Markov Decision Processes

Designing model-free algorithms for distributionally robust reinforcement learning (DRRL) poses fundamental challenges. The robust Bellman operator is nonlinear in the transition kernel, which makes one-sample Bellman updates biased, while the adversarial optimization underlying robustness makes robust evaluation computationally demanding. To address these difficulties, we consider the natural small-ambiguity regime under Kullback--Leibler ambiguity sets and propose an approximate DRRL framework based on a first-order expansion of the relevant robust functional. This yields an approximate robust Bellman equation that removes the adversarial optimization while remaining first-order accurate in the ambiguity radius. To learn the fixed point of this approximate equation, we propose Mean-Variance Stochastic Approximation (MVSA), a model-free algorithm that uses only one-sample updates. This is achieved via a lifted stochastic approximation dynamics and a two-time-scale design. We then prove convergence and a central limit theorem for MVSA: its main iterate satisfies a central limit theorem at the canonical $n^{-1/2}$ scale, with explicitly characterized asymptotic covariances. Finally, we validate our theoretical findings with a numerical experiment.

Reinforcement learning (RL) for exponential-utility optimization in discounted Markov decision processes (MDPs) lacks principled value-based algorithms. We address this gap in the fixed risk-aversion setting. Building on the Bellman-type equation for exponential utility studied in \cite{porteus1975optimality}, we derive two Q-value-style extensions and show that the associated operators are contractions in the $L_\infty$ and sup-log/Thompson metrics, respectively. We characterize their fixed points and prove that the induced greedy stationary policy is optimal for the exponential-utility objective among stationary policies. These structural results lead to two model-free algorithms: a two-timescale Q-learning--style algorithm, for which we establish almost-sure convergence and provide finite-time convergence rates via timescale separation, and a one-timescale algorithm governed by a sublinear power-law operator. Since the latter does not admit a global contraction in standard metrics, we prove its convergence using delicate arguments based on local Lipschitzness, monotonicity, homogeneity, and Dini derivatives, and provide a scalar finite-time analysis that highlights the challenges in obtaining convergence rates in the vector case. Our work provides a foundation for value-based RL under exponential-utility objectives.

We propose Drifting Field Policy (DFP), a non-ODE one-step generative policy built on the drifting model paradigm. We frame the policy update as a reverse-KL Wasserstein-2 gradient flow toward a soft target policy, so that each DFP update corresponds to a gradient step in probability space. By construction, this gradient is decomposed into an ascent toward higher action-value regions and a score matching with the anchor policy as a trust region. We further derive a simple, tractable surrogate of the otherwise intractable update loss, akin to behavior cloning on top-K critic-selected actions. We find empirically that this mechanism uniquely benefits the drifting backbone owing to its non-ODE parameterization. With one-step inference, DFP achieves state-of-the-art performance on several manipulation tasks across Robomimic and OGBench, outperforming ODE-based policies.

This paper presents a finite-time analysis for Monte Carlo Tree Search (MCTS) in Partially Observable Markov Decision Processes (POMDPs), with probabilistic concentration bounds in both discrete and continuous observation spaces. While MCTS-style solvers such as POMCP achieve empirical success in many applications, rigorous finite-time guarantees remain an open problem due to the nonstationarity and the interdependencies induced by heuristic action selection (e.g., UCB). In the discrete setting, we address these challenges by extending the polynomial exploration bonus to UCB in POMDP setting, yielding polynomial concentration bounds for the empirical value estimation at the root node. For continuous observation spaces, we introduce an abstract partitioning framework and propose a finite-time bound on partitioning loss. Under mild conditions, we prove highprobability bound on value estimates in POMDPs with continuous observation space. Specifically, we propose Voro-POMCPOW, a variant of POMCPOW with f inite-time guarantees that adaptively partitions the continuous observation space using Voronoi cells. This approach maintains a finite branching factor while preserving the original observation generator. Empirical validation demonstrates that the proposed Voro-POMCPOW shows competitive performance while providing theoretical guarantees. Although our analysis focuses on continuous POMDPs, the techniques developed herein are also applicable to continuous MDPs, closing another gap on the MDP side.

Partially Observable Markov Decision Processes (POMDPs) are systems in which one agent interacts with a stochastic environment, and receives only partial information about the current state. In a multi-environment POMDP (MEPOMDP), the initial state is unknown, and assumed to be adversarially chosen. In this work we focus on computing the optimal value and policy in MEPOMDPs with finite-horizon objectives. That problem is known to be PSPACE-complete in POMDPs. Our main results are as follows: (1) we establish that it is also PSPACE-complete in the more general setting of MEPOMDPs; (2) we present a practical algorithm and evaluate it on classical benchmarks, significantly outperforming the only previously known algorithm.

For continuous state-action space scenarios, classical reinforcement learning (RL) theory predominantly focuses on low-rank Markov decision processes (MDPs), which provide sample-efficient guarantees at the expense of restrictive structural assumptions. Kernel smoothing model-based approaches offer a promising alternative paradigm that instead leverages the smoothness of the MDP and employs non-parametric kernel smoothing estimates of transition dynamics. This paper proposes a new kernel-smoothing model-based approach for online reinforcement learning in finite-horizon settings under Lipschitz continuity assumptions on the MDP. By incorporating a Bernstein-style exploration bonus into the kernel smoothing framework, our method achieves a regret bound which improves upon the state-of-the-art regret bound in its dependence on the horizon. The theoretical advancement relies on a delicate analysis of the synergy between Bernstein-style bonuses and kernel smoothing, where a new tight Bernstein-type concentration inequality for martingales may be of independent interest.

Real-world clinical diagnosis is a complex process in which the doctor is required to obtain information from both interaction with the patient and conducting medical exams. Additionally, the doctor needs to adapt to different patient personas, as well as noisy and incomplete information that can happen at any time during the process. However, existing benchmarks for medical LLMs and methods for automatic diagnosis largely simplify this process by reducing it to single-turn question answering, noise-free conversations, or sequential exam making, etc., ignoring the interactive and uncertain nature of clinical diagnosis. In this paper, we aim to address this gap by formalizing clinical diagnosis as a Partially Observable Markov Decision Process (POMDP) with three action types: questioning the patient, ordering medical exams as tool calls, and issuing a diagnosis. We also introduce a systematic noise model comprising seven patient noise types and three exam noise types. Using our proposed environment, we train an effective diagnosis agent, \textbf{MedExAgent}, through a two-stage pipeline that first performs supervised finetuning on synthetic conversations structured after the Calgary-Cambridge model for clinical interviews, and then applies DAPO to optimize a composite reward capturing diagnostic accuracy, tool call quality, and exam cost including financial cost and patient discomfort. Through extensive experiments and ablation studies, we demonstrate that MedExAgent achieves diagnostic performance comparable to larger models while maintaining cost-efficient examination strategies.

Large Language Model (LLM) agents are deployed in complex environments -- such as massive codebases, enterprise databases, and conversational histories -- where the relevant state far exceeds their context windows. To navigate these spaces, an agent must iteratively explore the environment to find relevant information. However, without explicit infrastructure, an agent's working memory can degrade into lossy representations of the search state, resulting in redundant work (e.g. repetitive looping) and premature stopping. In this work, we formalize this challenge as the Context Gathering Decision Process (CGDP), a specialized Partially Observable Markov Decision Process, where an agent's objective is to adaptively refine its belief state to isolate the necessary information for a task. We model an LLM's behavior as approximate Thompson Sampling within this CGDP, and introduce a predicate-based method that decomposes an LLM's implicit search into explicit and modular operations. We then derive two plug-and-play interventions for iterative LLM agents: a persistent, predicate-based belief state that bounds context while preserving multi-hop reasoning, and a programmatic exhaustion gate that halts unproductive search without premature stopping. Across four methods and three question-answering domains, we empirically validate that replacing an LLM's implicit state with our CGDP-motivated belief state improves multi-hop reasoning by up to $11.4\%$; while the modular programmatic exhaustion detection saves up to $39\%$ of tokens without any degradation in agent performance. Ultimately, we argue that framing the LLM agent loop as a CGDP can guide the design of modular, non-interfering improvements to agentic search harnesses.

Distributional reinforcement learning (DRL) models the full return distribution, but existing finite-support or quantile-based methods rely on projections, while recent flow-based approaches can suffer from \emph{boundary mismatch} at the flow source or from \emph{high-variance} bootstrapping when current and successor noises are independent. We propose Path-Coupled Bellman Flows (PCBF), a continuous-time DRL method that learns return distributions with flow matching using \textbf{source-consistent Bellman-coupled paths}: the current path starts from the required base prior at $t{=}0$, reaches the Bellman target at $t{=}1$, and maintains a pathwise affine relation to the successor flow at intermediate times (without requiring time-$t$ marginals to satisfy a distributional Bellman fixed point for all $t$). PCBF couples current and successor return flows through shared base noise and uses a $λ$-parameterized control-variate target: $λ{=}0$ recovers an unbiased sample Bellman target, while $λ{>}0$ trades controlled bias for variance reduction. Experiments on analytically tractable MRPs, OGBench, and D4RL show improved distributional fidelity and training stability, and competitive offline RL performance.

In networks, effective dynamic treatment allocation requires deciding both whom to treat and also when, so as to amplify policy impact through spillovers. An early intervention at a well-connected node can trigger cascades that change which nodes are worth targeting in the next period. Existing treatment strategies under network interference are largely static while dynamic treatment frameworks typically ignore network structure altogether. We integrate these perspectives and propose Q-Ising, a three-stage pipeline that (i) estimates network adoption dynamics via a Bayesian dynamic Ising model from a single observed panel, (ii) augments treatment adoption histories with continuous posterior latent states, and (iii) learns a dynamic policy via offline reinforcement learning. The Bayesian mechanism enables uncertainty quantification over dynamic decisions, yielding posterior ensemble policies with interpretable spillover estimates. We provide a finite-sample regret upper bound that decomposes into standard offline-RL uncertainty, network abstraction error, and first stage error in Ising state estimation. We apply our method to data from Indian village microfinance networks and synthetic stochastic block models under simulated heterogeneous susceptible-infected-susceptible (SIS) dynamics and demonstrate that adaptive targeting outperforms static centrality benchmarks.

Regulatory approval of products in high-stakes domains such as drug development requires statistical evidence of safety and efficacy through large-scale randomized controlled trials. However, the high financial cost of these trials may deter developers who lack absolute certainty in their product's efficacy, ultimately stifling the development of `moonshot' products that could offer high social utility. To address this inefficiency, in this paper, we introduce a statistical protocol for experimentation where the product developer (the agent) conducts a randomized controlled trial sequentially and the regulator (the principal) partially subsidizes its cost. By modeling the protocol using a belief Markov decision process, we show that the agent's optimal strategy can be found efficiently using dynamic programming. Further, we show that the social utility is a piecewise linear and convex function over the subsidy level the principal selects, and thus the socially optimal subsidy can also be found efficiently using divide-and-conquer. Simulation experiments using publicly available data on antibiotic development and approval demonstrate that our statistical protocol can be used to increase social utility by more than 35$$\% relative to standard, non-sequential protocols.

We present a novel theoretical framework, Q-MMR, for off-policy evaluation in finite-horizon MDPs. Q-MMR learns a set of scalar weights, one for each data point, such that the reweighted rewards approximate the expected return under the target policy. The weights are learned inductively in a top-down manner via a moment matching objective against a value-function discriminator class. Notably, and perhaps surprisingly, a data-dependent finite-sample guarantee for general function approximation can be established under only the realizability of $Q^π$, with a dimension-free bound -- that is, the error does not depend on the statistical complexity of the function class. We also establish connections to several existing methods, such as importance sampling and linear FQE. Further theoretical analyses shed new light on the nature of coverage, a concept of fundamental importance to offline RL.

We study agency under partial observability in deterministic physical or simulated worlds, where apparent randomness arises from uncertainty over initial conditions, fixed law bits, and unrolled exogenous noise. We model sensing and actuation as bridge interfaces split between agent-controlled parameters and environment-controlled channel state, inducing a deterministic POMDP through a prior over latent microstates and many-to-one observation coarsening. Within this framework, we prove a separation between prediction, compression, and empowerment. Perfect prediction can be achieved either by identifying the hidden quotient relevant to the target family or by overwrite control that makes the future target action-determined; high empowerment alone is insufficient. Under refinable interfaces and sufficient memory, action-conditioned observation-compression progress reduces posterior uncertainty about the latent quotient, and when refinement requires steering world-side channel conditions, this creates target-conditioned interface empowerment. A bit-string specialization with a conserved information budget makes the resulting tradeoff explicit: prediction by identification requires internal capacity at least the relevant latent entropy, whereas overwrite control requires terminal action capacity over the controlled quotient. For modern AI agents, the results suggest a design principle rather than a theorem of inevitability: objectives should distinguish hidden-state identification, interface refinement, task-relevant controllability, and mere overwrite or distractor control. Human--AI alignment is partly an interface-design problem, where the relevant bridge is between human intent, agent internal state, external tools, and world-side channel conditions. This is a working draft: feedback and criticism is most welcome.

Online-safety regulation under the UK Online Safety Act and the EU Digital Services Act increasingly treats scalar metrics as compliance evidence. Once announced, such a metric also becomes an optimization target: a strategic platform can improve its score by routing recommendations through semantically equivalent content variants, without reducing true harm. We ask when such an audit metric can still certify a genuine reduction in harm. The protocol is modeled as a published transformation graph whose connected components form semantic classes, and the metric itself is treated as a security object. Three results follow. First, any metric that scores variants directly is manipulable as soon as two equivalent variants in a harmful class disagree in score. Second, the semantic-envelope lift, which assigns each variant the maximum score in its class, is the unique pointwise minimum among conservative classwise-constant repairs. Third, a class-stratified certificate, $H^\star(x) \le (1/\hatα) M_{\mathrm{Env}(m)}(x) + \barη$, holds for every platform strategy, with $\barη$ absorbing annotation and protocol error. We check the claims at three levels: exhaustive enumeration on a finite-state grid of mixed strategies, an SMT encoding in Z3 cross-replayed in cvc5, and a bounded single-player MDP encoded in PRISM-games. The fragile metric fails manipulation invariance and cannot support the same useful predeclared class-coverage certificate; under the envelope-level certificate, it produces large violations at every tested instance, with a large mean gaming gap across random catalogs at a fixed audit budget. The semantic-envelope metric exhibits no such violation in the tested instances.

Deterministic policy gradient (DPG) is widely utilized for continuous control; however, it inherently relies on the differentiability of the critic with respect to the action during policy updates. This assumption is violated in practical control problems involving sparse or discrete rewards, leading to ill-defined policy gradients and unstable learning. To address these challenges, we propose a principled alternative based on a smoothed Bellman equation formulated via Gaussian smoothing. Specifically, we define a novel action-value function based on a smoothed Bellman equation and derive the soft deterministic policy gradient (Soft-DPG). Our formulation eliminates explicit dependence on critic action-gradients and ensures that the gradient remains well-defined even for non-smooth Q-functions. We instantiate this framework into a deep reinforcement learning algorithm, which we call soft deep deterministic policy gradient (Soft DDPG). Empirical evaluations on standard continuous control benchmarks and their discretized-reward variants show that Soft DDPG remains competitive in dense-reward settings and provides clear gains in most discretized-reward environments, where standard DDPG is more sensitive to irregular critic landscapes.

We study the efficient computation of Shapley values for \emph{product games} -- cooperative games in which the coalition value factorizes as a product of per-player terms. Such games arise in machine learning explainability whenever the value function inherits a multiplicative structure from the underlying model, as in kernel methods with product kernels and tree-based models. Our key result is that the Shapley value of each player in a product game admits an exact one-dimensional integral representation: the weighted sum over exponentially many feature coalitions collapses to the integral of a degree-$(d-1)$ polynomial over $[0,1]$, where $d$ is the total number of features. This yields a Gauss--Legendre quadrature scheme that is \emph{provably exact} whenever the number of nodes satisfies $m_q \geq \lceil d/2 \rceil$, and otherwise provides a \emph{near-exact} approximation with error provably decaying geometrically in $m_q$. In practice, a few hundred nodes can achieve highly precise estimates even with thousands of features. Building on this formulation, we derive a numerically stable implementation via log-space evaluation, together with an efficient parallel implementation based on associative scan primitives that achieves $O(d\,m_q)$ total work and $O(\log d)$ parallel time. Experiments show that \textsc{QuadraSHAP} is the fastest numerically stable method across all tested configurations.

Off-policy, value-based reinforcement learning methods such as Q-learning are appealing because they can learn from arbitrary experience, including data collected by older policies or other agents. In practice, however, bootstrapping makes long-horizon learning brittle: estimation errors at later states propagate backward through temporal-difference (TD) updates and can compound over time. We propose long-horizon Q-learning (LQL), which introduces a principled backstop against compounding error when learning the optimal action-value function. LQL builds on a prior optimality tightening observation: any realized action sequence lower-bounds what the optimal policy can achieve in expectation, so acting optimally earlier should not be worse than following the observed actions for several steps before switching to optimal behavior. Our contribution is to turn this inequality into a practical stabilization mechanism for Q-learning by using a hinge loss to penalize violations of these bounds. Importantly, LQL computes these penalties using network outputs already produced for the TD error, requiring no auxiliary networks and no additional forward passes relative to Q-learning. When combined with multiple state-of-the-art methods on a range of online and offline-to-online benchmarks, LQL consistently outperforms both 1-step TD and n-step TD learning at similar runtime.

Robots operate under significant uncertainty, from quantifiable noise to unquantifiable unknowns, and must account for strict operational constraints, such as limited resources. In this paper, we consider the problem of synthesizing robust strategies to guide a robot's actions in fulfilling a given task, while ensuring the system never exhausts its resources. To solve this problem, we first model the robotic system as a Consumption Markov Decision Process with Set-valued Transitions(CMDPST), a unified framework modelling nondeterministic actions, quantifiable and unquantifiable uncertainty, and resource consumption. Then, we combine the CMDPST with the task specification, expressed as a Linear Temporal Logic over finite traces (LTLf ) formula. Lastly, we address the resource constrained optimal robust strategy synthesis problem, which aims to synthesize a strategy that maximizes the probability of satisfying the LTLf objective without resource exhaustion. Our solution involves two techniques: a direct unrolling-based method and a more efficient, optimized approach that leverages state-space pruning for better performance. Experiments on a warehouse transportation network show the effectiveness of the proposed solutions.

While reinforcement learning (RL) promises to revolutionize the control of complex nonlinear robotic systems, a profound gap persists between the heuristic success of model-free off-policy deep RL and the underlying theory, which remains largely confined to tabular or linearizable settings. We identify the cause of this gap as an emergent isolation of three traditions: (i) measure-theoretic MDP foundations on general spaces limit their analysis to exact dynamic programming and ignore all error sources of a learning process; (ii) deterministic error propagation analysis addresses the approximation error via concentrability coefficients without a finite-sample analysis of the estimation error; and (iii) PAC generalization bounds characterize the estimation errors of simplified topologies. We bridge these traditions with a unified theoretical framework for fitted Q-iteration (FQI) on general measurable Borel spaces. Our main result provides a finite-sample, adaptive-data performance bound by chaining measure-theoretic probability with Bellman-operator contraction in Banach spaces. We prove that sequential Rademacher complexity controls Bellman-regression generalization under policy-dependent data collection. We further extend this analysis to provide the first cumulative, pathwise online regret guarantee for FQI in continuous spaces. These results lay the necessary foundations for the formal analysis of many modern deep RL algorithms.

In the modern age of large-scale AI, federated learning has become an increasingly important tool for training large populations of AI agents; however, its computational and communication costs can rapidly fail to scale with the number of agents. This is precisely where decentralized agentic strategies shine: each agent acts autonomously, using only its own state together with a minimal summary of the ensemble, namely the mean-field. We derive the unique optimal decentralized policy in closed form. Optimality is characterized through a worst-client/minimax criterion: minimizing the under-performer regret, namely the maximal online cost incurred by the weakest agent in the ensemble. We further prove that the resulting decentralized policy asymptotically converges, in the large-population limit, to the Nash-optimal centralized policy, whose direct computation is not scalable. We use an online weighting mechanism to optimize the server-computed mixture of client predictions, thereby improving the mean prediction in addition to the previously optimized weakest-client prediction. Numerical experiments verify our theoretical guarantees and demonstrate that our decentralized policy typically outperforms natural greedy decentralized baselines.

We consider the stabilization of Vlasov--Poisson plasma dynamics, a central control problem in nuclear fusion. Our focus is the gap between what an ideal controller would use and what experiments can actually observe: while optimal policy may rely on the full phase-space state, practical feedback is typically limited to sparse macroscopic diagnostics. We therefore study imitation learning methods that distill a fully observed expert policy into controllers operating only on macroscopic measurements. We show the stability guarantees of the learned policy, where the error floor depends on the minimal behavior cloning loss achievable under the observation constraints. We further characterize this minimal loss in terms of a notion of entropy that quantifies the complexity of the initial distribution. Our results demonstrates the theoretical feasibility of learning stabilizing feedback policies for kinetic plasma dynamics from macroscopic observations, and exhibits the adaptivity of the learning approach to low-complexity structures. Through extensive numerical experiments, we validate our theory and show that the learned policies can stabilize the system using only macroscopic observations, within a significantly longer time horizon than non-adaptive baseline controllers.

A Tree Markov Decision Problem (T-MDP) is a finite-horizon MDP with a starting state $s_{1}$, in which every state is reachable from $s_{1}$ through exactly one state-action trajectory. T-MDPs arise naturally as abstractions of decision making in sequential games with perfect recall, against stationary opponents. We consider the problem of on-line learning in T-MDPs, both in the PAC and the regret-minimisation regimes. We show that well-known bandit algorithms -- \textsc{Lucb} and \textsc{Ucb} -- can be applied on T-MDPs by treating each policy as an arm. The apparent technical challenge in this approach is that the number of policies is exponential in the number of states. Our main innovation is in the design of confidence bounds based on data shared by the policies, so that the bandit algorithms can yet be implemented with polynomial memory and per-step computation. We obtain instance-dependent upper bounds on sample complexity and regret that sum a ``gap term'' from every terminal state, rather than every policy. Empirically, our algorithms consistently outperform available alternatives on a suite of hidden-information games.

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

Input Convex Neural Networks (ICNNs) are commonly used in a two-stage manner: one first trains a convex network and then minimizes it over its input in a downstream inference problem. Recent second-order-cone ICNNs (SOC-ICNNs) enrich ReLU-based ICNNs with quadratic and conic modules and admit an exact representation as value functions of second-order cone programs (SOCPs). This value-function structure enables an explicit convex-analytic treatment of SOC-ICNN inference. In this paper, we study the exact first-order and local second-order geometry of SOC-ICNNs from the dual viewpoint. We show that supporting slopes, subdifferentials, directional derivatives, and local Hessians can be recovered directly from optimal dual variables. These results provide the geometric primitives for white-box SOC-ICNN inference, going beyond black-box automatic differentiation. Numerical experiments validate the exact multiplier readout, the local Hessian formula, and the set-valued behavior at structurally degenerate inputs. We also provide a step-by-step tutorial showing how the readout mechanism instantiates a complete white-box inference loop. The code is available at https://anonymous.4open.science/r/SOC-ICNN-Theory-BEFC/.

We formally establish the equivalence between Observation Delay (OD) and Action Delay (AD) in cooperative partially observable multi-agent systems using observation-action histories. We show that both systems generate identical admissible joint-policy sets, and their induced state-action-observation trajectories are identical in distribution, leading to identical optimal solutions in Decentralized Partially Observable Markov Decision Processes (Dec-POMDPs). This formally generalizes existing infinite-horizon single-agent results to any-horizon partially observable cooperative multi-agent problems with decentralized policy execution, and allows any mixed-delay configuration to be reduced to a pure OD system. We further prove that in Transition-Independent MDPs (TI-MDPs), the observation-action history reduces to a tractable minimal local augmented state. However, we show through numerical experiments that although the optimal solution spaces are structurally isomorphic, the practical learning dynamics are fundamentally different. First, using the minimal local augmented state, the equivalence no longer holds when transitions are not independent. Second, operational constraints and causal credit-assignment errors in Temporal Difference (TD) algorithms induce different learning behaviors across regimes. Finally, leveraging this structural equivalence to bypass these learning challenges, we demonstrate successful multi-agent zero-shot policy transfer from OD to AD, paving the way for unified, efficient solution methods in complex delayed systems.

We study the $(\varepsilon, δ)$-PAC policy identification problem in finite-horizon episodic Markov Decision Processes. Existing approaches provide finite-time guarantees for approximate settings ($\varepsilon>0$) but suffer from high computational cost, rendering them hard to implement, and also suffer from suboptimal dependence on $\log(1/δ)$. We propose a randomized and computationally efficient algorithm for best policy identification that combines posterior sampling with an online learning algorithm to guide exploration in the MDP. Our method achieves asymptotic optimality in sample complexity, also in terms of posterior contraction rate, and runs in $O(S^2AH)$ per episode, matching standard model-based approaches. Unlike prior algorithms such as MOCA and PEDEL, our guarantees remain meaningful in the asymptotic regime and avoid sub-optimal polynomial dependence on $\log(1/δ)$. Our results provide both theoretical insights and practical tools for efficient policy identification in tabular MDPs.

Contextual MDPs are powerful tools with wide applicability in areas from biostatistics to machine learning. However, specializing them to offline datasets has been challenging due to a lack of robust, theoretically backed methods. Our work tackles this problem by introducing a new approach towards adaptive estimation and cost optimization of contextual MDPs. This estimator, to the best of our knowledge, is the first of its kind, and is endowed with strong optimality guarantees. We achieve this by overcoming the key technical challenges evolving from the endogenous properties of contextual MDPs; such as non-stationarity, or model irregularity. Our guarantees are established under complete generality by utilizing the relatively recent and powerful statistical technique of $T$-estimation (Baraud, 2011). We first provide a procedure for selecting an estimator given a sample from a contextual MDP and use it to derive oracle risk bounds under two distinct, but nevertheless meaningful, loss functions. We then consider the problem of determining the optimal control with the aid of the aforementioned density estimate and provide finite sample guarantees for the cost function.

Multi-agent reinforcement learning (MARL) holds great potential but faces robustness challenges due to environmental uncertainty. To address this, distributionally robust Markov games (RMGs) optimize worst-case performance when the environment deviates from the nominal model within a uncertainty set. Beyond robustness, an equally urgent goal for MARL is data efficiency -- sampling from vast state and action spaces that grow exponentially with the number of agents potentially leads to the curse of multiagency. However, current provably data-efficient algorithms for RMGs are limited to tabular settings with finite state and action spaces, which are only computationally manageable for small-scale problems, leaving RMGs with large-scale (or infinite) state spaces largely unexplored. The only existing work beyond tabular settings focuses on linear function approximation (LFA) for a restrictive class of RMGs using vanish minimal value assumption and still suffers from sample complexity with the curse of multiagency. In this work, we focuses on general RMGs with LFA. For uncertainty sets defined by total variation distance, we develop provably data-efficient algorithms that break the curse of multiagency in both the generative model setting and a newly proposed online interactive setting. To our knowledge, our results are the first to break the curse of multiagency of sample complexity for RMGs with large (possibly infinite) state spaces, regardless of the uncertainty set construction.

We present two new classes of causal models of decision-making agents. Our approach is motivated by the needs of modeling the economics of computing systems. These systems are composed of subsystems and can exhibit endogenous limits on cognitive resources and value discounting. Structural Causal Decision Models (SCDMs) expand on Structural Causal Influence Models. Like SCIMs, they explicitly represent the causal relationships between model variables and the payoffs of agent decisions. Additionally, agent decisions can be constrained by their causal antecedents, and SCDMs can have open root variables for which no probability distribution or structural equation is given. We show that SCDMs have a well-defined and computationally useful property of composability. Building on SCDMs, we then define a Structural Causal Decision Process (SCDP) as a recurring SCDM with a discount variable. SCDPs benefit from the useful composition properties of SCDMs. Moreover, SCDPs are strictly more expressive than POMDPs because they do not assume rational belief formation. Indeed, an SCDP can endogenously model the memory-formation process, and is thus useful for modeling resource rational agents in dynamic settings. SCDPs are also capable of modeling variable discounting, a tool used widely in social scientific modeling. We pose that SCDPs are a useful framework for policy simulation for the digital economy, mechanism design for information systems, and digital twin modeling of cyberinfrastructure.

Middle-mile logistics describes the problem of routing parcels through a network of hubs linked by trucks with finite capacity. We rephrase this as a multi-object goal-conditioned MDP. Our method combines graph neural networks with model-free RL, extracting small feature graphs from the environment state.

This paper develops a deep reinforcement learning based observer control policy for autonomous bearings-only tracking of a moving target. The observer manoeuvre problem is formulated as a belief Markov decision process, where the belief state is represented by the posterior of a cubature Kalman filter (CKF). The reward function is designed to address two conflicting objectives: minimising the absolute target position estimation error (Euclidean distance) and maintaining CKF estimation consistency (Mahalanobis distance). The reward is formulated as a geometric interpolation between the two objectives on the Pareto front, parametrised by a weighting factor $β\in [0,1]$. The policy is implemented as a deep Q-network (DQN) trained over 50,000 episodes. Performance is evaluated over 5,000 Monte Carlo episodes and compared against two baselines: the perpendicular-to-bearing heuristic and the D-optimal Fisher information maximisation criterion. The results show that the DQN policy at $β= 0.7$ achieves the best trade-off between accuracy and robustness: it matches the information-theoretic baseline on mean tracking accuracy while reducing the worst-case error by nearly a factor of ten, owing to the implicit filter-consistency regularisation provided by the Mahalanobis term in the reward.

The analysis of Temporal Difference (TD) learning in the average-reward setting faces notable theoretical difficulties because the Bellman operator is not contractive with respect to any norm. This complicates standard analyses of stochastic updates that are effective in discounted settings. Although a considerable body of literature addresses these challenges, existing theoretical approaches come with limitations. We introduce a novel algorithm designed explicitly for policy evaluation in the average-reward setting, utilizing sampling from two Markovian trajectories. Our proposed method overcomes previous limitations by guaranteeing convergence to the unique solution of a properly defined projected Bellman equation. Notably, and in contrast to earlier work, our convergence analysis is uniformly applicable to both linear function approximation and tabular settings and does not involve explicit dimension-dependent terms in its convergence bounds. These results align with what is known to hold in the discounted setting. Furthermore, our algorithm achieves improved dependence on the problem's condition number, reducing the sample complexity from quartic, as in prior literature, to quadratic scaling, and thus matching the efficiency seen in the discounted setting.

Off-policy temporal-difference (TD) learning with function approximation faces a structural tradeoff among stability, projection geometry, and variance control. Emphatic TD (ETD) improves the off-policy projection geometry through follow-on emphasis, but the follow-on trace can have high variance. We revisit this tradeoff through Bellman-error centering. Although centering naturally removes a common drift term from TD errors, we show that a naive centered emphatic extension introduces an auxiliary coupling that can destroy the positive-definiteness of the ETD key matrix. We propose \emph{Regularized Emphatic Temporal-Difference Learning} (RETD), which preserves the follow-on trace and regularizes only the auxiliary centering recursion, corresponding to lifting the lower-right block of the coupled key matrix from $1$ to $1+c$. We derive the RETD core matrix, prove convergence under a conservative sufficient regularization condition, and evaluate the method on diagnostic linear off-policy prediction tasks. The experiments show that RETD avoids the instability of naive centered emphatic learning, preserves favorable emphatic geometry, and exhibits a robust intermediate regime for the regularization parameter $c$ across the diagnostics.

Learning-based approaches to verifying unknown Markov decision processes (MDPs) often employ uncertain MDPs. These models use, for example, confidence intervals to capture transition uncertainty and allow synthesis of policies that are robust to this uncertainty. However, this approach typically quantifies uncertainty independently for individual transition probabilities, ignoring dependencies due to shared latent quantities. We propose to learn such models using parametric MDPs (pMDPs), where transition probabilities are expressions over a set of parameters. We project statistical uncertainty from empirical transition frequencies onto the pMDP's parameter space, yielding a probably approximately correct (PAC) uncertainty model for the underlying MDP that respects the algebraic dependencies between transitions. The resulting models are algorithmically challenging to solve, so we propose a hierarchy of sound polytopic outer approximations of the induced confidence set. We implement and evaluate our approach, demonstrating substantially tighter uncertainty estimates than classical interval-based uncertain MDP learning techniques.

Reinforcement learning (RL) is a fundamental framework for sequential decision-making, in which an agent learns an optimal policy through interactions with an unknown environment. In settings with function approximation, many existing RL algorithms achieve favorable sample complexity, but often rely on computationally intractable oracles. In this paper, we use supervised learning as a computational proxy to establish a clear hierarchy of commonly adopted RL oracles under low-rank Markov Decision Processes (MDPs). This hierarchy shows that policy evaluation is the most computationally efficient oracle, provided that supervised learning can be efficiently solved. Motivated by this observation, we propose a novel optimistic actor-critic algorithm that relies solely on the policy evaluation oracle. We prove that our algorithm outperforms the existing sample complexity guarantees for low-rank MDPs while avoiding computationally expensive planning or optimization oracles commonly assumed in prior works. We further extend our theoretical results to approximately low-rank MDPs and demonstrate that this setting captures a broad class of real-world environments. Finally, we validate our theoretical results with experiments on several standard Gym environments.

We model the interaction between a user and an AI driven recommendation system. The user initiates the process by conveying preference information through a costly and noisy message. The AI assistant, acting as a Bayesian agent, interprets the user's message to form a posterior belief about their true preferences and make product recommendations. In particular, it determines how many recommendations to present so as to maximize the user's expected utility from their final choice, while accounting for the search cost induced by the size of the recommendation set. We use mutual information based cost functions to model the two distinct costs incurred by the user during the interaction: (i) a communication cost, which increases with the precision of their preference message, and (ii) a search cost, which increases with the size of the recommendation set provided by the AI assistant. We study products and preferences which live in d dimensional space, and ask how the user's expected payoff can be maximized. For large d, we characterize how optimal message precision and recommendation set size depend on the cost parameters, under two distinct distributions from which recommendations can be sampled from the product universe: (i) Bayes' posterior belief, and (ii) an optimized tilted distribution. Under the posterior sampling scheme (i), we identify a hybrid regime, in which an efficient interaction policy requires jointly optimizing the amount of information (in bits) conveyed by the user and the number of recommendations provided by the AI assistant. In the tilted sampling scheme (ii), our results show that the optimal interaction policy uses only one of communication and search, favoring whichever of them is less costly.

While Bellman equations for basic reach, avoid, and reach-avoid problems are well studied, the relationship between value optimality and policy optimality becomes subtle in the undiscounted infinite-horizon setting, particularly for more complicated tasks. Greedily maximizing the Q-function can produce policies that indefinitely defer task completion for reach-avoid problems, or equivalently, Until specifications, even when the value function is optimal. Building upon recent results decomposing the value function for temporal logic (TL) into a graph of constituent value functions, we construct non-Markovian policies based on state history that avoid this pathology and prove their optimality with respect to the quantitative robustness score for nested Until, Globally, and Globally-Until specifications. We further show how the Q function can serve as a safety filter for complex TL specifications, extending prior results beyond simple avoid or reach-avoid tasks.

While representation and similarity learning have improved the sample efficiency of Reinforcement Learning (RL), they are rarely used to shape policy updates directly in the action space. To bridge this gap, a geometry-aware RL algorithm that explicitly incorporates value-based similarity into the policy update, State-Action Value Geometry Optimization (SAVGO), is proposed. In detail, SAVGO learns a joint state-action embedding space in which pairs with similar action-value estimates exhibit high cosine similarity, while dissimilar pairs are mapped to distinct directions. This learned geometry enables the generation of a similarity kernel over candidate actions sampled at each update, allowing policy improvement to be guided directly toward higher-value regions beyond local gradient-based updates. As a result, representation learning, value estimation, and policy optimization are unified within a single geometry-consistent objective, while preserving the scalability of off-policy actor-critic training. The proposed method is evaluated on standard MuJoCo continuous-control benchmarks, demonstrating improvements over strong baselines on challenging high-dimensional tasks. Ablation studies are done to analyze the contributions of value-geometry learning and similarity-based policy updates.

Reinforcement learning (RL) in large environments often suffers from severe computational bottlenecks, as conventional regret minimization algorithms require repeated, costly calls to planning and statistical estimation oracles. While recent advances have explored offline oracle-efficient algorithms, their computational complexity typically scales with the cardinality of the state and action spaces, rendering them intractable for large-scale or continuous environments. In this paper, we address this fundamental limitation by studying offline oracle-efficient episodic RL through the lens of log-barrier and log-determinant regularization. Specifically, for tabular Markov Decision Processes (MDPs), we propose a novel algorithm that achieves the optimal $\tilde{O}(\sqrt{T})$ regret bound while requiring only $O(H\log\log T)$ calls to both the offline statistical estimation and planning oracles when $T$ is known and $O(H\log T)$ calls when $T$ is unknown. Crucially, this oracle complexity is entirely independent of the size of the state and action spaces. This strict independence drastically reduces the planning oracle complexity, representing a substantial improvement over existing offline oracle-efficient algorithms (Qian et al., 2024). Furthermore, we demonstrate the versatility of our framework by generalizing the algorithm to linear MDPs featuring infinite state spaces and arbitrary action spaces. We prove that this generalized approach successfully attains meaningful sub-linear regret. Consequently, our work yields the first doubly oracle-efficient (i.e., efficient with respect to both statistical estimation and policy optimization) regret minimization algorithm capable of solving MDPs with infinite state and action spaces, significantly expanding the boundaries of computationally tractable RL.

Deploying reinforcement learning policies in the real world requires adapting to time-varying environments. We study this problem in the contextual Markov Decision Process (cMDP) framework, where a family of environments is indexed by a low-dimensional context unknown at test time. The standard approach decomposes the problem: train a so-called "universal policy" which assumes knowledge of the true context, then pair it with a context estimator which approximates context using the observed trajectory. We identify a simple, counterintuitive trick that substantially improves the estimator: randomly delete a fraction of the training buffer after each round. This works because data is collected across multiple rounds using progressively better policies, and older trajectories come from a different distribution than what the estimator will face at deployment time; random deletion creates an implicit exponential decay on older data while preserving diversity without requiring any explicit identification of which samples are stale. This reduces robustness gap by 30% for MLPs and by 6% on average for recurrent networks. Strikingly, it allows a narrow MLP with 5x fewer parameters to outperform a wide MLP trained without deletion. To understand when and why deletion helps, we analyze regularized empirical risk minimization with a mismatch between the train distribution and the distribution at deployment; in this idealized setting, we prove that removing a single uniformly random training point decreases expected test loss in expectation under mild conditions. For ridge regression we make this quantitative: deletion helps when the regularization coefficient is moderate and the signal-to-noise ratio (SNR) is sufficiently low, and, crucially, this SNR threshold gives a direct measure of how large the distribution mismatch between training and deployment must be for deletion to be beneficial.

Recent advances in large language models (LLMs) have increasingly relied on reinforcement learning (RL) to improve their reasoning capabilities. Three types of approaches have been widely adopted: The first relies on a deep neural network to estimate the value function of the learning policy in order to reduce the variance of the policy gradient. However, estimating and maintaining such a value network incurs substantial computational and memory overhead. The second avoids training a value network by approximating the value function using sample averages. However, it samples a large number of reasoning traces per prompt for accurate value function approximation, making it computationally expensive. The third samples only a single reasoning trajectory per prompt, which reduces computational cost but suffers from poor sample efficiency. This paper focuses on a practical, resource-constrained setting in which only a small number of reasoning traces can be sampled per prompt, while low-variance gradient estimation remains essential for high-quality policy learning. To address this challenge, we bring classical nonparametric statistical methods, which are both computationally and statistically efficient, to LLM reasoning. We employ kernel smoothing as a concrete example for value function estimation and the subsequent policy optimization. Numerical and theoretical results demonstrate that our proposal achieves accurate value and gradient estimation, leading to improved policy optimization.

This paper investigates the continuous-time counterpart of the Q-function for entropy-regularized mean-field control (MFC) with controlled common noise, coined as q-function by Jia and Zhou (2023) in the single agent's model. We first show that, under discretely sampled actions, the value function in the exploratory formulation converges to the one in the relaxed control formulation as the time grid refines. Leveraging the relaxed control formulation, we derive the exploratory Hamilton-Jacobi-Bellman (HJB) equation, in which the controlled common noise gives rise to an additional nonlinear functional of policy, rendering the policy iteration intricate. Under certain concavity condition, we establish the existence and uniqueness of the optimal one-step policy iteration via a first-order condition using the partial linear functional derivative with respect to policy. The policy improvement at each iteration is verified by relating to an entropy-regularized optimization problem over the space of policies. In the mean-field setting, we introduce the integrated q-function (Iq-function) defined on the state distribution and the policy, and it is shown that an optimal policy is identified as a two-layer fixed point to the argmax operator of the Iq-function. Finally, we provide the explicit characterization of an optimal policy as a Gaussian distribution in the general linear-quadratic (LQ) setting.

Reinforcement Learning (RL) algorithms exhibit high sample complexity, particularly when applied to Decentralized Partially Observable Markov Decision Processes (Dec-POMDPs). As a response, projects such as SampleFactory, EnvPool, Brax, and IsaacLab migrate parallel execution of classic environments such as MuJoCo and Atari into C++ thread pools or the GPU to decrease the computational cost of environment steps. We are interested in optimizing the decision-level of human-AI joint operations, so we introduce a compute-efficient Dec-POMDP engine natively architected in C++ called Hide-And-Seek-Engine. By employing Data-Oriented Design (DOD) principles, explicit 64-byte cache-line alignment to remove false sharing, and a zero-copy PyTorch memory bridge using pinned memory and Direct Memory Access (DMA), our engine sustains throughput of up to 33,000,000 steps per second (SPS) in a single-agent, 1024-environment, decentralized observations on an AMD Ryzen 9950X (16 cores). Ten agents reduces FPS to 7M SPS with generating random actions contributing 1/3rd the total runtime for reference. The engine achieves a throughput increase of approximately 3,500$\times$ over the baseline single threaded vectorized NumPy implementation and successfully trains cooperative multi-agent policies via PPO, DQN, and SAC in minutes, validating both its performance and generality.

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

Contact variability, sensing uncertainty, and external disturbances make grasp execution stochastic. Expected-quality objectives ignore tail outcomes and often select grasps that fail under adverse contact realizations. Risk-sensitive POMDPs address this failure mode, but many use particle-filter beliefs that scale poorly, obstruct gradient-based optimization, and estimate Conditional Value-at-Risk (CVaR) with high-variance approximations. We instead formulate grasp acquisition as variational inference over latent contact parameters and object pose, representing the belief with a differentiable Gaussian mixture. We use Gumbel-Softmax component selection and location-scale reparameterization to express samples as smooth functions of the belief parameters, enabling pathwise gradients through a differentiable CVaR surrogate for direct optimization of tail robustness. In simulation, our variational neural belief improves robust grasp success under contact-parameter uncertainty and exogenous force perturbations while reducing planning time by roughly an order of magnitude relative to particle-filter model-predictive control. On a serial-chain robot arm with a multifingered hand, we validate grasp-and-lift success under object-pose uncertainty against a Gaussian baseline. Both methods succeed on the tested perturbations, but our controller terminates in fewer steps and less wall-clock time while achieving a higher tactile grasp-quality proxy. Our learned belief also calibrates risk more accurately, keeping mean absolute calibration error below 0.14 across tested simulation regimes, compared with 0.58 for a Cross-Entropy Method planner.

We study city-scale control of electric-vehicle (EV) ride-hailing fleets where dispatch, repositioning, and charging decisions must respect charger and feeder limits under uncertain, spatially correlated demand and travel times. We formulate the problem as a hex-grid semi-Markov decision process (semi-MDP) with mixed actions -- discrete actions for serving, repositioning, and charging, together with continuous charging power -- and variable action durations. To guarantee physical feasibility during both training and deployment, the policy learns over high-level intentions produced by a masked, temperature-annealed actor. These intentions are projected at every decision step through a time-limited rolling mixed-integer linear program (MILP) that strictly enforces state-of-charge, port, and feeder constraints. To mitigate distributional shifts, we optimize a Soft Actor-Critic (SAC) agent against a Wasserstein-1 ambiguity set with a graph-aligned Mahalanobis ground metric that captures spatial correlations. The robust backup uses the Kantorovich-Rubinstein dual, a projected subgradient inner loop, and a primal-dual risk-budget update. Our architecture combines a two-layer Graph Convolutional Network (GCN) encoder, twin critics, and a value network that drives the adversary. Experiments on a large-scale EV fleet simulator built from NYC taxi data show that PD-RSAC achieves the highest net profit, reaching $1.22M, compared with $0.58M-$0.70M for strong heuristic, single-agent RL, and multi-agent RL baselines, including Greedy, SAC, MAPPO, and MADDPG, while maintaining zero feeder-limit violations.

Traffic signal control (TSC) plays a central role in reducing congestion and maintaining urban mobility. This dissertation introduces DGLight, a critic-guided reinforcement-learning framework for adapting a pretrained large language model to TSC. DGLight first trains a CoLight-based Deep Q-Network critic to estimate traffic-aware action values from structured intersection states, then uses the frozen critic to score candidate language-model actions and optimize the policy with Group Relative Policy Optimization (GRPO). The resulting controller maps traffic states to interpretable reasoning traces and signal decisions while learning from dense per-state supervision rather than raw cumulative environment rewards. Experiments on TSC benchmarks covering Jinan and Hangzhou show that DGLight is the strongest overall method among the compared LLM-based controllers, remains competitive with strong RL baselines, and transfers well to city datasets not used to fit the critic. Qualitative examples further show that the model's generated reasoning is interpretable and aligned with the chosen signal phase. The project code is available $\href{https://github.com/yyccbb/FYP_LLMTSC}{here}$.

Fraud can pose a challenge in many resource allocation domains, including social service delivery and credit provision. For example, agents may misreport private information in order to gain benefits or access to credit. To mitigate this, a principal can design strategic audits to verify claims and penalize misreporting. In this paper, we introduce a general model of audit policy design as a principal-agent game with multiple agents, where the principal commits to an audit policy, and agents collectively choose an equilibrium that minimizes the principal's utility. We examine both adaptive and non-adaptive settings, depending on whether the principal's policy can be responsive to the distribution of agent reports. Our work provides efficient algorithms for computing optimal audit policies in both settings and extends these results to a setting with limited audit budgets.

This paper explores the application of the Soft Actor-Critic (SAC) algorithm within a Distributional Reinforcement Learning setting and introduces an implementation of such algorithm named Cramér-based Distributional Soft Actor-Critic (C-DSAC). The novel approach employs distributional reinforcement learning to represent state-action values, and minimizes the squared Cramér distance for learning the distribution. Empirical results across various robotic benchmarks indicate that our algorithm surpasses the performance of baseline SAC and contemporary distributional methods, with the performance advantage becoming increasingly pronounced in high-complexity environments. To explain the efficiency of the new approach, we conduct an analysis showing that its superior performance is partly due to \textit{confidence-driven} Q-value updates: High-variance target distributions (low confidence in target) lead to more conservative model updates, thereby attenuating the impact of overestimated values. This work deepens the understanding of distributional reinforcement learning, offering insights into the algorithmic mechanisms governing convergence and value estimation.

Aligning denoising generative models with human preferences or verifiable rewards remains a key challenge. While policy-gradient online reinforcement learning (RL) offers a principled post-training framework, its direct application is hindered by the intractable likelihoods of these models. Prior work therefore either optimizes an induced Markov decision process (MDP) over sampling trajectories, which is stable but inefficient, or uses likelihood surrogates based on the diffusion evidence lower bound (ELBO), which have so far underperformed on visual generation. Our key insight is that the ELBO-based approach can, in fact, be made both stable and efficient. By reducing surrogate variance and controlling gradient steps, we show that this approach can beat MDP-based methods. To this end, we introduce Variational GRPO (V-GRPO), a method that integrates ELBO-based surrogates with the Group Relative Policy Optimization (GRPO) algorithm, alongside a set of simple yet essential techniques. Our method is easy to implement, aligns with pretraining objectives, and avoids the limitations of MDP-based methods. V-GRPO achieves state-of-the-art performance in text-to-image synthesis, while delivering a $2\times$ speedup over MixGRPO and a $3\times$ speedup over DiffusionNFT.

Reward models in RLHF are trained to score only the final token of a response - a choice that discards rich signal from every intermediate position and produces models whose token-level outputs are noise. We argue this is a missed opportunity: a well-trained reward model's output at any token should represent the conditional expectation of the final reward given the response so far. We introduce Temporally Coherent Reward Modeling (TCRM), which induces this property via two regularization terms on top of the standard Bradley-Terry loss, with minimizers provably equal to conditional expectations. The regularizers correspond to Monte Carlo and TD value-learning objectives, establishing a direct connection to RL value functions. TCRM requires zero changes to architecture, data, or inference, yet unlocks three capabilities from one principle: interpretable token-level reward trajectories (middle-token pairwise accuracy improved from 50% to 88.9%, final-token accuracy preserved); state-of-the-art PRM performance on ProcessBench (44.9% average F1) among models trained only on outcome data; and unified reward/value modeling in PPO, reducing peak GPU memory by 27% and step time by 19% with matching LLM quality.

Here, we explore the problem of error propagation mitigation in modular digital twins as a sequential decision process. Building on a companion study that used a Hidden Markov Model (HMM) to infer latent error regimes from surrogate-physics residuals, we develop a Markov Decision Process (MDP) in which the inferred regimes serve as states, corrective interventions serve as actions, and a scalar reward that takes into consideration the cost-benefit tradeoff between system fidelity and maintenance expense. The baseline transition matrix is extracted from the HMM-learned parameters. We then extend the formulation to a Partially Observable MDP (POMDP) that accounts for the imperfect nature of regime classification by maintaining a belief distribution updated via Bayesian filtering, with the HMM confusion matrix serving as the observation model. Both formulations are solved via dynamic programming and validated through Gillespie stochastic simulation. We then benchmark two model-free reinforcement learning algorithms, Q-learning and REINFORCE, to assess whether effective policies can be learned without explicit model knowledge. A systematic comparison of different intervention policies demonstrates that the MDP policy achieves the highest cumulative reward and fraction of time in nominal operation, while the POMDP recovers approximately 95% of MDP performance under realistic observation noise. Sensitivity analyses across observation quality, repair probability, and discount factor confirm the robustness of these conclusions, and the major gaps in the policy hierarchy are statistically significant at $p < 0.001$. The gap between MDP and POMDP performance quantifies the value of information providing a principled criterion for investing in improved classification accuracy.

We study the problem of adaptive control of the stochastic linear quadratic regulator (LQR) with constraints that must be satisfied at every time step. Prior work on the multidimensional problem has shown $\tilde{O}(T^{2/3})$ regret and satisfaction of robust constraints, leaving open the question of whether $\tilde{O}(\sqrt{T})$ regret can be attained in the constrained LQR setting. We contribute to this problem by showing $\tilde{O}(\sqrt{T})$ regret and satisfaction of chance constraints. This type of constraints allow us to handle unbounded noise and also enable analytical techniques not directly applicable to robust constraints. Our proposed algorithm for this problem uses an SDP to select an optimistic policy, and then "scales back" this policy until it is verifiably-safe. Our theoretical analysis establishes regret and constraint guarantees via a key lemma that bounds the system covariance in terms of the chosen policy. This covariance-based analysis is in contrast with the cost-to-go based analysis that is typically used in adaptive LQR.

Simultaneous localization and mapping (SLAM) is a foundational state estimation problem in robotics in which a robot accurately constructs a map of its environment while also localizing itself within this construction. We study the active SLAM problem through the lens of optimal stochastic control, thereby recasting it as a decision-making problem under partial information. After reviewing several commonly studied models, we present a general stochastic control formulation of active SLAM together with a rigorous treatment of motion, sensing, and map representation. We introduce a new exploration stage cost that encodes the geometry of the state when evaluating information-gathering actions. This formulation, constructed as a nonstandard partially observable Markov decision process (POMDP), is then analyzed to derive rigorously justified approximate solutions that are near-optimal. To enable this analysis, the associated regularity conditions are studied under general assumptions that apply to a wide range of robotics applications. For a particular case, we conduct an extensive numerical study in which standard learning algorithms are used to learn near-optimal policies.

Policies for Partially Observable Markov Decision Processes (POMDPs) are often designed using a nominal system model. In practice, this model can deviate from the true system during deployment due to factors such as calibration drift or sensor degradation, leading to unexpected performance degradation. This work studies policy robustness against deviations in the POMDP observation model. We introduce the Policy Observation Robustness Problem: to determine the maximum tolerable deviation in a POMDP's observation model that guarantees the policy's value remains above a specified threshold. We analyze two variants: the sticky variant, where deviations are dependent on state and actions, and the non-sticky variant, where they can be history-dependent. We show that the Policy Observation Robustness Problem can be formulated as a bi-level optimization problem in which the inner optimization is monotonic in the size of the observation deviation. This enables efficient solutions using root-finding algorithms in the outer optimization. For the non-sticky variant, we show that when policies are represented with finite-state controllers (FSCs) it is sufficient to consider observations which depend on nodes in the FSC rather than full histories. We present Robust Interval Search, an algorithm with soundness and convergence guarantees, for both the sticky and non-sticky variants. We show this algorithm has polynomial time complexity in the non-sticky variant and at most exponential time complexity in the sticky variant. We provide experimental results validating and demonstrating the scalability of implementations of Robust Interval Search to POMDP problems with tens of thousands of states. We also provide case studies from robotics and operations research which demonstrate the practical utility of the problem and algorithms.

We propose a learning-augmented framework for accelerating max-flow computation and image segmentation by integrating Graph Neural Networks (GNNs) with the Ford-Fulkerson algorithm. Rather than predicting initial flows, our method learns edge importance probabilities to guide augmenting path selection. We introduce a Message Passing GNN (MPGNN) that jointly learns node and edge embeddings through coupled updates, capturing both global structure and local flow dynamics such as residual capacity and bottlenecks. Given an input image, we propose a method to construct a grid-based flow network with source and sink nodes, extract features, and perform a single GNN inference to assign edge probabilities reflecting their likelihood of belonging to high-capacity cuts. These probabilities are stored in a priority queue and used to guide a modified Ford-Fulkerson procedure, prioritizing augmenting paths via an Edmonds-Karp-style search with bottleneck-aware tie-breaking. This avoids repeated inference over residual graphs while leveraging learned structure throughout optimization. We further introduce a bidirectional path construction strategy centered on high-probability edges and provide a theoretical framework relating prediction quality to efficiency via a weighted permutation distance metric. Our method preserves max-flow/min-cut optimality while reducing the number of augmentations in practice. We also outline a hybrid extension combining flow warm-starting with edge-priority prediction, establishing a foundation for learning-guided combinatorial optimization in image segmentation.

Autonomous systems that rely on learned perception can make unsafe decisions when sensor readings are misclassified. We study shielding for this setting: given a proposed action, a shield blocks actions that could violate safety. We consider the common case where system dynamics are known but perception uncertainty must be estimated from finite labeled data. From these data we build confidence intervals for the probabilities of perception outcomes and use them to model the system as a finite Interval Partially Observable Markov Decision Process with discrete states and actions. We then propose an algorithm to compute a conservative set of beliefs over the underlying state that is consistent with the observations seen so far. This enables us to construct a runtime shield that comes with a finite-horizon guarantee: with high probability over the training data, if the true perception uncertainty rates lie within the learned intervals, then every action admitted by the shield satisfies a stated lower bound on safety. Experiments on four case studies show that our shielding approach (and variants derived from it) improves the safety of the system over state-of-the-art baselines.

We propose SmoothCruiser, a new planning algorithm for estimating the value function in entropy-regularized Markov decision processes and two-player games, given a generative model of the environment. SmoothCruiser makes use of the smoothness of the Bellman operator promoted by the regularization to achieve problem-independent sample complexity of order O~(1/epsilon^4) for a desired accuracy epsilon, whereas for non-regularized settings there are no known algorithms with guaranteed polynomial sample complexity in the worst case.

High-fidelity measurements of continuum physical fields are essential for scientific discovery and engineering design but remain challenging under sparse and constrained sensing. Conventional reconstruction methods typically rely on fixed sensor layouts, which cannot adapt to evolving physical states. We propose LASER, a unified, closed-loop framework that formulates active sensing as a Partially Observable Markov Decision Process (POMDP). At its core, LASER employs a continuum field latent world model that captures the underlying physical dynamics and provides intrinsic reward feedback. This enables a reinforcement learning policy to simulate ''what-if'' sensing scenarios within a latent imagination space. By conditioning sensor movements on predicted latent states, LASER navigates toward potentially high-information regions beyond current observations. Our experiments demonstrate that LASER consistently outperforms static and offline-optimized strategies, achieving high-fidelity reconstruction under sparsity across diverse continuum fields.

Safe Reinforcement Learning from Human Feedback (Safe RLHF) has recently achieved empirical success in developing helpful and harmless large language models by decoupling human preferences regarding helpfulness and harmlessness. Existing approaches typically rely on fitting fixed horizon reward models from human feedback and have only been validated empirically. In this paper, we formulate safe RLHF as an infinite horizon discounted Con- strained Markov Decision Process (CMDP), since humans may interact with the model over a continuing sequence of interactions rather than within a single finite episode. We propose two Safe RLHF algorithms that do not require reward model fitting and, in contrast to prior work assuming fixed-length trajectories, support flexible trajectory lengths for training. Both algo- rithms are based on the primal-dual method and achieve global convergence guarantees with polynomial rates in terms of policy gradient iterations, trajectory sample lengths, and human preference queries. To the best of our knowledge, this is the first work to study infinite horizon discounted CMDP under human feedback and establish global, non-asymptotic convergence.