Regret Minimization in RL

Classical reinforcement learning (RL) assumes that a reward is observed for every visited state-action pair. However, in real-world applications such as autonomous driving, such fine-grained feedback can be costly or difficult to collect, whereas trajectory-level feedback may be too sparse for efficient learning. To provide a general feedback model bridging these two extremes and handle large state spaces, we study RL with segment reward feedback under linear function approximation. Our work answers how the granularity of segment feedback and the choice of segmentation influence learning. For equal-length segments with known transitions, we design algorithms $\bitssegd$ and $\edlinucbsegd$ for binary and sum feedback types, respectively. They adopt posterior sampling with planning to achieve computational efficiency and the E-optimal experimental design to attain near-optimality. Nearly matching lower bounds are established. For equal-length segments with unknown transitions, we develop a unified $\seglsvits$ framework with two instantiations for binary and sum feedback, which carefully integrates the posterior estimated reward parameters into least-squares value iteration. These results reveal a fundamental insight: under binary feedback, increasing the number of segments significantly reduces the regret through an exponential factor, while surprisingly, under sum feedback, the granularity of segments does not affect learning much. Finally, to investigate whether segmenting according to state-action features can further expedite learning, we design an algorithm $\uneqsegbitsd$ that allows arbitrary segmentations. The resulting regret bound shows that under the usual elliptical potential analysis, the influence of state-action features on the regret appears only through logarithmic factors, and equal segmentation achieves the best performance.

We study episodic adversarial linear constrained Markov decision processes (CMDPs) with unknown transitions, where both the loss and constraint functions may vary adversarially across episodes. The best previous algorithm achieves $\widetilde{\mathcal{O}}(K^{3/4})$ regret and cumulative constraint violation, leaving a gap to the optimal $\widetilde{\mathcal{O}}(\sqrt{K})$ dependence on the number of episodes $K$. We close this gap by proposing a new primal dual algorithm that achieves $\widetilde{\mathcal{O}}(\sqrt{K})$ regret and cumulative constraint violation without assuming Slater's condition. We further extend the algorithm to achieve the same $\widetilde{\mathcal{O}}(\sqrt{K})$ guarantees for regret and hard constraint violation, which does not allow constraint violations to cancel across episodes. The main challenge is that learning linear CMDPs requires uniform concentration over a value function class with a controlled covering number, whereas standard techniques in constrained online learning, such as policy mixing, can make this class more complex. Our algorithm combines adaptive Follow the Regularized Leader (FTRL), contracted value estimation, and an exponential Lyapunov function. An adaptive dual regularizer offsets the dependence on the dual weights in the primal regret bound, removing the need for policy mixing. We further show that the normalization in the FTRL update bounds the policy parameters independently of the magnitudes of the dual weights, which explains why the resulting policy class remains compatible with uniform concentration. Under feature access, the computational complexity is independent of the size of the state space.

We study infinite-horizon average-reward constrained Markov decision processes (CMDPs) under the weakly communicating assumption. Existing high-probability guarantees for this setting either require computationally inefficient algorithms or have suboptimal dependence on the number of interactions $T$. We propose, to the best of our knowledge, the first computationally efficient algorithm that achieves $\widetilde{\mathcal{O}}(\sqrt{T})$ regret and cumulative constraint violation with high probability in the tabular setting. The $\sqrt{T}$ dependence is optimal up to logarithmic factors. Our approach incorporates cumulative constraint violation into the state and defines a reshaped reward through differences of a Huber potential. The added state determines the penalty on further violations while the reward function remains fixed on the augmented state space. Since the added state has known deterministic dynamics, only the original transition kernel needs to be estimated. The bounded slope of the Huber potential keeps the per-step reward bounded, and the potential differences telescope to relate the reshaped return to the original cumulative reward and the terminal potential. These properties allow us to apply finite-horizon approximation and optimistic value iteration with clipping, as used in unconstrained average-reward MDPs, without worsening the regret rate in $T$.

We study decentralized multi-player reinforcement learning in episodic tabular Markov decision processes (MDPs) under three forms of information asymmetry: (A) unobserved actions with common rewards, (B) observed actions with independent rewards, and (C) unobserved actions with independent rewards. Players cannot communicate during learning but may agree on a protocol a priori. For Problems A and B we propose \texttt{mQ-learning} and \texttt{mQ-learning-intervals}, achieving $\tilde{O}(\sqrt{H^4 S A_{\text{joint}}\, T})$ regret, where $H$ is the horizon, $S$ the state count, $T = KH$ the total steps, and $A_{\text{joint}} = \prod_{i=1}^M |\mathcal{A}_i|$ the joint action space across $M$ players. For Problem C we give \texttt{mEXC} and \texttt{mEXC-Bellman}, two-phase explore-then-commit algorithms with regret $\tilde{O}(H (S A_{\text{joint}})^{1/3} T^{2/3})$. Against the centralized joint-action benchmark, decentralized learning under information asymmetry matches the single-agent Q-learning rate of \cite{jin2018q} up to logarithmic factors. Because $A_{\text{joint}}$ grows exponentially in $M$, the bounds are most meaningful for small $M$ or small per-player action sets.

Average-reward reinforcement-learning regret is known up to logarithmic factors, but the numerical content of published guarantees is difficult to compare because probability mode, structural parameter, logarithmic normalization, prior information, and planning assumptions differ. We introduce a constant-aware comparison protocol and derive an explicit finite lower certificate for communicating MDPs. The construction is a binary tree of two-state blocks; its proof uses exact trajectory-level Bernoulli KL divergence and keeps action budget, diameter, occupancy, navigation cost, and terminal bias explicit. A common closed-form envelope improves the published coefficient $0.015$ across a finite frontier: $0.0200$ in a moderate regime and up to $0.0291$ under stronger action, diameter, and horizon conditions, a $94\%$ increase. The limiting coefficient is $\frac1{32}\sqrt{(A-3)/A}$. For upper bounds, we give an auditable composition rule for a span-constrained optimistic learner, but do not claim a coefficient while adaptive directional-variance and planning certificates remain open. We also formalize valid expectation conversion and constant comparability. Controlled diagnostics test diameter dependence, bonus-by-width interactions, span misspecification, and the finite lower certificate on its exact family.

This paper studies the policy gradient update for a multi-arm bandit problem in diffusion environment that is described by a stochastic differential equation (SDE) under the continuous-time reinforcement learning framework by Wang et al. (2020), Jia and Zhou (2022b). With the logit parameterization for the stochastic policy, we show that it converges almost surely to the optimal arm under an arbitrary constant learning rate. Furthermore, we derive the non-asymptotic regret upper bound when the constant learning rate is below a time-invariant threshold; and the regret bound has order $O(\log T)$. We improve the analysis in Lattimore (2026a) for the same SDE by constructing a novel Lyapunov function and demonstrate the transparency of analyzing policy gradient using the tools in SDEs. In addition, the same Lyapunov function is also helpful in analyzing the discrete-time policy gradient algorithm.

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

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

Optimal Reinforcement Learning (RL) algorithms typically rely on carefully constructed count-based uncertainty estimates to drive exploration. Although theoretically sound, such estimates are hard to compute in practical settings and therefore offer limited insight for designing exploration heuristics. Meanwhile, ensembling has emerged as a practical approach, but remains without theoretical justification. Building on a recent ensemble-based method for Multi-Armed Bandits, we propose a quantile-based ensemble method for finite-horizon Markov Decision Processes (MDPs). Our simple count-free approach achieves optimal variance-dependent regret bounds, providing theoretical grounding for ensemble-based exploration in RL.

Reinforcement learning with verifiable rewards (RLVR) has enabled progress on reasoning-intensive tasks by relying on task-specific verifiers that provide automated correctness signals. However, many realistic language tasks are difficult to equip with reliable verifiers, motivating a growing reliance on reinforcement learning from human feedback (RLHF). In this setting, we argue that a closer examination of how human feedback should be interpreted is essential. We introduce Regret-based Preference Optimization $(\textbf{RePO})$, which reframes RLHF through $\textit{regret minimization}$ rather than reward maximization. Human preferences are often shaped by $\textit{prospective}$ anticipation of outcomes and $\textit{counterfactual}$ comparisons to alternative behaviors, rather than by immediate, outcome-independent utility. $\textbf{RePO}$ captures this structure by modeling preferences as behavior-conditioned assessments of relative suboptimality. Experiments on mathematical reasoning benchmarks and human preference datasets demonstrate consistent performance gains, indicating that $\textbf{RePO}$ is an effective and human-aligned approach for training large language models.

We study KL-regularized contextual bandits and episodic reinforcement learning (RL) under general function approximation with model misspecification. Existing guarantees rely on realizability and therefore do not extend to misspecified models, where classical regret bounds may fail. This work introduces KL misspecification formulations for contextual bandits and episodic RL and analyzes regression-based algorithms with Gibbs policy updates. High-probability KL-regret guarantees with explicit misspecification terms are established, recovering the standard realizable KL-regularized setting as a special case.

We study sequential decision-making in partially observable environments against strategic, adaptive opponents, modeled as partially observable Markov games (POMGs). The central challenge is to learn latent dynamics from partial observations while facing an adversary whose behavior depends on the learner's strategy, making standard regret notions inadequate. We prove that an epoch-based optimistic maximum-likelihood algorithm achieves $\tilde{O}(\sqrt{T})$ policy regret for fixed problem parameters, with explicit dependence on the horizon, adversary memory, confidence radius, and the aggregate Eluder dimension of the observable-operator class. The algorithm selects one policy per geometrically growing epoch using confidence sets built cumulatively from past data, which keeps the cost of comparing adversary responses across policies logarithmic in $T$. We also prove a lower bound matching the $\sqrt{T}$ and aggregate-Eluder-dimension dependence, up to problem-dependent and logarithmic factors. Finally, we extend the framework to horizon-adaptive guarantees and adversaries with geometric fading memory.

Decision-making from offline datasets typically warm-starts a policy or score model from fixed offline data and then refines it with limited online interaction. Offline data reduces uncertainty, but it does not remove the need for exploration; it changes what remains to be explored. We formalise this residual uncertainty by the conditional mutual information $I(χ;τ_{1:T}\mid\mathcal{D}_N)$ between a learning target $χ$ and the online trajectories after conditioning on the offline dataset. This view leads naturally to information-directed sampling (IDS), a family parameterised by $η\ge 0$ that selects actions by trading off instantaneous regret against information gain. We prove a generic offline-to-online Bayesian regret bound for IDS through a ratio certificate: any information-ratio bound satisfied by a reference Thompson-sampling policy over the same randomised policy class is inherited by IDS. In a known-dynamics Bayesian linear-reward model, the conditional mutual information has a log-determinant form, and vanilla IDS ($η=0$) satisfies $\widetilde O\!\left(Hd\min\left\{\sqrt T,\,T\sqrt{C^\dagger_{β,\mathrm{IDS}_0}(N,T)/N}\right\}\right),$ where the coverage coefficient is tied to the visitation distribution induced by vanilla IDS itself. We also identify a warm-start regime with a dominated but informative probe in which vanilla IDS selects the probe while Thompson sampling never does, giving a constant-factor Bayesian regret separation. Controlled bandit experiments and D4RL offline-to-online RL experiments validate this mechanism: IDS is most beneficial when offline data is informative but leaves biased or low-probability residual uncertainty that targeted online actions can resolve, a regime shared by offline RL, offline black-box optimization, and Bayesian optimization.

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.

Human feedback often arrives as preferences rather than calibrated numeric rewards, motivating reinforcement learning from preferential feedback, also referred to as reinforcement learning from human feedback (RLHF). We present a rigorous theoretical study of preference-only learning in episodic kernel MDPs. In each episode, the learner deploys two policies from a common start state and receives a single binary label indicating which trajectory is preferred, modeled by a Bradley--Terry--Luce link on the difference of cumulative (unobserved) rewards. Under kernel-based assumptions on the reward and transition functions (one of the most general models amenable to theoretical analysis) we develop preference-based value estimation and confidence sets tailored to end-of-episode comparisons. We prove high-probability regret bounds that scale sublinearly in the number of episodes, implying that the value of the learned policy converges to that of the optimal policy.

We study reinforcement learning for episodic Markov Decision Processes (MDPs) whose transitions are modelled by a multinomial logistic (MNL) model. Existing algorithms for MNL mixture MDPs yield a regret of $\smash{\tilde{O}(dH^2\sqrt{T})}$ (Li et al., 2024), where $d$ is the feature dimension, $H$ the episode length, and $T$ the number of episodes. Inspired by the logistic bandit literature (Abeille et al., 2021; Faury et al., 2022; Boudart et al., 2026), we introduce a problem-dependent constant $\barσ\_T \leq 1/2$, measuring the normalised average variance of the optimal downstream value function along the learner's trajectory. We propose an algorithm achieving a regret of $\smash{\tilde{O}(dH^2\barσ\_T\sqrt{T})}$, which recovers the existing bound in the worst case and improves upon it for structured MDPs. For instance, for KL-constrained robust MDPs, $\barσ\_T = O(H^{-1})$, reducing the horizon dependence by a factor $H$. We further establish a matching $\smash{Ω(dH^2\barσ\_T\sqrt{T})}$ lower bound, proving minimax optimality (up to logarithmic factors) and fully characterising the regret complexity of MNL mixture MDPs for the first time.

We study episodic reinforcement learning with fixed reward and transition functions, but with episode-dependent admissible action sets that are observed at the start of each episode. Performance is measured by cumulative regret against the episode-wise optimal value, $\sum_{k=1}^K [V^{*,M^k} - V^{π^k,M^k}]$, where $M^k$ represents the action context in the $k$-th episode. We show that the MVP algorithm naturally extends to this framework and enjoys strong theoretical guarantees. In particular, we establish a minimax regret bound of $\widetilde{O}(\sqrt{SAH^3K\log L})$ for adversarial contexts, where $L$ denotes the number of possible contexts. This result implies a regret bound of $\widetilde{O}(\sqrt{SAH^3K})$ for stochastic contexts. We further translate the stochastic regret guarantee into a sample complexity bound of $\widetilde{O}(SAH^3/ε^2)$ for a fixed context distribution. In addition, we derive a gap-dependent regret bound of $$\widetilde O\left( \inf_{p\in [0,1)} \left( \frac{1}{Δ_{\min}^{p}} + pKΔ_{\min}^{p} \right)\log K \cdot \mathrm{poly}(S,A,H) \right),$$ where $Δ_{\min}^{p}$ is the global $p$-trimmed positive-gap floor over suboptimal $(h,s,a)$ triples. This bound can substantially improve upon the minimax rate when the relevant suboptimality gaps are large.

We study infinite-horizon average-reward constrained Markov decision processes (CMDPs) under the weakly communicating assumption. Our contributions are twofold. First, we establish strong duality for weakly communicating average-reward CMDPs over stationary policies with finite state and action spaces. Despite the absence of a linear programming formulation and the resulting nonconvexity under the weakly communicating setting, we show that strong duality still holds by carefully exploiting the geometric structure of the occupation measure set. Second, building on this result, we propose a primal--dual clipped value iteration algorithm for learning weakly communicating average-reward linear CMDPs. Our algorithm achieves regret and constraint violation bounds of $\widetilde{\mathcal{O}}(T^{2/3})$, improving upon the best known bounds, where $T$ denotes the number of interactions. Our approach extends clipped value iteration to the constrained setting and adapts it to a finite-horizon approximation, which stabilizes the dual variable and is crucial for achieving improved regret bounds. To analyze this, we develop a novel approach based on strong duality that enables the decomposition of the composite Lagrangian regret into separate bounds on regret and constraint violation.

We present the first theoretical guarantees for differentially private online reinforcement learning (RL) with general function approximation, extending beyond prior work restricted to tabular and linear settings. Our approach combines a batched policy update scheme with the exponential mechanism, together with a novel regret analysis. We show that, even under general function approximation, the regret in the model-free setting under differential privacy matches the state of the art for the linear case, scaling as $\widetilde{O}(K^{3/5})$, where $K$ denotes the number of episodes. As an important by-product, we also establish the first regret bound for online RL with batch update that depends on the standard complexity measure of coverability, complementing existing results based on a newly introduced Eluder-Condition class. In addition, we uncover fundamental gaps in recent results for private RL with linear function approximation, thereby clarifying its landscape.

We study the distribution of regret in stochastic multi-armed bandits and episodic reinforcement learning through a unified framework. We formalize a distributional regret bound as a probabilistic guarantee that holds uniformly over all confidence levels $δ\in (0,1]$, thereby characterizing the regret distribution across the full range of $δ$. We present a simple UCBVI-style algorithm with exploration bonus $\min\{c_{1,k}/N, c_{2,k}/\sqrt{N}\}$, where $N$ denotes the visit count and $(c_{1,k},c_{2,k})$ are user-specified parameters. For arbitrary parameter sequences, we derive general gap-independent and gap-dependent distributional regret bounds, yielding a principled characterization of how the parameters control the trade-off between expected performance, tail risk, and instance-dependent behavior. In particular, our bounds achieve optimal trade-offs between expected and distributional regret in both minimax and instance-dependent regimes. As a special case, for multi-armed bandits with $A$ arms and horizon $T$, we obtain a distributional regret bound of order $\mathcal{O}(\sqrt{AT}\log(1/δ))$, confirming the conjecture of Lattimore & Szepesvári (2020, Section 17.1) for the first time.

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.

Reinforcement learning from human feedback (RLHF) is a central post-training tool for aligning large language models, but its training reward is only a learned proxy for true human utility. This creates a decision problem under objective misspecification: the policy is optimized against an estimated reward, while deployment performance is governed by an unobserved population preference. The resulting gap leads to reward over-optimization, where proxy reward keeps improving after true quality deteriorates. We propose distributionally robust regret optimization (DRRO) for RLHF with a Wasserstein ambiguity set over reward laws, using promptwise $\ell_p$ distances between reward vectors as transport costs. Unlike standard distributionally robust optimization, which pessimizes worst-case value, DRRO pessimizes worst-case regret relative to the best policy under the same plausible reward perturbation. We show that the expressive-policy problem decomposes into promptwise regret problems. For each prompt, the inner adversary has a dual-norm closed form; under the $\ell_1$ transport cost used by our algorithm, the optimizer has a water-filling structure. These results lead to a practical policy-gradient algorithm that adds a simple sampled bonus to GRPO-style training. Theory and experiments both show that DRRO is less over-pessimistic than standard DRO and mitigates over-optimization more effectively than existing baselines.

We study model-free reinforcement learning (RL) in non-stationary finite-horizon episodic Markov decision processes (MDPs) without prior knowledge of the non-stationarity. We focus on the piecewise stationary (PS) setting, where both rewards and transition dynamics can change at unknown times. We first revisit existing state-of-the-art approaches and identify theoretical and practical limitations that change the current landscape of performance guarantees. To characterize the difficulty of the problem, we establish the first minimax lower bounds for PS-RL in tabular and linear MDPs. We then introduce Detection Augmented Reinforcement Learning (DARLING), a modular wrapper for PS-RL that applies to both tabular and linear MDPs, without knowledge of the changes. In tabular MDPs, under change-point separability and reachability conditions, DARLING improves the best known dynamic regret bounds and matches our minimax lower bound. In linear MDPs, DARLING matches the minimax lower bound when the relevant reachability parameters are known, and our analysis clarifies the structural obstacles that distinguish this setting from the tabular case. Finally, through extensive experimentation across diverse non-stationary benchmarks, we show that DARLING consistently surpasses the state-of-the-art methods.

We analyze the Bayesian regret of the Gaussian process posterior sampling reinforcement learning (GP-PSRL) algorithm. Posterior sampling is a heuristic for decision-making under uncertainty that has been used to develop successful algorithms for a variety of continuous control problems. However, theoretical work on GP-PSRL is limited. All known regret bounds either have a sub-optimal growth rate, require strong smoothness assumptions, or fail to properly account for the fact that the set of possible system states is unbounded. Through a recursive application of the Borell-Tsirelson-Ibragimov-Sudakov inequality, we show that, with high probability, the states actually visited by the algorithm are contained within a ball of near-constant radius. We then use the chaining method to control the regret suffered by GP-PSRL under weak smoothness conditions. Our main result is a Bayesian regret bound of the order $\widetilde{\mathcal{O}}(H\sqrt{γ_TT})$, where $H$ is the horizon, $T$ is the number of time steps and $γ_T$ is the expected information gain. With this result, we resolve the limitations with prior theoretical work on PSRL, and provide the theoretical foundation and tools for analyzing PSRL in complex settings.

This work studies online episodic tabular Markov decision processes (MDPs) with known transitions and develops best-of-both-worlds algorithms that achieve refined data-dependent regret bounds in the adversarial regime and variance-dependent regret bounds in the stochastic regime. We quantify MDP complexity using a first-order quantity and several new data-dependent measures for the adversarial regime, including a second-order quantity and a path-length measure, as well as variance-based measures for the stochastic regime. To adapt to these measures, we develop algorithms based on global optimization and policy optimization, both built on optimistic follow-the-regularized-leader with log-barrier regularization. For global optimization, our algorithms achieve first-order, second-order, and path-length regret bounds in the adversarial regime, and in the stochastic regime, they achieve a variance-aware gap-independent bound and a variance-aware gap-dependent bound that is polylogarithmic in the number of episodes. For policy optimization, our algorithms achieve the same data- and variance-dependent adaptivity, up to a factor of the episode horizon, by exploiting a new optimistic $Q$-function estimator. Finally, we establish regret lower bounds in terms of data-dependent complexity measures for the adversarial regime and a variance measure for the stochastic regime, implying that the regret upper bounds achieved by the global-optimization approach are nearly optimal.

We study a structured class of Markov Decision Processes, known as Exo-MDPs, in which the state space is partitioned into exogenous and endogenous components. Exogenous states evolve stochastically, independent of the agent's actions, while endogenous states evolve deterministically based on both state components and actions. Exo-MDPs capture many operations research settings, including inventory control, resource management, and ride-sharing. Our first contribution is structural: we establish a representational equivalence between discrete MDPs, Exo-MDPs, and discrete linear mixture MDPs. Our second contribution is statistical. We characterize the minimax regret of learning in Exo-MDPs when the effective dimension r is small relative to the endogenous state and action spaces. When the exogenous states are unobserved, we prove matching upper and lower regret bounds of order $Θ(Hr \sqrt{K})$ over $K$ episodes of horizon $H$, where $r$ is the effective dimension of the Exo-MDP. When exogenous states are observed, the minimax regret improves to $Θ(H\sqrt{ r K})$, revealing a $Θ(\sqrt{r})$ statistical gap due to observation of the exogenous states. These results show that Exo-MDPs decouple sample complexity from action space and endogenous state space. We validate these insights with experiments on inventory control and resource allocation.

This paper develops a framework for learning in discounted infinite-horizon Markov decision processes (MDPs) with Borel state and action spaces, whose rewards and transitions depend on an unknown parameter.To analyze sampling-based adaptive learning algorithms in this setting, we introduce a canonical probability space that explicitly incorporates sampled parameters into the history of the process. As a performance criterion, we adopt the per-period suboptimality gap used in discounted-MDP regret analysis and specialize it to our parametrized Bayesian setting. Since this quantity captures the remaining loss in future performance from the current period onward, we refer to it as residual regret. We use the expected residual regret to connect discounted-MDP regret analysis with asymptotic discount optimality from adaptive control and the temporal-difference error perspective from reinforcement learning. We then focus on Thompson sampling (TS) in discounted infinite-horizon MDPs. Under assumptions that extend those used in prior work on finite state and action spaces to the Borel setting, we show that the expected residual regret for TS converges to zero exponentially fast. We further show that, under mild conditions ensuring the existence of the relevant limits, the ample-path residual regret converges to zero almost surely and TS achieves complete learning.

Most learning algorithms with formal regret guarantees essentially rely on trying all possible behaviors, which is problematic when some errors cannot be recovered from. Instead, we allow the learning agent to ask for help from a mentor and to transfer knowledge between similar states. We show that this combination enables the agent to learn both safely and effectively. Under standard online learning assumptions, we provide an algorithm whose regret and number of mentor queries are both sublinear in the time horizon for Markov decision processes with irreversible dynamics and infinite state spaces. Our proof involves a sequence of three reductions, making our result more general than a single algorithm. Conceptually, our result may be the first formal proof that it is possible for an agent to obtain high reward while becoming self-sufficient in an unknown, unbounded, and high-stakes environment without resets.