Optimal Policies

In evidence-based policymaking, typically one experimental sample is observed, then a learned policy recommendation is implemented at scale. Policies learned from the experimental data can perform well in expected welfare, yet random sampling in the experiment can produce recommendations with poor welfare outcomes. In this paper, we ask: how should policy learning algorithms balance expected welfare against sampling risk? Our main contribution is to show that algorithmic stability plays a central role in characterizing and navigating the tradeoff. Intuitively, if a policy learning algorithm's recommendation remains stable when one experimental unit is replaced, then that algorithm has limited sampling risk. We propose a method for policy learning called policy-vote bagging, which learns treatment decisions on many subsamples then averages their votes into treatment probabilities. Relative to using one subsample, averaging across subsamples preserves expected welfare and improves expected utility for a risk-averse researcher. We derive sharp bounds linking estimation accuracy, subsample size, and welfare variation, including an exact guarantee under CARA utility.

In multi-agent environments, coordinating agents to prevent interference and ensure robust individual performance is a critical challenge. Previous research on social laws for multi-agent systems has primarily focused on deterministic, goal-based settings. This paper extends the concept of social laws to stochastic, reward-based environments, proposing a formalism for defining and verifying their robustness under various conditions. We introduce the notion of $α$-robustness, a measure of the guaranteed utility each agent retains while pursuing its optimal single agent policy, assuming all agents obey the social law. We then present an approach for robustness verification of social laws in stochastic settings, based on a reduction to solving a series of Markov decision processes. Empirical evaluations on toy environments illustrate the potential of our framework.

We introduce the first Probably Approximately Correct (PAC) learning framework for general-sum concurrent stochastic games (CSGs) with transition uncertainty, while addressing the challenge of Nash equilibrium (NE) existence. Our algorithm maintains data-driven $L^1$ confidence sets over transition kernels and solves a robust CSG to compute a social-welfare optimal $\varepsilon$-NE, using a robust MDP-based exploration mechanism to drive joint state-action coverage. Crucially, we introduce a Nash margin characterisation that enables principled reasoning about equilibrium existence: the framework either returns an $\varepsilon$-approximate NE whose social-welfare value is $\varepsilon$-close to optimal, or provides a sound certificate that no exact NE exists. Under a minimum reachability condition $p_{\mathrm{reach}}>0$ over relevant state-action pairs, the algorithm terminates after a polynomial number of trajectory samples, with sample complexity $\widetilde{O}\left( {R_{\max}^2 H^4 |S|^2 |A| / (p_{\mathrm{reach}} \varepsilon^2)} \right)$. Empirical results on benchmark CSGs demonstrate near-optimal performance, correct handling of equilibrium (non-)existence, and sample complexity consistent with theory.

Whittle index policies offer a scalable method for restless multi-armed bandits, but under partial observability even determining the indifference subsidy at a single belief requires solving an infinite-horizon belief-state problem with no closed-form value function. Liu [10] addresses this difficulty by linearizing the unknown decision boundary, leading to a linear system and a closed-form approximate Whittle index. However, the resulting threshold uses only a one-step active--passive comparison and does not account for longer-horizon continuation values. We extend this framework to a \emph{$t$-step lookahead threshold policy}. For each subsidy $m$, the threshold is defined by the active-minus-passive advantage under $t$-step finite-horizon value iteration. At $t=1$, the threshold is $m$-independent and recovers the linear threshold of Liu [10]; for $t>1$, it becomes subsidy-dependent through the induced first-crossing structure and tracks the exact decision boundary more closely. The proposed algorithm does not require indexability as an input and includes an indexability verification. Under the original Whittle indexability, we prove that the $t$-step approximate Whittle index converges geometrically to the exact Whittle index, $$|\widehat W_t(ω)-W(ω)|=O(β^t).$$ Numerically, all 2,715 tested three-state instances are verified as indexable according to the proposed criterion. The P95 index error decreases from $2.18\times10^{-2}$ at $t=1$ to $8.93\times10^{-4}$ at $t=8$. In an exact-comparable instance with $β=0.9999$, $t=2$ already recovers the exact Whittle-index ordering. Moderate-depth threshold policies also outperform the one-step baseline and remain close to the optimal dynamic-programming benchmark, while runtime grows mildly with $t$.

Online platforms increasingly compare many adaptive decision policies---ranking systems, recommendation algorithms, pricing rules, and language-model agents---while each reward-bearing interaction can be costly or risky. A direct A/B/n design gives each of $J$ policies its own horizon-$T$ trajectory and therefore uses $JT$ outcomes. We introduce Tree-Coupled A/B Testing (\TCAB), an exact feedback-sharing design for arbitrary history-dependent contextual-bandit policies. At each round, a predictable tree connects the current policy histories; every parent--child context--action law is maximally coupled, and one reward is shared within each component of matched tree edges. Every policy retains exactly its standalone finite-horizon trajectory law, even though the policies are deliberately dependent. If $D_{e,t}$ records a mismatch on tree edge $e$ at round $t$, the number of reward queries satisfies the pathwise identity $N(T)=T+\sum_{t,e}D_{e,t}$ and hence equals $T$ plus cumulative tree-edge total variation in expectation. This cost is conditionally optimal among exact edge-local designs on the selected tree, and a current-round minimum-spanning tree is myopically optimal among tree designs. For fixed $J$, sublinear pseudo-regret of every policy and almost-sure uniqueness of the oracle action imply $\mathbb{E}[N(T)]=T+o(T)$, versus $JT$ for independent runs. We also obtain finite-sample variance bounds for pairwise policy contrasts. Experiments on reward-model evaluation, multiple-choice language-model evaluation, and adaptive search policies demonstrate substantial improvements in the cost--precision frontier.

We study a Restart POMDP (Partially Observable Markov Decision Process) on a general Borel state space, where the controller either lets the hidden state evolve unobserved or restarts the system and observes the new state. Exploiting a sufficient-statistic representation consisting of the last observed state and the elapsed time since restart, we reduce the problem to a fully observed MDP. Under a natural one-step cost deterioration condition, we prove that optimal policies have a threshold structure in the elapsed time for both the discounted and total undiscounted cost criteria. When the state space is partially ordered and the kernel is stochastically monotone, we further show that the optimal threshold is nonincreasing in the state. For the average cost criterion, under additional assumptions of geometric ergodicity and domination of the transient gain, we establish analogous threshold results via the vanishing discount approach, after showing the uniform boundedness of the optimal thresholds and relative value functions.

We study policy-based reinforcement learning under the $μ$-resets interaction protocol of Kakade and Langford [KL02]. This interaction protocol enables the learner to sample trajectories from a given exploratory reset distribution $μ$, in addition to the starting distribution. We resolve the question raised by [KLS25] on the role of policy realizability for the sample complexity of this problem. Critically, the dependence on horizon $H$ is governed by the notion of coverage assumed of the reset distribution. Under bounded all-policy concentrability, we show a $\exp(Ω(H))$ sample complexity lower bound; with bounded pushforward concentrability, we show the dependence on horizon is tightly characterized as $\exp(Θ(\sqrt H))$.

Contextual bandits offer a natural framework for sample-efficient personalization, but practical deployment remains difficult under sparse, biased interaction data, unreliable uncertainty estimates, and severe cold starts. We study whether pre-trained tabular foundation models with in-context learning can be turned into randomized policies for online decision making. We propose BC-ICL (Bootstrap-conditioned action selection using ICL), which at each round draws a bootstrap resample of the interaction history, conditions a frozen pre-trained ICL model on that resample, scores all actions, and selects the action with the highest sampled score. We further introduce an arm-context conditioning architecture that promotes shared statistical strength across actions and helps avoid common bootstrap failure modes of isolated-arm bandits. Empirically, this policy delivers strong early-round regret and regret performance on standard contextual bandit suites, outperforming established baselines under a strict online protocol.

Distributionally robust Markov decision processes provide a principled framework for sequential decision making under model uncertainty. We study how many samples are necessary and sufficient to learn an $\varepsilon$-optimal robust policy under the average-reward criterion. A generative model provides samples from the nominal transition kernel, whereas policy performance is evaluated over $(s,a)$-rectangular total-variation uncertainty sets of radius at most $σ$. Let $H_0$ and $H_σ$ denote the nominal and robust optimal bias spans, respectively. We identify $σH_0$ as the perturbation scale separating high- and low-tolerance regimes. Our matching upper and lower bounds show that, up to logarithmic factors, the minimax total sample complexity is $NSA \asymp \frac{SA}{\varepsilon^2}\begin{cases} \min\{H_0,H_σ\}, & \varepsilon\gtrsimσH_0,\\ \min\{H_0,H_σ\}+σH_σ^2, & \varepsilon\lesssimσH_0. \end{cases}$ Here $S$ and $A$ are the numbers of states and actions, and $N$ is the number of samples per state-action pair. The sample complexity consists of a linear-span term that resembles the nominal AMDP results and a robustness-specific term that appears only in the low-tolerance regime. We attain these rates using reduction-based plug-in procedures that select the reduction---nominal or robust---and its discount factor: a span-informed procedure that makes these choices using known span parameters, and a span-agnostic procedure that calibrates both choices from data.

Many social services assign scarce resources, such as housing assistance or hospital interventions, to people who arrive one at a time: each arrival must receive a decision immediately, and the long-run usage of every resource must stay within its capacity. We study how to learn such an assignment policy from logged observational data. The standard pipeline is decision-blind: fit one outcome model per arm by regression, price each capacitated resource from the fitted models, and assign each arrival the arm whose predicted outcome minus price is largest. We instead train the outcome models end-to-end, differentiating an off-policy estimate of the deployed policy's value through the dual prices themselves. We study two formulations: an exact nonconvex one, and a convex relaxation whose optimum always satisfies the capacity constraints in expectation and which is suboptimal by at most a term linear in the smoothing temperature and logarithmic in the number of arms. Every method is evaluated in a queueing simulation with resources replenished at their capacity rates. Across six datasets, the two end-to-end variants take the top slots on a deployment-adjusted value index at every delay cost, including zero; when capacities are binding, decision-blind baselines frequently violate them and incur much longer queueing delays. On the largest dataset, a hospital cohort of seventy thousand patients, end-to-end training also achieves significantly higher policy value, a margin that survives a capacity-matched neural baseline. Flexible decision-blind regression remains the stronger pure predictor where ground truth is measurable; end-to-end training is best suited to settings where resources are genuinely scarce and feasibility matters.

We study contextual dynamic pricing with arbitrary covariate sequences and bounded, possibly nonbinary purchase quantities. Demand follows a semiparametric surplus-index model with an unknown linear valuation parameter and an unknown Hölder-smooth response. We impose neither concavity nor strong unimodality on revenue and allow nonunique optimal prices. We develop a pilot-corrected layered decision-partitioning policy that combines directional pilot estimation, local polynomial learning, predictable data assignment, and global action elimination. Pilot correction removes the first-order effect of valuation-parameter error, while permanent labels enable concentration under adaptive sampling. The policy attains the minimax smoothness-dependent horizon rate up to logarithmic factors; a matching lower bound already holds for a constant-context binary-demand subclass.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Tsitsiklis proved convergence of Monte Carlo optimistic policy iteration under a uniform update structure and identified nonuniform update frequencies as a delicate obstruction. We give a certified negative answer for the natural scalar-stepsize, unnormalized asynchronous state-value recursion with fixed nonuniform state-selection probabilities. In a three-state, two-action discounted MDP, the nonuniform update frequencies induce a diagonally scaled greedy-policy mean field with a certified nonconstant attracting hybrid periodic orbit. With a bounded unbiased geometric-horizon estimator and Robbins--Monro stepsizes, the original stochastic recursion remains trapped near the cycle with positive probability and therefore fails to converge. The example pinpoints a geometric obstruction: uniform sampling gives radial residual contraction, whereas scalar nonuniform sampling anisotropically distorts the residual dynamics and can generate switched attracting cycles.

In this paper, we develop a continuous-time model-free reinforcement learning algorithm to learn deterministic equilibrium policies in general time-inconsistent control problems. Utilizing the extended Hamilton-Jacobi-Bellman system, we recast the original time-inconsistent problem into an equivalent two-stage problem. In the first stage, for given auxiliary functions, we employ the deterministic policy gradient approach to learn an optimal policy in an auxiliary time-consistent control problem. In the second stage, given the updated policy, we exploit the inner fixed point iterations and some martingale characterizations to learn the auxiliary functions. As a theoretical contribution, we provide some mild model assumptions and establish the convergence of inner fixed point iterations. By repeating this actor-critic style of iterations across two stages, our algorithm aims to learn the equilibrium under different sources of time-inconsistency in a unified manner. The superior effectiveness of the proposed algorithm are illustrated in two classical financial applications with time-inconsistency: mean-variance portfolio management and optimal tracking portfolio under non-exponential discounting.

The asymptotic behaviour of Monte Carlo optimistic policy iteration (MC-O-PI) is a long-standing open question. When the model of the environment is unknown, as is common in practice, the only known condition that guarantees convergence to optimality is impractical. In its canonical form, this condition requires that the episodes used for policy evaluation be initialised uniformly over the entire state-action space. This paper strictly relaxes that requirement. Specifically, we prove that initial-visit MC-O-PI converges to optimality even when updates are uniform only over the actions within each state. This allows episodes to start in different states at arbitrary frequencies; a realistic implementation when the state space is large or unknown but the action space in each state is manageable. The proof departs from the classical analysis of Tsitsiklis whose central commutativity argument no longer applies when states are updated at different frequencies. Instead, we first show that the mean-field dynamics of MC-O-PI generate monotonically improving policies when updates are uniform over the actions in each state, and then prove that noise cannot consistently prevent this improvement by extending the lock-in argument of the combined stability-ODE method. This approach suggests a new way to study optimistic policy-iteration algorithms in general.

Scaling recommender systems via large language models (LLMs) has become a prominent trend in the industry. However, aligning the LLM's semantic space with the recommender's ID space via post-training (e.g., SFT and RL) remains challenging. Existing LLM4Rec paradigms are bottlenecked by two main issues: (1) the difficulty of measuring and improving chain-of-thought (CoT) quality in open-domain recommendation during SFT, and (2) the neglect of the trade-off between LLM semantic rewards and recommendation preference rewards during RL alignment. Inspired by these challenges, we present Taiji, a novel LLM-as-Enhancer framework designed for industrial recommender systems. To overcome the SFT bottleneck, we utilize reverse-engineered reasoning and open-ended rejection sampling to generate high-quality, domain-specific CoT data. To resolve the RL alignment issue, we propose Pareto Optimal Policy Optimization (POPO), which adaptively adjusts cross-domain reward weights. Theoretically, it achieves an optimal trade-off between the semantic world knowledge of LLMs and the collaborative ID features representing online user preferences. Extensive offline evaluations and online A/B tests validate the effectiveness of Taiji. Deployed on Kuaishou's advertising platform since May 2026, Taiji currently serves over 400 million users daily, yielding significant commercial revenue and demonstrating its robust scalability in web-scale environments.

Unmanned Aerial Vehicles (UAVs) equipped with radar sensors are deployed for target search missions in diverse environments, where targets exhibit characteristic signatures (e.g., respiration micro-motion in human search) detectable through occlusions. A fundamental challenge arises from shifts in radar statistics as the UAV moves through a dynamic and potentially non-stationary environment, rendering any fixed signal-processing strategy suboptimal; yet perception and adaptation must run onboard a resource-constrained aerial node in real time. Since no single detector performs well across all conditions, we adopt a multi-policy paradigm and formulate UAV target search as an online policy selection problem over a library of specialized detectors, with performance measured by regret, the cumulative loss gap relative to the best policy in each scene. The setting couples in-scene stochastic noise with inter-scene shifts. Whereas prior methods capture only one regime, we account for both through the Stochastically Extended Adversary (SEA) framework, without requiring oracle knowledge of scene dynamics. Because adaptation must run at the UAV, we instantiate SEA through \textsc{SEArch}, a lightweight optimistic Follow the Regularized Leader (OFTRL) selector with an adaptive learning rate, achieving regret $O(\barσ_T \sqrt{T} + \sqrt{J})$, where $\barσ_T$ captures radar measurement noise and $J$ is the number of scene transitions over the mission horizon $T$. To enable rapid adaptation under frequent scene changes, we further introduce \textsc{W-SEArch}, a windowed variant that restarts every $w$ rounds and achieves regret $O(\barσ_I \sqrt{w})$ under at most one transition per window. Experiments show up to 30% regret reduction compared to non-adaptive baselines across a range of non-stationary settings.

Decision-focused learning (DFL) trains predictive models by optimizing downstream decision quality rather than standalone prediction accuracy. For contextual linear optimization, most existing DFL methods assume offline data and full observations of the objective cost vector. We develop an on-policy learning method for sequential contextual linear optimization under partial feedback, generalizing the standard bandit feedback setting. Our method learns a stochastic predict-then-optimize policy that samples a cost-vector prediction from a conditional distribution and solves the resulting downstream linear optimization problem. To update this distributional model, we introduce a two-component hybrid gradient estimator. The first component is a score function estimator, which provides an unbiased but potentially high-variance policy gradient estimate. The second is a decision-focused plug-in component that uses an auxiliary nuisance estimate of the latent cost vector to exploit the downstream optimization structure, becoming more informative as the estimate improves. We prove an $\mathcal{O}(T^{-1/2})$ bound on the average squared policy-gradient norm, matching the standard non-convex SGD rate. Experiments on top-$k$ selection, shortest path, combinatorial pricing, and a real-data energy-scheduling benchmark show that the hybrid gradient approach achieves lower cumulative regret than contextual-bandit-style baselines across all benchmarks, using both Gaussian and richer conditional generative models. Code is available at https://github.com/Joeyetinghan/on-policy-bandit-dfl.

When many reinforcement-learning policies achieve near-optimal return, a post-hoc auditor may have to distinguish among many behaviorally distinct but return-equivalent policies. We formalize this phenomenon through an occupancy-measure analogue of Rashomon capacity: the metric entropy of the near-optimal occupancy region, computed relative to an audited deployment class. Because occupancy measures identify behavior only up to occupancy equivalence, we formulate auditing at the occupancy-class level and distinguish exact local-query oracles from noisy sample-query oracles. Our main exact-query result is conditional: if the audited class contains a $2/H$-separated near-optimal packing whose local signatures are $b$-sparse, then exact local-query auditing requires $Ω(M/b)$ queries; when the packing realizes deployment-class capacity and $b=O(1)$, this becomes $Ω(2^{\Hopt^\cF(\eps)})$. We give a finite discounted hidden-branch MDP attaining this bound and show the exact Bayes success law. For noisy hidden-trigger testing, we prove a mixture lower bound of order $M/β$, where $β$ is the per-sample KL signal, yielding $Ω(2^{\Hopt^\cF(\eps)}/(ρ^2Δ^2))$ for capacity-order packings with $β=O(ρ^2Δ^2)$. We also provide a static target-recognition information lower bound, a transcript-compatible oracle-cover verification upper bound, and a canonical occupancy regularizer whose regularized audited capacity collapses when a trusted reference occupancy is available. Controlled benchmarks distinguish positive sparse-signature instances from high-capacity negative controls where exact auditing is easy, and map the noisy-trigger law to post-processed continuous-control and visual-RL auditing regimes.

Reinforcement learning problems typically define the goal as maximizing the expected value of a scalar reward function. But, pairwise preferences are often easier to specify than scalar rewards, and they express certain goals that scalar rewards cannot. Methods for reinforcement learning with pairwise preferences have thus received growing interest. Unfortunately, these methods are inefficient in problems with long time horizons, and they lack guarantees on the performance of Markov policies relative to history-dependent policies, which bridge the theory and practice of reinforcement learning. We therefore propose the \textit{Markov decision contest} as a new problem model for reinforcement learning with pairwise preferences. We prove that stationary Markov policies are optimal among all history-dependent policies, that solving a Markov decision contest exactly is in P, and that a simple iterative algorithm converges to an optimal policy at a sublinear rate. Lastly, in a set of high-dimensional decision problems with long time horizons, we show that our approximate algorithm is significantly more learning-efficient than prior work.

Language models fine-tuned with reinforcement learning typically optimize for task reward, ignoring multi-agent strategic structure. Because these agents condition on natural language game-state descriptions and emit actions through free-form generation, strategic failure modes -- exploiting weaker opponents, coordinating on harmful equilibria, and externalizing costs are inseparable from the language interface itself. We propose Safe Equilibrium Policy Optimization (\sepo{}), a training objective that augments expected payoff with explicit penalties for exploitability, collusion risk, and externality cost. We implement \sepo{} as a reward signal for Group Relative Policy Optimization (GRPO), applied to Gemma4 E4B-it and Qwen3.5-4B after supervised fine-tuning (SFT). Evaluated across five strategic domains: Iterated Prisoner's Dilemma, repeated auctions, two negotiation variants, and Kuhn Poker. \sepo{} achieves zero exploit-pool advantage in Kuhn Poker for both models, outperforms the base model on safety in four domains, and corrects the over-cooperative behavior introduced by SFT. In negotiation, \sepo{} achieves a positive-safety outcome and only the positive normalized relative advantage of any negotiation configuration. Ablation experiments confirm that per-rollout exploit computation is necessary: a shared constant penalty cancels in GRPO advantage normalization (constant control-variate property), producing zero gradient. To support further research in strategic safety for agents, we release our \href{https://anonymous.4open.science/r/sepo-2668/README.md}{code} and SFT datasets.

Critical sequential decisions are rarely single-timescale: a strategic decision causally shapes the context in which every subsequent tactical choice is made; standard bandit and reinforcement-learning theory does not capture this causal coupling between timescales. We formalise the problem class as Nested Contextual Causal Bandits (NCCBs), a hierarchical SCM where each level's action sets the next level's context distribution, and propose Nested Causal Thompson Sampling (NCTS), which draws one mechanism-factorised belief per episode and acts recursively under it. Our main theoretical result is a causal PAC-Bayesian excess-risk bound that certifies any candidate deployment policy from historic data alone, off-policy and anytime, answering the deployment question: can we trust this agent here, and at what risk? Experiments on a hierarchical SCM show that, against a matched RFF-GP joint regression on the same function class, the factorised SCM-mechanism posterior transfers significantly better zero-shot under exogenous distribution shifts, the recursive meta-to-inner commit significantly dominates the joint-commit alternative in distribution, and the certificate significantly contracts as offline data accumulates. Combining these results, we establish progressive certified handover, a safe-deployment method: each timescale flips from a legacy controller to NCTS when gains can be certified, independently of the others.

We study contextual bandits in the stochastic i.i.d.\ setting, where a learner observes contexts drawn from an unknown distribution, selects actions from a finite set $A$, and aims to identify an approximately optimal policy from a given class based on bandit feedback. Motivated by bandit multiclass classification with zero-one rewards, we focus on the \emph{$s$-sparse} setting in which, for every context, the reward vector has $L_1$-norm at most $s \ll |A|$. Our main result is the design of algorithms that, with high probability, output an $ε$-optimal policy compared to policy class $Π$ using $\tilde{O} ((s/ε^2 + |A|/ε)\log |Π|/δ)$ samples. We extend this bound to general Natarajan classes and complement it with a matching lower bound (up to logarithmic factors), thereby closing a substantial gap left by prior work (Erez et al., 2024, 2025), which incurred an additional $Θ(|A|^9)$ dependence. We obtain these results via two complementary approaches. First, we analyze contextual bandits through the lens of contextual decision making with structured observations, designing an exploration-by-optimization algorithm whose sample complexity is governed by the \emph{decision-estimation coefficient} (DEC; Foster et al., 2021, 2022). We show that, with $s$-sparse rewards, the induced model class admits a sharp DEC bound that scales with $s$ and directly yields the optimal rate. Since this approach is largely information-theoretic and involves solving complex min-max optimization problems, we also develop a second, more specialized algorithmic method based on a low-variance exploration technique. This approach leads to concrete, tractable algorithms and naturally extends to contextual combinatorial semi-bandits, leading to improved sample complexity guarantees for bandit multiclass list classification.

Reinforcement learning with multinomial logistic (MNL) function approximation has become an important framework due to its flexibility and broad applicability. While existing studies have established regret guarantees under worst-case analysis, they do not capture how performance depends on the variability of the interaction between the learner and the environment. In this paper, we develop a new theoretical analysis for MNL-based Markov decision processes that yields explicit variance-adaptive regret bounds. Our algorithm is computationally efficient and achieves the instance-wise optimal rate of regret, narrowing the gap between upper and lower bounds. Our numerical experiments validate that our method learns optimal policies more efficiently than conventional approaches.

Reinforcement learning algorithms are commonly analyzed (and designed) under the Markov assumption. This is unrealistic, as most environments encountered in practice are either partially observable, or require function approximation that restricts the agent to access non-Markovian state features. We consider the problem of learning an optimal reactive policy in a finite environment with deterministic observations (or equivalently, hard state aggregation). We introduce a new algorithm, Committed Q-learning, and prove almost-sure convergence to the optimal reactive policy under an intuitive assumption we call rewire-robustness. This assumption is strictly weaker than the $q_\star$-realizability condition used in prior work. Our algorithm is a variant of classical Q-learning in which the behavior policy commits to a single action upon entering a feature, and only resamples actions when the observed feature changes. A crucial part of our analysis is the introduction of quasi-Markov environments.

We study the problem of learning to bid when the bidder's value is dynamic, i.e., when the current value depends on past outcomes. Specifically, we consider a bidder participating in repeated second-price auctions whose value depends on the time elapsed since their last successful bid, with auctions arriving in continuous time and only aggregated feedback revealed at the end of the horizon. Such a bidder must (1) balance the immediate benefit of winning the current auction against its impact on future values and (2) learn unknown environmental parameters. We derive regret bounds for a class of learning methods that combine plug-in estimators with a differential-equation characterization of the optimal policy, and show that a specific confidence bound algorithm learns the optimal policy with a near optimal regret of $\widetilde{O}(\log N)$ for piecewise linear primitives, and $\widetilde{O}(N^{1/3})$ for general, smooth primitives, achieving these regrets without explicit randomization. These theoretical results are supported by numerical experiments.

Humans increasingly make decisions with AI assistance; for example, doctors may follow AI-recommended diagnostic tests and base their diagnoses on the results. A natural question is which tests should AI recommend to balance short-term decision quality and long-term human learning when different features (e.g., test results) are correlated. While prior work establishes that stationary policies that recommend the same tests repeatedly are optimal when features are independent, we prove that feature correlations lead such policies to perform arbitrarily poorly. Instead, we prove that any optimal policy must follow an explore-then-commit structure; initially, the AI should offer diverse tests so humans can learn accurate feature coefficients, then the AI should commit to a single set of tests, with exploration length that depends on the degree of feature correlation. We prove that computing the optimal policy is NP-hard and derive a dynamic programming-based algorithm that finds the optimal policy for finite horizons. We additionally develop an approximation that plans for shorter horizons and appends a stationary suffix, achieving near-optimal performance. Our empirical results complement our theory by showing that stronger feature correlation leads to longer exploration phases.

Reinforcement learning (RL) for reachability specifications is fundamental in sequential decision-making, yet theoretical guarantees remain less explored. A recent work achieves asymptotic convergence to optimal policies. However, this approach provides limited insight into convergence dynamics. In this work, we present an alternative approach that provides deeper theoretical insights into convergence. Our approach builds on PAC learning with assumptions. PAC learning guarantees near-optimal policies with high confidence in finite time but requires knowing internal MDP parameters like minimum transition probability. We argue that while these parameters are unknown in RL, they can be iteratively refined and estimated with increasing accuracy. By iteratively satisfying PAC conditions, we show that exact optimality can be achieved in the limit. Empirical evaluations on standard benchmarks validate our theoretical insights into convergence dynamics.

Pure exploration in episodic Reinforcement Learning has primarily focused on Best Policy Identification (BPI), which seeks to identify a (near)-optimal policy with high confidence. Motivated by practical settings where a ``good enough'' policy suffices, we study an alternate objective of Good Policy Identification (GPI). For a given reward threshold $μ_0$, GPI only requires identifying a policy with expected reward in an episode at least $μ_0$ if such a policy exists (positive instance), or declaring None if no such policy exists (negative instance). We formalize GPI under the fixed-confidence setting. We require the output to be correct with probability $\geq 1-δ$, and seek to minimize the expected sample complexity, which is the expected number of episodes explored for the output. We propose a novel algorithm BEE-GPI, and derive theoretically-grounded upper bounds on its sample complexity for positive and negative instances. Notably, for positive instances, the coefficient of $\log 1/δ$ in our upper bound is $O(H^2/(V^* - μ_0)^2)$, where $H$ is the episode length and $V^*$ is the optimal expected reward in an episode. The coefficient does not depend on the action and state space sizes otherwise, in sharp contrast to the sample complexity in BPI. We further establish lower bound results to show the near-optimality of BEE-GPI and the necessity of the $1/(V^* -μ)^2$ term. Numerical experiments further validate the efficiency of our approach.