while cutting the underserved low-income population by 96% versus a random baseline. In West Java it closes the low-access gap while crowding in the most private capital. More interestingly, a naive market-chasing heuristic, which is mildly sub-optimal in the U.S., could yield catastrophic outcomes in Indonesia, by underserving every low-access region, because chasing attractive markets and serving the underserved goals diverge once the planner acts through private developers.

Offline-to-online reinforcement learning is promising for generalizable robotic manipulation, yet its full-stack complexity obscures reproduction and diagnosis. Within such systems, value estimation plays a central role in prioritizing heterogeneous data for policy improvement. Despite its importance, the central question remains underexplored: how value-function reliability shapes policy optimization in offline-to-online reinforcement learning. To answer this question, we propose Robo-ValueRL, a unified framework that enables reliable value estimation and systematically traces its downstream effects on policy pretraining and online improvement. Concretely, Robo-ValueRL learns a history-conditioned value estimator and evaluates its reliability through global-progress and local-preference metrics. These resulting value estimates are propagated into quality-conditioned consistency-policy pretraining and a residual adaptation module on online rollouts, providing a unified testbed for analyzing how value reliability shapes downstream policy performance. Across 240 hours of offline demonstrations and over 3,000 online rollout trajectories, our extensive experiments show that downstream performance is strongly associated with value reliability. Reliable value functions provide better action-quality estimates, allowing value-guided offline RL to scale more effectively than quality-agnostic behavior cloning, and stabilize online improvement by prioritizing high-quality rollout data. Integrating reliable value guidance through offline pretraining with online improvement, our system achieves 86% success on millimeter-level precise chip insertion and 84% on generalizable block disassembly. We hope these findings highlight the importance of value-guided data utilization for effective policy improvement from heterogeneous robotic experience.

We study general-utility Markov decision processes (GUMDPs) with risk-aware objectives. In this framework, an agent aims to optimize a risk measure of the distribution of objective values, where the objective function depends on the frequency of visitation of states induced by the agent's policy. First, we motivate, propose, and formalize risk-aware GUMDPs, which enable agents and decision makers to trade off expected performance by risk aversion while benefiting from the rich set of objectives that can be cast under the framework of GUMDPs. We focus our attention on the entropic risk measure (ERM). Second, we show how we can solve risk-aware GUMDPs with ERM objectives by resorting to online planning techniques. In particular, we propose an approach based on Monte Carlo Tree Search (MCTS) to provably solve risk-aware GUMDPs up to any desired accuracy. Third, we provide a set of experimental results showcasing that our approach is successful when optimizing for a spectrum of risk-aware behaviors in the context of GUMDPs under diverse tasks (standard MDPs, maximum state entropy exploration, imitation learning, and multi-objective MDPs).