Daily Papers

Current Reinforcement Learning (RL) methods often suffer from sample-inefficiency, resulting from blind exploration strategies that neglect causal relationships among states, actions, and rewards. Although recent causal approaches aim to address this problem, they lack grounded modeling of reward-guided causal understanding of states and actions for goal-orientation, thus impairing learning efficiency. To tackle this issue, we propose a novel method named Causal Information Prioritization (CIP) that improves sample efficiency by leveraging factored MDPs to infer causal relationships between different dimensions of states and actions with respect to rewards, enabling the prioritization of causal information. Specifically, CIP identifies and leverages causal relationships between states and rewards to execute counterfactual data augmentation to prioritize high-impact state features under the causal understanding of the environments. Moreover, CIP integrates a causality-aware empowerment learning objective, which significantly enhances the agent's execution of reward-guided actions for more efficient exploration in complex environments. To fully assess the effectiveness of CIP, we conduct extensive experiments across 39 tasks in 5 diverse continuous control environments, encompassing both locomotion and manipulation skills learning with pixel-based and sparse reward settings. Experimental results demonstrate that CIP consistently outperforms existing RL methods across a wide range of scenarios.

Despite the success of neural-based combinatorial optimization methods for end-to-end heuristic learning, out-of-distribution generalization remains a challenge. In this paper, we present a novel formulation of Combinatorial Optimization Problems (COPs) as Markov Decision Processes (MDPs) that effectively leverages common symmetries of COPs to improve out-of-distribution robustness. Starting from a direct MDP formulation of a constructive method, we introduce a generic way to reduce the state space, based on Bisimulation Quotienting (BQ) in MDPs. Then, for COPs with a recursive nature, we specialize the bisimulation and show how the reduced state exploits the symmetries of these problems and facilitates MDP solving. Our approach is principled and we prove that an optimal policy for the proposed BQ-MDP actually solves the associated COPs. We illustrate our approach on five classical problems: the Euclidean and Asymmetric Traveling Salesman, Capacitated Vehicle Routing, Orienteering and Knapsack Problems. Furthermore, for each problem, we introduce a simple attention-based policy network for the BQ-MDPs, which we train by imitation of (near) optimal solutions of small instances from a single distribution. We obtain new state-of-the-art results for the five COPs on both synthetic and realistic benchmarks. Notably, in contrast to most existing neural approaches, our learned policies show excellent generalization performance to much larger instances than seen during training, without any additional search procedure.

General function approximation is a powerful tool to handle large state and action spaces in a broad range of reinforcement learning (RL) scenarios. However, theoretical understanding of non-stationary MDPs with general function approximation is still limited. In this paper, we make the first such an attempt. We first propose a new complexity metric called dynamic Bellman Eluder (DBE) dimension for non-stationary MDPs, which subsumes majority of existing tractable RL problems in static MDPs as well as non-stationary MDPs. Based on the proposed complexity metric, we propose a novel confidence-set based model-free algorithm called SW-OPEA, which features a sliding window mechanism and a new confidence set design for non-stationary MDPs. We then establish an upper bound on the dynamic regret for the proposed algorithm, and show that SW-OPEA is provably efficient as long as the variation budget is not significantly large. We further demonstrate via examples of non-stationary linear and tabular MDPs that our algorithm performs better in small variation budget scenario than the existing UCB-type algorithms. To the best of our knowledge, this is the first dynamic regret analysis in non-stationary MDPs with general function approximation.

We study regret minimization for reinforcement learning (RL) in Latent Markov Decision Processes (LMDPs) with context in hindsight. We design a novel model-based algorithmic framework which can be instantiated with both a model-optimistic and a value-optimistic solver. We prove an O(mathsf{Var^star M Gamma S A K}) regret bound where O hides logarithm factors, M is the number of contexts, S is the number of states, A is the number of actions, K is the number of episodes, Gamma le S is the maximum transition degree of any state-action pair, and Var^star is a variance quantity describing the determinism of the LMDP. The regret bound only scales logarithmically with the planning horizon, thus yielding the first (nearly) horizon-free regret bound for LMDP. This is also the first problem-dependent regret bound for LMDP. Key in our proof is an analysis of the total variance of alpha vectors (a generalization of value functions), which is handled with a truncation method. We complement our positive result with a novel Omega(mathsf{Var^star M S A K}) regret lower bound with Gamma = 2, which shows our upper bound minimax optimal when Gamma is a constant for the class of variance-bounded LMDPs. Our lower bound relies on new constructions of hard instances and an argument inspired by the symmetrization technique from theoretical computer science, both of which are technically different from existing lower bound proof for MDPs, and thus can be of independent interest.

We propose a new method for optimistic planning in infinite-horizon discounted Markov decision processes based on the idea of adding regularization to the updates of an otherwise standard approximate value iteration procedure. This technique allows us to avoid contraction and monotonicity arguments typically required by existing analyses of approximate dynamic programming methods, and in particular to use approximate transition functions estimated via least-squares procedures in MDPs with linear function approximation. We use our method to recover known guarantees in tabular MDPs and to provide a computationally efficient algorithm for learning near-optimal policies in discounted linear mixture MDPs from a single stream of experience, and show it achieves near-optimal statistical guarantees.

We present metrics for measuring state similarity in Markov decision processes (MDPs) with infinitely many states, including MDPs with continuous state spaces. Such metrics provide a stable quantitative analogue of the notion of bisimulation for MDPs, and are suitable for use in MDP approximation. We show that the optimal value function associated with a discounted infinite horizon planning task varies continuously with respect to our metric distances.

Reasoning remains a challenging task for large language models (LLMs), especially within the logically constrained environment of automated theorem proving (ATP), due to sparse rewards and the vast scale of proofs. These challenges are amplified in benchmarks like PutnamBench, which contains university-level problems requiring complex, multi-step reasoning. To address this, we introduce self-generated goal-conditioned MDPs (sG-MDPs), a new framework in which agents generate and pursue their subgoals based on the evolving proof state. Given this more structured generation of goals, the resulting problem becomes more amenable to search. We then apply Monte Carlo Tree Search (MCTS)-like algorithms to solve the sG-MDP, instantiating our approach in Bourbaki (7B), a modular system that can ensemble multiple 7B LLMs for subgoal generation and tactic synthesis. On PutnamBench, Bourbaki (7B) solves 26 problems, achieving new state-of-the-art results with models at this scale.

Multiple types of inference are available for probabilistic graphical models, e.g., marginal, maximum-a-posteriori, and even marginal maximum-a-posteriori. Which one do researchers mean when they talk about ``planning as inference''? There is no consistency in the literature, different types are used, and their ability to do planning is further entangled with specific approximations or additional constraints. In this work we use the variational framework to show that, just like all commonly used types of inference correspond to different weightings of the entropy terms in the variational problem, planning corresponds exactly to a different set of weights. This means that all the tricks of variational inference are readily applicable to planning. We develop an analogue of loopy belief propagation that allows us to perform approximate planning in factored-state Markov decisions processes without incurring intractability due to the exponentially large state space. The variational perspective shows that the previous types of inference for planning are only adequate in environments with low stochasticity, and allows us to characterize each type by its own merits, disentangling the type of inference from the additional approximations that its practical use requires. We validate these results empirically on synthetic MDPs and tasks posed in the International Planning Competition.

Reinforcement learning (RL) tasks are typically framed as Markov Decision Processes (MDPs), assuming that decisions are made at fixed time intervals. However, many applications of great importance, including healthcare, do not satisfy this assumption, yet they are commonly modelled as MDPs after an artificial reshaping of the data. In addition, most healthcare (and similar) problems are offline by nature, allowing for only retrospective studies. To address both challenges, we begin by discussing the Semi-MDP (SMDP) framework, which formally handles actions of variable timings. We next present a formal way to apply SMDP modifications to nearly any given value-based offline RL method. We use this theory to introduce three SMDP-based offline RL algorithms, namely, SDQN, SDDQN, and SBCQ. We then experimentally demonstrate that only these SMDP-based algorithms learn the optimal policy in variable-time environments, whereas their MDP counterparts do not. Finally, we apply our new algorithms to a real-world offline dataset pertaining to warfarin dosing for stroke prevention and demonstrate similar results.

Robust Markov Decision Processes (MDPs) and risk-sensitive MDPs are both powerful tools for making decisions in the presence of uncertainties. Previous efforts have aimed to establish their connections, revealing equivalences in specific formulations. This paper introduces a new formulation for risk-sensitive MDPs, which assesses risk in a slightly different manner compared to the classical Markov risk measure (Ruszczy\'nski 2010), and establishes its equivalence with a class of regularized robust MDP (RMDP) problems, including the standard RMDP as a special case. Leveraging this equivalence, we further derive the policy gradient theorem for both problems, proving gradient domination and global convergence of the exact policy gradient method under the tabular setting with direct parameterization. This forms a sharp contrast to the Markov risk measure, known to be potentially non-gradient-dominant (Huang et al. 2021). We also propose a sample-based offline learning algorithm, namely the robust fitted-Z iteration (RFZI), for a specific regularized RMDP problem with a KL-divergence regularization term (or equivalently the risk-sensitive MDP with an entropy risk measure). We showcase its streamlined design and less stringent assumptions due to the equivalence and analyze its sample complexity.

Sequential decision making, commonly formalized as Markov Decision Process (MDP) optimization, is a important challenge in artificial intelligence. Two key approaches to this problem are reinforcement learning (RL) and planning. This paper presents a survey of the integration of both fields, better known as model-based reinforcement learning. Model-based RL has two main steps. First, we systematically cover approaches to dynamics model learning, including challenges like dealing with stochasticity, uncertainty, partial observability, and temporal abstraction. Second, we present a systematic categorization of planning-learning integration, including aspects like: where to start planning, what budgets to allocate to planning and real data collection, how to plan, and how to integrate planning in the learning and acting loop. After these two sections, we also discuss implicit model-based RL as an end-to-end alternative for model learning and planning, and we cover the potential benefits of model-based RL. Along the way, the survey also draws connections to several related RL fields, like hierarchical RL and transfer learning. Altogether, the survey presents a broad conceptual overview of the combination of planning and learning for MDP optimization.

Reinforcement learning from Human Feedback (RLHF) learns from preference signals, while standard Reinforcement Learning (RL) directly learns from reward signals. Preferences arguably contain less information than rewards, which makes preference-based RL seemingly more difficult. This paper theoretically proves that, for a wide range of preference models, we can solve preference-based RL directly using existing algorithms and techniques for reward-based RL, with small or no extra costs. Specifically, (1) for preferences that are drawn from reward-based probabilistic models, we reduce the problem to robust reward-based RL that can tolerate small errors in rewards; (2) for general arbitrary preferences where the objective is to find the von Neumann winner, we reduce the problem to multiagent reward-based RL which finds Nash equilibria for factored Markov games under a restricted set of policies. The latter case can be further reduce to adversarial MDP when preferences only depend on the final state. We instantiate all reward-based RL subroutines by concrete provable algorithms, and apply our theory to a large class of models including tabular MDPs and MDPs with generic function approximation. We further provide guarantees when K-wise comparisons are available.

Markov Decision Processes (MDPs) are a formal framework for modeling and solving sequential decision-making problems. In finite-time horizons such problems are relevant for instance for optimal stopping or specific supply chain problems, but also in the training of large language models. In contrast to infinite horizon MDPs optimal policies are not stationary, policies must be learned for every single epoch. In practice all parameters are often trained simultaneously, ignoring the inherent structure suggested by dynamic programming. This paper introduces a combination of dynamic programming and policy gradient called dynamic policy gradient, where the parameters are trained backwards in time. For the tabular softmax parametrisation we carry out the convergence analysis for simultaneous and dynamic policy gradient towards global optima, both in the exact and sampled gradient settings without regularisation. It turns out that the use of dynamic policy gradient training much better exploits the structure of finite-time problems which is reflected in improved convergence bounds.

This paper studies the fundamental limits of reinforcement learning (RL) in the challenging partially observable setting. While it is well-established that learning in Partially Observable Markov Decision Processes (POMDPs) requires exponentially many samples in the worst case, a surge of recent work shows that polynomial sample complexities are achievable under the revealing condition -- A natural condition that requires the observables to reveal some information about the unobserved latent states. However, the fundamental limits for learning in revealing POMDPs are much less understood, with existing lower bounds being rather preliminary and having substantial gaps from the current best upper bounds. We establish strong PAC and regret lower bounds for learning in revealing POMDPs. Our lower bounds scale polynomially in all relevant problem parameters in a multiplicative fashion, and achieve significantly smaller gaps against the current best upper bounds, providing a solid starting point for future studies. In particular, for multi-step revealing POMDPs, we show that (1) the latent state-space dependence is at least Omega(S^{1.5}) in the PAC sample complexity, which is notably harder than the Theta(S) scaling for fully-observable MDPs; (2) Any polynomial sublinear regret is at least Omega(T^{2/3}), suggesting its fundamental difference from the single-step case where O(T) regret is achievable. Technically, our hard instance construction adapts techniques in distribution testing, which is new to the RL literature and may be of independent interest.

The general sequential decision-making problem, which includes Markov decision processes (MDPs) and partially observable MDPs (POMDPs) as special cases, aims at maximizing a cumulative reward by making a sequence of decisions based on a history of observations and actions over time. Recent studies have shown that the sequential decision-making problem is statistically learnable if it admits a low-rank structure modeled by predictive state representations (PSRs). Despite these advancements, existing approaches typically involve oracles or steps that are computationally intractable. On the other hand, the upper confidence bound (UCB) based approaches, which have served successfully as computationally efficient methods in bandits and MDPs, have not been investigated for more general PSRs, due to the difficulty of optimistic bonus design in these more challenging settings. This paper proposes the first known UCB-type approach for PSRs, featuring a novel bonus term that upper bounds the total variation distance between the estimated and true models. We further characterize the sample complexity bounds for our designed UCB-type algorithms for both online and offline PSRs. In contrast to existing approaches for PSRs, our UCB-type algorithms enjoy computational tractability, last-iterate guaranteed near-optimal policy, and guaranteed model accuracy.

This paper introduces Multi-Agent MDP Homomorphic Networks, a class of networks that allows distributed execution using only local information, yet is able to share experience between global symmetries in the joint state-action space of cooperative multi-agent systems. In cooperative multi-agent systems, complex symmetries arise between different configurations of the agents and their local observations. For example, consider a group of agents navigating: rotating the state globally results in a permutation of the optimal joint policy. Existing work on symmetries in single agent reinforcement learning can only be generalized to the fully centralized setting, because such approaches rely on the global symmetry in the full state-action spaces, and these can result in correspondences across agents. To encode such symmetries while still allowing distributed execution we propose a factorization that decomposes global symmetries into local transformations. Our proposed factorization allows for distributing the computation that enforces global symmetries over local agents and local interactions. We introduce a multi-agent equivariant policy network based on this factorization. We show empirically on symmetric multi-agent problems that globally symmetric distributable policies improve data efficiency compared to non-equivariant baselines.

A key component of model-based reinforcement learning (RL) is a dynamics model that predicts the outcomes of actions. Errors in this predictive model can degrade the performance of model-based controllers, and complex Markov decision processes (MDPs) can present exceptionally difficult prediction problems. To mitigate this issue, we propose predictable MDP abstraction (PMA): instead of training a predictive model on the original MDP, we train a model on a transformed MDP with a learned action space that only permits predictable, easy-to-model actions, while covering the original state-action space as much as possible. As a result, model learning becomes easier and more accurate, which allows robust, stable model-based planning or model-based RL. This transformation is learned in an unsupervised manner, before any task is specified by the user. Downstream tasks can then be solved with model-based control in a zero-shot fashion, without additional environment interactions. We theoretically analyze PMA and empirically demonstrate that PMA leads to significant improvements over prior unsupervised model-based RL approaches in a range of benchmark environments. Our code and videos are available at https://seohong.me/projects/pma/

We consider finite-horizon Markov Decision Processes where parameters, such as transition probabilities, are unknown and estimated from data. The popular distributionally robust approach to addressing the parameter uncertainty can sometimes be overly conservative. In this paper, we propose a new formulation, Bayesian risk Markov Decision Process (BR-MDP), to address parameter uncertainty in MDPs, where a risk functional is applied in nested form to the expected total cost with respect to the Bayesian posterior distribution of the unknown parameters. The proposed formulation provides more flexible risk attitutes towards parameter uncertainty and takes into account the availability of data in future times stages. To solve the proposed formulation with the conditional value-at-risk (CVaR) risk functional, we propose an efficient approximation algorithm by deriving an analytical approximation of the value function and utilizing the convexity of CVaR. We demonstrate the empirical performance of the BR-MDP formulation and proposed algorithms on a gambler's betting problem and an inventory control problem.

We present the OMG-CMDP! algorithm for regret minimization in adversarial Contextual MDPs. The algorithm operates under the minimal assumptions of realizable function class and access to online least squares and log loss regression oracles. Our algorithm is efficient (assuming efficient online regression oracles), simple and robust to approximation errors. It enjoys an O(H^{2.5} T|S||A| ( mathcal{R(O) + H log(delta^{-1}) )}) regret guarantee, with T being the number of episodes, S the state space, A the action space, H the horizon and R(O) = R(O_{sq}^F) + R(O_{log}^P) is the sum of the regression oracles' regret, used to approximate the context-dependent rewards and dynamics, respectively. To the best of our knowledge, our algorithm is the first efficient rate optimal regret minimization algorithm for adversarial CMDPs that operates under the minimal standard assumption of online function approximation.

MDPs with low-rank transitions -- that is, the transition matrix can be factored into the product of two matrices, left and right -- is a highly representative structure that enables tractable learning. The left matrix enables expressive function approximation for value-based learning and has been studied extensively. In this work, we instead investigate sample-efficient learning with density features, i.e., the right matrix, which induce powerful models for state-occupancy distributions. This setting not only sheds light on leveraging unsupervised learning in RL, but also enables plug-in solutions for convex RL. In the offline setting, we propose an algorithm for off-policy estimation of occupancies that can handle non-exploratory data. Using this as a subroutine, we further devise an online algorithm that constructs exploratory data distributions in a level-by-level manner. As a central technical challenge, the additive error of occupancy estimation is incompatible with the multiplicative definition of data coverage. In the absence of strong assumptions like reachability, this incompatibility easily leads to exponential error blow-up, which we overcome via novel technical tools. Our results also readily extend to the representation learning setting, when the density features are unknown and must be learned from an exponentially large candidate set.

Many control tasks exhibit similar dynamics that can be modeled as having common latent structure. Hidden-Parameter Markov Decision Processes (HiP-MDPs) explicitly model this structure to improve sample efficiency in multi-task settings. However, this setting makes strong assumptions on the observability of the state that limit its application in real-world scenarios with rich observation spaces. In this work, we leverage ideas of common structure from the HiP-MDP setting, and extend it to enable robust state abstractions inspired by Block MDPs. We derive instantiations of this new framework for both multi-task reinforcement learning (MTRL) and meta-reinforcement learning (Meta-RL) settings. Further, we provide transfer and generalization bounds based on task and state similarity, along with sample complexity bounds that depend on the aggregate number of samples across tasks, rather than the number of tasks, a significant improvement over prior work that use the same environment assumptions. To further demonstrate the efficacy of the proposed method, we empirically compare and show improvement over multi-task and meta-reinforcement learning baselines.

In reinforcement learning (RL), rewards of states are typically considered additive, and following the Markov assumption, they are independent of states visited previously. In many important applications, such as coverage control, experiment design and informative path planning, rewards naturally have diminishing returns, i.e., their value decreases in light of similar states visited previously. To tackle this, we propose submodular RL (SubRL), a paradigm which seeks to optimize more general, non-additive (and history-dependent) rewards modelled via submodular set functions which capture diminishing returns. Unfortunately, in general, even in tabular settings, we show that the resulting optimization problem is hard to approximate. On the other hand, motivated by the success of greedy algorithms in classical submodular optimization, we propose SubPO, a simple policy gradient-based algorithm for SubRL that handles non-additive rewards by greedily maximizing marginal gains. Indeed, under some assumptions on the underlying Markov Decision Process (MDP), SubPO recovers optimal constant factor approximations of submodular bandits. Moreover, we derive a natural policy gradient approach for locally optimizing SubRL instances even in large state- and action- spaces. We showcase the versatility of our approach by applying SubPO to several applications, such as biodiversity monitoring, Bayesian experiment design, informative path planning, and coverage maximization. Our results demonstrate sample efficiency, as well as scalability to high-dimensional state-action spaces.

A recent line of works showed regret bounds in reinforcement learning (RL) can be (nearly) independent of planning horizon, a.k.a.~the horizon-free bounds. However, these regret bounds only apply to settings where a polynomial dependency on the size of transition model is allowed, such as tabular Markov Decision Process (MDP) and linear mixture MDP. We give the first horizon-free bound for the popular linear MDP setting where the size of the transition model can be exponentially large or even uncountable. In contrast to prior works which explicitly estimate the transition model and compute the inhomogeneous value functions at different time steps, we directly estimate the value functions and confidence sets. We obtain the horizon-free bound by: (1) maintaining multiple weighted least square estimators for the value functions; and (2) a structural lemma which shows the maximal total variation of the inhomogeneous value functions is bounded by a polynomial factor of the feature dimension.

In this work, we consider the problem of collaborative multi-user reinforcement learning. In this setting there are multiple users with the same state-action space and transition probabilities but with different rewards. Under the assumption that the reward matrix of the N users has a low-rank structure -- a standard and practically successful assumption in the offline collaborative filtering setting -- the question is can we design algorithms with significantly lower sample complexity compared to the ones that learn the MDP individually for each user. Our main contribution is an algorithm which explores rewards collaboratively with N user-specific MDPs and can learn rewards efficiently in two key settings: tabular MDPs and linear MDPs. When N is large and the rank is constant, the sample complexity per MDP depends logarithmically over the size of the state-space, which represents an exponential reduction (in the state-space size) when compared to the standard ``non-collaborative'' algorithms.

This paper investigates conservative exploration in reinforcement learning where the performance of the learning agent is guaranteed to be above a certain threshold throughout the learning process. It focuses on the tabular episodic Markov Decision Process (MDP) setting that has finite states and actions. With the knowledge of an existing safe baseline policy, an algorithm termed as StepMix is proposed to balance the exploitation and exploration while ensuring that the conservative constraint is never violated in each episode with high probability. StepMix features a unique design of a mixture policy that adaptively and smoothly interpolates between the baseline policy and the optimistic policy. Theoretical analysis shows that StepMix achieves near-optimal regret order as in the constraint-free setting, indicating that obeying the stringent episode-wise conservative constraint does not compromise the learning performance. Besides, a randomization-based EpsMix algorithm is also proposed and shown to achieve the same performance as StepMix. The algorithm design and theoretical analysis are further extended to the setting where the baseline policy is not given a priori but must be learned from an offline dataset, and it is proved that similar conservative guarantee and regret can be achieved if the offline dataset is sufficiently large. Experiment results corroborate the theoretical analysis and demonstrate the effectiveness of the proposed conservative exploration strategies.

We study risk-sensitive Reinforcement Learning (RL), where we aim to maximize the Conditional Value at Risk (CVaR) with a fixed risk tolerance tau. Prior theoretical work studying risk-sensitive RL focuses on the tabular Markov Decision Processes (MDPs) setting. To extend CVaR RL to settings where state space is large, function approximation must be deployed. We study CVaR RL in low-rank MDPs with nonlinear function approximation. Low-rank MDPs assume the underlying transition kernel admits a low-rank decomposition, but unlike prior linear models, low-rank MDPs do not assume the feature or state-action representation is known. We propose a novel Upper Confidence Bound (UCB) bonus-driven algorithm to carefully balance the interplay between exploration, exploitation, and representation learning in CVaR RL. We prove that our algorithm achieves a sample complexity of Oleft(H^7 A^2 d^4{tau^2 epsilon^2}right) to yield an epsilon-optimal CVaR, where H is the length of each episode, A is the capacity of action space, and d is the dimension of representations. Computational-wise, we design a novel discretized Least-Squares Value Iteration (LSVI) algorithm for the CVaR objective as the planning oracle and show that we can find the near-optimal policy in a polynomial running time with a Maximum Likelihood Estimation oracle. To our knowledge, this is the first provably efficient CVaR RL algorithm in low-rank MDPs.

We study variance-dependent regret bounds for Markov decision processes (MDPs). Algorithms with variance-dependent regret guarantees can automatically exploit environments with low variance (e.g., enjoying constant regret on deterministic MDPs). The existing algorithms are either variance-independent or suboptimal. We first propose two new environment norms to characterize the fine-grained variance properties of the environment. For model-based methods, we design a variant of the MVP algorithm (Zhang et al., 2021a). We apply new analysis techniques to demonstrate that this algorithm enjoys variance-dependent bounds with respect to the norms we propose. In particular, this bound is simultaneously minimax optimal for both stochastic and deterministic MDPs, the first result of its kind. We further initiate the study on model-free algorithms with variance-dependent regret bounds by designing a reference-function-based algorithm with a novel capped-doubling reference update schedule. Lastly, we also provide lower bounds to complement our upper bounds.

This paper introduces an approach for learning to solve continuous constraint satisfaction problems (CCSP) in robotic reasoning and planning. Previous methods primarily rely on hand-engineering or learning generators for specific constraint types and then rejecting the value assignments when other constraints are violated. By contrast, our model, the compositional diffusion continuous constraint solver (Diffusion-CCSP) derives global solutions to CCSPs by representing them as factor graphs and combining the energies of diffusion models trained to sample for individual constraint types. Diffusion-CCSP exhibits strong generalization to novel combinations of known constraints, and it can be integrated into a task and motion planner to devise long-horizon plans that include actions with both discrete and continuous parameters. Project site: https://diffusion-ccsp.github.io/

We consider the problem of learning in a non-stationary reinforcement learning (RL) environment, where the setting can be fully described by a piecewise stationary discrete-time Markov decision process (MDP). We introduce a variant of the Restarted Bayesian Online Change-Point Detection algorithm (R-BOCPD) that operates on input streams originating from the more general multinomial distribution and provides near-optimal theoretical guarantees in terms of false-alarm rate and detection delay. Based on this, we propose an improved version of the UCRL2 algorithm for MDPs with state transition kernel sampled from a multinomial distribution, which we call R-BOCPD-UCRL2. We perform a finite-time performance analysis and show that R-BOCPD-UCRL2 enjoys a favorable regret bound of Oleft(D O A T K_T logleft (frac{T{delta} right) + K_T log frac{K_T{delta}}{minlimits_ell : KLleft( {theta^{(ell+1)}}midmathbf{theta^{(ell)}}right)}}right), where D is the largest MDP diameter from the set of MDPs defining the piecewise stationary MDP setting, O is the finite number of states (constant over all changes), A is the finite number of actions (constant over all changes), K_T is the number of change points up to horizon T, and theta^{(ell)} is the transition kernel during the interval [c_ell, c_{ell+1}), which we assume to be multinomially distributed over the set of states O. Interestingly, the performance bound does not directly scale with the variation in MDP state transition distributions and rewards, ie. can also model abrupt changes. In practice, R-BOCPD-UCRL2 outperforms the state-of-the-art in a variety of scenarios in synthetic environments. We provide a detailed experimental setup along with a code repository (upon publication) that can be used to easily reproduce our experiments.

Robust reinforcement learning is essential for deploying reinforcement learning algorithms in real-world scenarios where environmental uncertainty predominates. Traditional robust reinforcement learning often depends on rectangularity assumptions, where adverse probability measures of outcome states are assumed to be independent across different states and actions. This assumption, rarely fulfilled in practice, leads to overly conservative policies. To address this problem, we introduce a new time-constrained robust MDP (TC-RMDP) formulation that considers multifactorial, correlated, and time-dependent disturbances, thus more accurately reflecting real-world dynamics. This formulation goes beyond the conventional rectangularity paradigm, offering new perspectives and expanding the analytical framework for robust RL. We propose three distinct algorithms, each using varying levels of environmental information, and evaluate them extensively on continuous control benchmarks. Our results demonstrate that these algorithms yield an efficient tradeoff between performance and robustness, outperforming traditional deep robust RL methods in time-constrained environments while preserving robustness in classical benchmarks. This study revisits the prevailing assumptions in robust RL and opens new avenues for developing more practical and realistic RL applications.

Mirror descent value iteration (MDVI), an abstraction of Kullback-Leibler (KL) and entropy-regularized reinforcement learning (RL), has served as the basis for recent high-performing practical RL algorithms. However, despite the use of function approximation in practice, the theoretical understanding of MDVI has been limited to tabular Markov decision processes (MDPs). We study MDVI with linear function approximation through its sample complexity required to identify an varepsilon-optimal policy with probability 1-delta under the settings of an infinite-horizon linear MDP, generative model, and G-optimal design. We demonstrate that least-squares regression weighted by the variance of an estimated optimal value function of the next state is crucial to achieving minimax optimality. Based on this observation, we present Variance-Weighted Least-Squares MDVI (VWLS-MDVI), the first theoretical algorithm that achieves nearly minimax optimal sample complexity for infinite-horizon linear MDPs. Furthermore, we propose a practical VWLS algorithm for value-based deep RL, Deep Variance Weighting (DVW). Our experiments demonstrate that DVW improves the performance of popular value-based deep RL algorithms on a set of MinAtar benchmarks.

Evaluations of Deep Reinforcement Learning (DRL) methods are an integral part of scientific progress of the field. Beyond designing DRL methods for general intelligence, designing task-specific methods is becoming increasingly prominent for real-world applications. In these settings, the standard evaluation practice involves using a few instances of Markov Decision Processes (MDPs) to represent the task. However, many tasks induce a large family of MDPs owing to variations in the underlying environment, particularly in real-world contexts. For example, in traffic signal control, variations may stem from intersection geometries and traffic flow levels. The select MDP instances may thus inadvertently cause overfitting, lacking the statistical power to draw conclusions about the method's true performance across the family. In this article, we augment DRL evaluations to consider parameterized families of MDPs. We show that in comparison to evaluating DRL methods on select MDP instances, evaluating the MDP family often yields a substantially different relative ranking of methods, casting doubt on what methods should be considered state-of-the-art. We validate this phenomenon in standard control benchmarks and the real-world application of traffic signal control. At the same time, we show that accurately evaluating on an MDP family is nontrivial. Overall, this work identifies new challenges for empirical rigor in reinforcement learning, especially as the outcomes of DRL trickle into downstream decision-making.

We study reward-free reinforcement learning (RL) with linear function approximation, where the agent works in two phases: (1) in the exploration phase, the agent interacts with the environment but cannot access the reward; and (2) in the planning phase, the agent is given a reward function and is expected to find a near-optimal policy based on samples collected in the exploration phase. The sample complexities of existing reward-free algorithms have a polynomial dependence on the planning horizon, which makes them intractable for long planning horizon RL problems. In this paper, we propose a new reward-free algorithm for learning linear mixture Markov decision processes (MDPs), where the transition probability can be parameterized as a linear combination of known feature mappings. At the core of our algorithm is uncertainty-weighted value-targeted regression with exploration-driven pseudo-reward and a high-order moment estimator for the aleatoric and epistemic uncertainties. When the total reward is bounded by 1, we show that our algorithm only needs to explore tilde O( d^2varepsilon^{-2}) episodes to find an varepsilon-optimal policy, where d is the dimension of the feature mapping. The sample complexity of our algorithm only has a polylogarithmic dependence on the planning horizon and therefore is ``horizon-free''. In addition, we provide an Omega(d^2varepsilon^{-2}) sample complexity lower bound, which matches the sample complexity of our algorithm up to logarithmic factors, suggesting that our algorithm is optimal.

We study infinite-horizon average-reward Markov decision processes (AMDPs) in the context of general function approximation. Specifically, we propose a novel algorithmic framework named Local-fitted Optimization with OPtimism (LOOP), which incorporates both model-based and value-based incarnations. In particular, LOOP features a novel construction of confidence sets and a low-switching policy updating scheme, which are tailored to the average-reward and function approximation setting. Moreover, for AMDPs, we propose a novel complexity measure -- average-reward generalized eluder coefficient (AGEC) -- which captures the challenge of exploration in AMDPs with general function approximation. Such a complexity measure encompasses almost all previously known tractable AMDP models, such as linear AMDPs and linear mixture AMDPs, and also includes newly identified cases such as kernel AMDPs and AMDPs with Bellman eluder dimensions. Using AGEC, we prove that LOOP achieves a sublinear mathcal{O}(poly(d, sp(V^*)) Tbeta ) regret, where d and beta correspond to AGEC and log-covering number of the hypothesis class respectively, sp(V^*) is the span of the optimal state bias function, T denotes the number of steps, and mathcal{O} (cdot) omits logarithmic factors. When specialized to concrete AMDP models, our regret bounds are comparable to those established by the existing algorithms designed specifically for these special cases. To the best of our knowledge, this paper presents the first comprehensive theoretical framework capable of handling nearly all AMDPs.

We study multi-agent reinforcement learning (MARL) for the general-sum Markov Games (MGs) under the general function approximation. In order to find the minimum assumption for sample-efficient learning, we introduce a novel complexity measure called the Multi-Agent Decoupling Coefficient (MADC) for general-sum MGs. Using this measure, we propose the first unified algorithmic framework that ensures sample efficiency in learning Nash Equilibrium, Coarse Correlated Equilibrium, and Correlated Equilibrium for both model-based and model-free MARL problems with low MADC. We also show that our algorithm provides comparable sublinear regret to the existing works. Moreover, our algorithm combines an equilibrium-solving oracle with a single objective optimization subprocedure that solves for the regularized payoff of each deterministic joint policy, which avoids solving constrained optimization problems within data-dependent constraints (Jin et al. 2020; Wang et al. 2023) or executing sampling procedures with complex multi-objective optimization problems (Foster et al. 2023), thus being more amenable to empirical implementation.

Value function factorization has achieved great success in multi-agent reinforcement learning by optimizing joint action-value functions through the maximization of factorized per-agent utilities. To ensure Individual-Global-Maximum property, existing works often focus on value factorization using monotonic functions, which are known to result in restricted representation expressiveness. In this paper, we analyze the limitations of monotonic factorization and present ConcaveQ, a novel non-monotonic value function factorization approach that goes beyond monotonic mixing functions and employs neural network representations of concave mixing functions. Leveraging the concave property in factorization, an iterative action selection scheme is developed to obtain optimal joint actions during training. It is used to update agents' local policy networks, enabling fully decentralized execution. The effectiveness of the proposed ConcaveQ is validated across scenarios involving multi-agent predator-prey environment and StarCraft II micromanagement tasks. Empirical results exhibit significant improvement of ConcaveQ over state-of-the-art multi-agent reinforcement learning approaches.

Generating simulations to train intelligent agents in game-playing and robotics from natural language input, from user input or task documentation, remains an open-ended challenge. Existing approaches focus on parts of this challenge, such as generating reward functions or task hyperparameters. Unlike previous work, we introduce FACTORSIM that generates full simulations in code from language input that can be used to train agents. Exploiting the structural modularity specific to coded simulations, we propose to use a factored partially observable Markov decision process representation that allows us to reduce context dependence during each step of the generation. For evaluation, we introduce a generative simulation benchmark that assesses the generated simulation code's accuracy and effectiveness in facilitating zero-shot transfers in reinforcement learning settings. We show that FACTORSIM outperforms existing methods in generating simulations regarding prompt alignment (e.g., accuracy), zero-shot transfer abilities, and human evaluation. We also demonstrate its effectiveness in generating robotic tasks.

In online reinforcement learning (RL), instead of employing standard structural assumptions on Markov decision processes (MDPs), using a certain coverage condition (original from offline RL) is enough to ensure sample-efficient guarantees (Xie et al. 2023). In this work, we focus on this new direction by digging more possible and general coverage conditions, and study the potential and the utility of them in efficient online RL. We identify more concepts, including the L^p variant of concentrability, the density ratio realizability, and trade-off on the partial/rest coverage condition, that can be also beneficial to sample-efficient online RL, achieving improved regret bound. Furthermore, if exploratory offline data are used, under our coverage conditions, both statistically and computationally efficient guarantees can be achieved for online RL. Besides, even though the MDP structure is given, e.g., linear MDP, we elucidate that, good coverage conditions are still beneficial to obtain faster regret bound beyond O(T) and even a logarithmic order regret. These results provide a good justification for the usage of general coverage conditions in efficient online RL.

Planning under partial observability is an essential capability of autonomous robots. The Partially Observable Markov Decision Process (POMDP) provides a powerful framework for planning under partial observability problems, capturing the stochastic effects of actions and the limited information available through noisy observations. POMDP solving could benefit tremendously from massive parallelization of today's hardware, but parallelizing POMDP solvers has been challenging. They rely on interleaving numerical optimization over actions with the estimation of their values, which creates dependencies and synchronization bottlenecks between parallel processes that can quickly offset the benefits of parallelization. In this paper, we propose Vectorized Online POMDP Planner (VOPP), a novel parallel online solver that leverages a recent POMDP formulation that analytically solves part of the optimization component, leaving only the estimation of expectations for numerical computation. VOPP represents all data structures related to planning as a collection of tensors and implements all planning steps as fully vectorized computations over this representation. The result is a massively parallel solver with no dependencies and synchronization bottlenecks between parallel computations. Experimental results indicate that VOPP is at least 20X more efficient in computing near-optimal solutions compared to an existing state-of-the-art parallel online solver.

Combinatorial optimization (CO) problems are often NP-hard and thus out of reach for exact algorithms, making them a tempting domain to apply machine learning methods. The highly structured constraints in these problems can hinder either optimization or sampling directly in the solution space. On the other hand, GFlowNets have recently emerged as a powerful machinery to efficiently sample from composite unnormalized densities sequentially and have the potential to amortize such solution-searching processes in CO, as well as generate diverse solution candidates. In this paper, we design Markov decision processes (MDPs) for different combinatorial problems and propose to train conditional GFlowNets to sample from the solution space. Efficient training techniques are also developed to benefit long-range credit assignment. Through extensive experiments on a variety of different CO tasks with synthetic and realistic data, we demonstrate that GFlowNet policies can efficiently find high-quality solutions.

We resolve the open question regarding the sample complexity of policy learning for maximizing the long-run average reward associated with a uniformly ergodic Markov decision process (MDP), assuming a generative model. In this context, the existing literature provides a sample complexity upper bound of widetilde O(|S||A|t_{mix}^2 epsilon^{-2}) and a lower bound of Omega(|S||A|t_{mix} epsilon^{-2}). In these expressions, |S| and |A| denote the cardinalities of the state and action spaces respectively, t_{mix} serves as a uniform upper limit for the total variation mixing times, and epsilon signifies the error tolerance. Therefore, a notable gap of t_{mix} still remains to be bridged. Our primary contribution is the development of an estimator for the optimal policy of average reward MDPs with a sample complexity of widetilde O(|S||A|t_{mix}epsilon^{-2}). This marks the first algorithm and analysis to reach the literature's lower bound. Our new algorithm draws inspiration from ideas in Li et al. (2020), Jin and Sidford (2021), and Wang et al. (2023). Additionally, we conduct numerical experiments to validate our theoretical findings.

We seek to learn an effective policy for a Markov Decision Process (MDP) with continuous states via Q-Learning. Given a set of basis functions over state action pairs we search for a corresponding set of linear weights that minimizes the mean Bellman residual. Our algorithm uses a Kalman filter model to estimate those weights and we have developed a simpler approximate Kalman filter model that outperforms the current state of the art projected TD-Learning methods on several standard benchmark problems.

We study the problem of optimally projecting the transition matrix of a finite ergodic multivariate Markov chain onto a lower-dimensional state space. Specifically, we seek to construct a projected Markov chain that optimizes various information-theoretic criteria under cardinality constraints. These criteria include entropy rate, information-theoretic distance to factorizability, independence, and stationarity. We formulate these tasks as best subset selection problems over multivariate Markov chains and leverage the submodular (or supermodular) structure of the objective functions to develop efficient greedy-based algorithms with theoretical guarantees. We extend our analysis to k-submodular settings and introduce a generalized version of the distorted greedy algorithm, which may be of independent interest. Finally, we illustrate the theory and algorithms through extensive numerical experiments with publicly available code on multivariate Markov chains associated with the Bernoulli-Laplace and Curie-Weiss model.

Representations are at the core of all deep reinforcement learning (RL) methods for both Markov decision processes (MDPs) and partially observable Markov decision processes (POMDPs). Many representation learning methods and theoretical frameworks have been developed to understand what constitutes an effective representation. However, the relationships between these methods and the shared properties among them remain unclear. In this paper, we show that many of these seemingly distinct methods and frameworks for state and history abstractions are, in fact, based on a common idea of self-predictive abstraction. Furthermore, we provide theoretical insights into the widely adopted objectives and optimization, such as the stop-gradient technique, in learning self-predictive representations. These findings together yield a minimalist algorithm to learn self-predictive representations for states and histories. We validate our theories by applying our algorithm to standard MDPs, MDPs with distractors, and POMDPs with sparse rewards. These findings culminate in a set of preliminary guidelines for RL practitioners.

POMDPs capture a broad class of decision making problems, but hardness results suggest that learning is intractable even in simple settings due to the inherent partial observability. However, in many realistic problems, more information is either revealed or can be computed during some point of the learning process. Motivated by diverse applications ranging from robotics to data center scheduling, we formulate a Hindsight Observable Markov Decision Process (HOMDP) as a POMDP where the latent states are revealed to the learner in hindsight and only during training. We introduce new algorithms for the tabular and function approximation settings that are provably sample-efficient with hindsight observability, even in POMDPs that would otherwise be statistically intractable. We give a lower bound showing that the tabular algorithm is optimal in its dependence on latent state and observation cardinalities.

We consider learning in an adversarial Markov Decision Process (MDP) where the loss functions can change arbitrarily over K episodes and the state space can be arbitrarily large. We assume that the Q-function of any policy is linear in some known features, that is, a linear function approximation exists. The best existing regret upper bound for this setting (Luo et al., 2021) is of order mathcal O(K^{2/3}) (omitting all other dependencies), given access to a simulator. This paper provides two algorithms that improve the regret to mathcal O(sqrt K) in the same setting. Our first algorithm makes use of a refined analysis of the Follow-the-Regularized-Leader (FTRL) algorithm with the log-barrier regularizer. This analysis allows the loss estimators to be arbitrarily negative and might be of independent interest. Our second algorithm develops a magnitude-reduced loss estimator, further removing the polynomial dependency on the number of actions in the first algorithm and leading to the optimal regret bound (up to logarithmic terms and dependency on the horizon). Moreover, we also extend the first algorithm to simulator-free linear MDPs, which achieves mathcal O(K^{8/9}) regret and greatly improves over the best existing bound mathcal O(K^{14/15}). This algorithm relies on a better alternative to the Matrix Geometric Resampling procedure by Neu & Olkhovskaya (2020), which could again be of independent interest.

In this paper, we propose a novel factored agent architecture designed to overcome the limitations of traditional single-agent systems in agentic AI. Our approach decomposes the agent into two specialized components: (1) a large language model (LLM) that serves as a high level planner and in-context learner, which may use dynamically available information in user prompts, (2) a smaller language model which acts as a memorizer of tool format and output. This decoupling addresses prevalent issues in monolithic designs, including malformed, missing, and hallucinated API fields, as well as suboptimal planning in dynamic environments. Empirical evaluations demonstrate that our factored architecture significantly improves planning accuracy and error resilience, while elucidating the inherent trade-off between in-context learning and static memorization. These findings suggest that a factored approach is a promising pathway for developing more robust and adaptable agentic AI systems.

Reinforcement learning often needs to deal with the exponential growth of states and actions when exploring optimal control in high-dimensional spaces (often known as the curse of dimensionality). In this work, we address this issue by learning the inherent structure of action-wise similar MDP to appropriately balance the performance degradation versus sample/computational complexity. In particular, we partition the action spaces into multiple groups based on the similarity in transition distribution and reward function, and build a linear decomposition model to capture the difference between the intra-group transition kernel and the intra-group rewards. Both our theoretical analysis and experiments reveal a surprising and counter-intuitive result: while a more refined grouping strategy can reduce the approximation error caused by treating actions in the same group as identical, it also leads to increased estimation error when the size of samples or the computation resources is limited. This finding highlights the grouping strategy as a new degree of freedom that can be optimized to minimize the overall performance loss. To address this issue, we formulate a general optimization problem for determining the optimal grouping strategy, which strikes a balance between performance loss and sample/computational complexity. We further propose a computationally efficient method for selecting a nearly-optimal grouping strategy, which maintains its computational complexity independent of the size of the action space.

The standard formulation of Markov decision processes (MDPs) assumes that the agent's decisions are executed immediately. However, in numerous realistic applications such as robotics or healthcare, actions are performed with a delay whose value can even be stochastic. In this work, we introduce stochastic delayed execution MDPs, a new formalism addressing random delays without resorting to state augmentation. We show that given observed delay values, it is sufficient to perform a policy search in the class of Markov policies in order to reach optimal performance, thus extending the deterministic fixed delay case. Armed with this insight, we devise DEZ, a model-based algorithm that optimizes over the class of Markov policies. DEZ leverages Monte-Carlo tree search similar to its non-delayed variant EfficientZero to accurately infer future states from the action queue. Thus, it handles delayed execution while preserving the sample efficiency of EfficientZero. Through a series of experiments on the Atari suite, we demonstrate that although the previous baseline outperforms the naive method in scenarios with constant delay, it underperforms in the face of stochastic delays. In contrast, our approach significantly outperforms the baselines, for both constant and stochastic delays. The code is available at http://github.com/davidva1/Delayed-EZ .

This paper investigates approaches to enhance the reasoning capabilities of Large Language Model (LLM) agents using Reinforcement Learning (RL). Specifically, we focus on multi-turn tool-use scenarios, which can be naturally modeled as Markov Decision Processes (MDPs). While existing approaches often train multi-turn LLM agents with trajectory-level advantage estimation in bandit settings, they struggle with turn-level credit assignment across multiple decision steps, limiting their performance on multi-turn reasoning tasks. To address this, we introduce a fine-grained turn-level advantage estimation strategy to enable more precise credit assignment in multi-turn agent interactions. The strategy is general and can be incorporated into various RL algorithms such as Group Relative Preference Optimization (GRPO). Our experimental evaluation on multi-turn reasoning and search-based tool-use tasks with GRPO implementations highlights the effectiveness of the MDP framework and the turn-level credit assignment in advancing the multi-turn reasoning capabilities of LLM agents in complex decision-making settings. Our method achieves 100% success in tool execution and 50% accuracy in exact answer matching, significantly outperforming baselines, which fail to invoke tools and achieve only 20-30% exact match accuracy.

As a seemingly self-explanatory task, problem-solving has been a significant component of science and engineering. However, a general yet concrete formulation of problem-solving itself is missing. With the recent development of AI-based problem-solving agents, the demand for process-level verifiability is rapidly increasing yet underexplored. To fill these gaps, we present a principled formulation of problem-solving as a deterministic Markov decision process; a novel framework, FPS (Formal Problem-Solving), which utilizes existing FTP (formal theorem proving) environments to perform process-verified problem-solving; and D-FPS (Deductive FPS), decoupling solving and answer verification for better human-alignment. The expressiveness, soundness and completeness of the frameworks are proven. We construct three benchmarks on problem-solving: FormalMath500, a formalization of a subset of the MATH500 benchmark; MiniF2F-Solving and PutnamBench-Solving, adaptations of FTP benchmarks MiniF2F and PutnamBench. For faithful, interpretable, and human-aligned evaluation, we propose RPE (Restricted Propositional Equivalence), a symbolic approach to determine the correctness of answers by formal verification. We evaluate four prevalent FTP models and two prompting methods as baselines, solving at most 23.77% of FormalMath500, 27.47% of MiniF2F-Solving, and 0.31% of PutnamBench-Solving.

In recent years, masked diffusion models (MDMs) have emerged as a promising alternative approach for generative modeling over discrete domains. Compared to autoregressive models (ARMs), MDMs trade off complexity at training time with flexibility at inference time. At training time, they must learn to solve an exponentially large number of infilling problems, but at inference time, they can decode tokens in essentially arbitrary order. In this work, we closely examine these two competing effects. On the training front, we theoretically and empirically demonstrate that MDMs indeed train on computationally intractable subproblems compared to their autoregressive counterparts. On the inference front, we show that a suitable strategy for adaptively choosing the token decoding order significantly enhances the capabilities of MDMs, allowing them to sidestep hard subproblems. On logic puzzles like Sudoku, we show that adaptive inference can boost solving accuracy in pretrained MDMs from <7% to approx 90%, even outperforming ARMs with 7times as many parameters and that were explicitly trained via teacher forcing to learn the right order of decoding.

The Distributionally Robust Markov Decision Process (DRMDP) is a popular framework for addressing dynamics shift in reinforcement learning by learning policies robust to the worst-case transition dynamics within a constrained set. However, solving its dual optimization oracle poses significant challenges, limiting theoretical analysis and computational efficiency. The recently proposed Robust Regularized Markov Decision Process (RRMDP) replaces the uncertainty set constraint with a regularization term on the value function, offering improved scalability and theoretical insights. Yet, existing RRMDP methods rely on unstructured regularization, often leading to overly conservative policies by considering transitions that are unrealistic. To address these issues, we propose a novel framework, the d-rectangular linear robust regularized Markov decision process (d-RRMDP), which introduces a linear latent structure into both transition kernels and regularization. For the offline RL setting, where an agent learns robust policies from a pre-collected dataset in the nominal environment, we develop a family of algorithms, Robust Regularized Pessimistic Value Iteration (R2PVI), employing linear function approximation and f-divergence based regularization terms on transition kernels. We provide instance-dependent upper bounds on the suboptimality gap of R2PVI policies, showing these bounds depend on how well the dataset covers state-action spaces visited by the optimal robust policy under robustly admissible transitions. This term is further shown to be fundamental to d-RRMDPs via information-theoretic lower bounds. Finally, numerical experiments validate that R2PVI learns robust policies and is computationally more efficient than methods for constrained DRMDPs.

Sphere packing, Hilbert's eighteenth problem, asks for the densest arrangement of congruent spheres in n-dimensional Euclidean space. Although relevant to areas such as cryptography, crystallography, and medical imaging, the problem remains unresolved: beyond a few special dimensions, neither optimal packings nor tight upper bounds are known. Even a major breakthrough in dimension n=8, later recognised with a Fields Medal, underscores its difficulty. A leading technique for upper bounds, the three-point method, reduces the problem to solving large, high-precision semidefinite programs (SDPs). Because each candidate SDP may take days to evaluate, standard data-intensive AI approaches are infeasible. We address this challenge by formulating SDP construction as a sequential decision process, the SDP game, in which a policy assembles SDP formulations from a set of admissible components. Using a sample-efficient model-based framework that combines Bayesian optimisation with Monte Carlo Tree Search, we obtain new state-of-the-art upper bounds in dimensions 4-16, showing that model-based search can advance computational progress in longstanding geometric problems. Together, these results demonstrate that sample-efficient, model-based search can make tangible progress on mathematically rigid, evaluation limited problems, pointing towards a complementary direction for AI-assisted discovery beyond large-scale LLM-driven exploration.

Many resource management problems require sequential decision-making under uncertainty, where the only uncertainty affecting the decision outcomes are exogenous variables outside the control of the decision-maker. We model these problems as Exo-MDPs (Markov Decision Processes with Exogenous Inputs) and design a class of data-efficient algorithms for them termed Hindsight Learning (HL). Our HL algorithms achieve data efficiency by leveraging a key insight: having samples of the exogenous variables, past decisions can be revisited in hindsight to infer counterfactual consequences that can accelerate policy improvements. We compare HL against classic baselines in the multi-secretary and airline revenue management problems. We also scale our algorithms to a business-critical cloud resource management problem -- allocating Virtual Machines (VMs) to physical machines, and simulate their performance with real datasets from a large public cloud provider. We find that HL algorithms outperform domain-specific heuristics, as well as state-of-the-art reinforcement learning methods.

We introduce new differentially private (DP) mechanisms for gradient-based machine learning (ML) with multiple passes (epochs) over a dataset, substantially improving the achievable privacy-utility-computation tradeoffs. We formalize the problem of DP mechanisms for adaptive streams with multiple participations and introduce a non-trivial extension of online matrix factorization DP mechanisms to our setting. This includes establishing the necessary theory for sensitivity calculations and efficient computation of optimal matrices. For some applications like >!! 10,000 SGD steps, applying these optimal techniques becomes computationally expensive. We thus design an efficient Fourier-transform-based mechanism with only a minor utility loss. Extensive empirical evaluation on both example-level DP for image classification and user-level DP for language modeling demonstrate substantial improvements over all previous methods, including the widely-used DP-SGD . Though our primary application is to ML, our main DP results are applicable to arbitrary linear queries and hence may have much broader applicability.

We define a new class of set functions that in addition to being monotone and subadditive, also admit a very limited form of submodularity defined over a permutation of the ground set. We refer to this permutation as a submodular order. This class of functions includes monotone submodular functions as a sub-family. To understand the importance of this structure in optimization problems we consider the problem of maximizing function value under various types of constraints. To demonstrate the modeling power of submodular order functions we show applications in two different settings. First, we apply our results to the extensively studied problem of assortment optimization. While the objectives in assortment optimization are known to be non-submodular (and non-monotone) even for simple choice models, we show that they are compatible with the notion of submodular order. Consequently, we obtain new and in some cases the first constant factor guarantee for constrained assortment optimization in fundamental choice models. As a second application of submodular order functions, we show an intriguing connection to the maximization of monotone submodular functions in the streaming model. We recover some best known guarantees for this problem as a corollary of our results.

We study the design of sample-efficient algorithms for reinforcement learning in the presence of rich, high-dimensional observations, formalized via the Block MDP problem. Existing algorithms suffer from either 1) computational intractability, 2) strong statistical assumptions that are not necessarily satisfied in practice, or 3) suboptimal sample complexity. We address these issues by providing the first computationally efficient algorithm that attains rate-optimal sample complexity with respect to the desired accuracy level, with minimal statistical assumptions. Our algorithm, MusIK, combines systematic exploration with representation learning based on multi-step inverse kinematics, a learning objective in which the aim is to predict the learner's own action from the current observation and observations in the (potentially distant) future. MusIK is simple and flexible, and can efficiently take advantage of general-purpose function approximation. Our analysis leverages several new techniques tailored to non-optimistic exploration algorithms, which we anticipate will find broader use.

In dynamic programming (DP) and reinforcement learning (RL), an agent learns to act optimally in terms of expected long-term return by sequentially interacting with its environment modeled by a Markov decision process (MDP). More generally in distributional reinforcement learning (DRL), the focus is on the whole distribution of the return, not just its expectation. Although DRL-based methods produced state-of-the-art performance in RL with function approximation, they involve additional quantities (compared to the non-distributional setting) that are still not well understood. As a first contribution, we introduce a new class of distributional operators, together with a practical DP algorithm for policy evaluation, that come with a robust MDP interpretation. Indeed, our approach reformulates through an augmented state space where each state is split into a worst-case substate and a best-case substate, whose values are maximized by safe and risky policies respectively. Finally, we derive distributional operators and DP algorithms solving a new control task: How to distinguish safe from risky optimal actions in order to break ties in the space of optimal policies?

Reinforcement learning (RL) post-training is crucial for LLM alignment and reasoning, but existing policy-based methods, such as PPO and DPO, can fall short of fixing shortcuts inherited from pre-training. In this work, we introduce Qsharp, a value-based algorithm for KL-regularized RL that guides the reference policy using the optimal regularized Q function. We propose to learn the optimal Q function using distributional RL on an aggregated online dataset. Unlike prior value-based baselines that guide the model using unregularized Q-values, our method is theoretically principled and provably learns the optimal policy for the KL-regularized RL problem. Empirically, Qsharp outperforms prior baselines in math reasoning benchmarks while maintaining a smaller KL divergence to the reference policy. Theoretically, we establish a reduction from KL-regularized RL to no-regret online learning, providing the first bounds for deterministic MDPs under only realizability. Thanks to distributional RL, our bounds are also variance-dependent and converge faster when the reference policy has small variance. In sum, our results highlight Qsharp as an effective approach for post-training LLMs, offering both improved performance and theoretical guarantees. The code can be found at https://github.com/jinpz/q_sharp.

Designing control policies whose performance level is guaranteed to remain above a given threshold in a span of environments is a critical feature for the adoption of reinforcement learning (RL) in real-world applications. The search for such robust policies is a notoriously difficult problem, related to the so-called dynamic model of transition function uncertainty, where the environment dynamics are allowed to change at each time step. But in practical cases, one is rather interested in robustness to a span of static transition models throughout interaction episodes. The static model is known to be harder to solve than the dynamic one, and seminal algorithms, such as robust value iteration, as well as most recent works on deep robust RL, build upon the dynamic model. In this work, we propose to revisit the static model. We suggest an analysis of why solving the static model under some mild hypotheses is a reasonable endeavor, based on an equivalence with the dynamic model, and formalize the general intuition that robust MDPs can be solved by tackling a series of static problems. We introduce a generic meta-algorithm called IWOCS, which incrementally identifies worst-case transition models so as to guide the search for a robust policy. Discussion on IWOCS sheds light on new ways to decouple policy optimization and adversarial transition functions and opens new perspectives for analysis. We derive a deep RL version of IWOCS and demonstrate it is competitive with state-of-the-art algorithms on classical benchmarks.

We study reinforcement learning with function approximation for large-scale Partially Observable Markov Decision Processes (POMDPs) where the state space and observation space are large or even continuous. Particularly, we consider Hilbert space embeddings of POMDP where the feature of latent states and the feature of observations admit a conditional Hilbert space embedding of the observation emission process, and the latent state transition is deterministic. Under the function approximation setup where the optimal latent state-action Q-function is linear in the state feature, and the optimal Q-function has a gap in actions, we provide a computationally and statistically efficient algorithm for finding the exact optimal policy. We show our algorithm's computational and statistical complexities scale polynomially with respect to the horizon and the intrinsic dimension of the feature on the observation space. Furthermore, we show both the deterministic latent transitions and gap assumptions are necessary to avoid statistical complexity exponential in horizon or dimension. Since our guarantee does not have an explicit dependence on the size of the state and observation spaces, our algorithm provably scales to large-scale POMDPs.

Reinforcement learning (RL) theory has largely focused on proving minimax sample complexity bounds. These require strategic exploration algorithms that use relatively limited function classes for representing the policy or value function. Our goal is to explain why deep RL algorithms often perform well in practice, despite using random exploration and much more expressive function classes like neural networks. Our work arrives at an explanation by showing that many stochastic MDPs can be solved by performing only a few steps of value iteration on the random policy's Q function and then acting greedily. When this is true, we find that it is possible to separate the exploration and learning components of RL, making it much easier to analyze. We introduce a new RL algorithm, SQIRL, that iteratively learns a near-optimal policy by exploring randomly to collect rollouts and then performing a limited number of steps of fitted-Q iteration over those rollouts. Any regression algorithm that satisfies basic in-distribution generalization properties can be used in SQIRL to efficiently solve common MDPs. This can explain why deep RL works, since it is empirically established that neural networks generalize well in-distribution. Furthermore, SQIRL explains why random exploration works well in practice. We leverage SQIRL to derive instance-dependent sample complexity bounds for RL that are exponential only in an "effective horizon" of lookahead and on the complexity of the class used for function approximation. Empirically, we also find that SQIRL performance strongly correlates with PPO and DQN performance in a variety of stochastic environments, supporting that our theoretical analysis is predictive of practical performance. Our code and data are available at https://github.com/cassidylaidlaw/effective-horizon.

Robust Markov decision processes (MDPs) address the challenge of model uncertainty by optimizing the worst-case performance over an uncertainty set of MDPs. In this paper, we focus on the robust average-reward MDPs under the model-free setting. We first theoretically characterize the structure of solutions to the robust average-reward Bellman equation, which is essential for our later convergence analysis. We then design two model-free algorithms, robust relative value iteration (RVI) TD and robust RVI Q-learning, and theoretically prove their convergence to the optimal solution. We provide several widely used uncertainty sets as examples, including those defined by the contamination model, total variation, Chi-squared divergence, Kullback-Leibler (KL) divergence and Wasserstein distance.

In this paper, we study the problem of (finite horizon tabular) Markov decision processes (MDPs) with heavy-tailed rewards under the constraint of differential privacy (DP). Compared with the previous studies for private reinforcement learning that typically assume rewards are sampled from some bounded or sub-Gaussian distributions to ensure DP, we consider the setting where reward distributions have only finite (1+v)-th moments with some v in (0,1]. By resorting to robust mean estimators for rewards, we first propose two frameworks for heavy-tailed MDPs, i.e., one is for value iteration and another is for policy optimization. Under each framework, we consider both joint differential privacy (JDP) and local differential privacy (LDP) models. Based on our frameworks, we provide regret upper bounds for both JDP and LDP cases and show that the moment of distribution and privacy budget both have significant impacts on regrets. Finally, we establish a lower bound of regret minimization for heavy-tailed MDPs in JDP model by reducing it to the instance-independent lower bound of heavy-tailed multi-armed bandits in DP model. We also show the lower bound for the problem in LDP by adopting some private minimax methods. Our results reveal that there are fundamental differences between the problem of private RL with sub-Gaussian and that with heavy-tailed rewards.

Most offline reinforcement learning (RL) algorithms return a target policy maximizing a trade-off between (1) the expected performance gain over the behavior policy that collected the dataset, and (2) the risk stemming from the out-of-distribution-ness of the induced state-action occupancy. It follows that the performance of the target policy is strongly related to the performance of the behavior policy and, thus, the trajectory return distribution of the dataset. We show that in mixed datasets consisting of mostly low-return trajectories and minor high-return trajectories, state-of-the-art offline RL algorithms are overly restrained by low-return trajectories and fail to exploit high-performing trajectories to the fullest. To overcome this issue, we show that, in deterministic MDPs with stochastic initial states, the dataset sampling can be re-weighted to induce an artificial dataset whose behavior policy has a higher return. This re-weighted sampling strategy may be combined with any offline RL algorithm. We further analyze that the opportunity for performance improvement over the behavior policy correlates with the positive-sided variance of the returns of the trajectories in the dataset. We empirically show that while CQL, IQL, and TD3+BC achieve only a part of this potential policy improvement, these same algorithms combined with our reweighted sampling strategy fully exploit the dataset. Furthermore, we empirically demonstrate that, despite its theoretical limitation, the approach may still be efficient in stochastic environments. The code is available at https://github.com/Improbable-AI/harness-offline-rl.

Constraint handling remains a key bottleneck in quantum combinatorial optimization. While slack-variable-based encodings are straightforward, they significantly increase qubit counts and circuit depth, challenging the scalability of quantum solvers. In this work, we investigate a suite of Lagrangian-based optimization techniques including dual ascent, bundle methods, cutting plane approaches, and augmented Lagrangian formulations for solving constrained combinatorial problems on quantum simulators and hardware. Our framework is applied to three representative NP-hard problems: the Travelling Salesman Problem (TSP), the Multi-Dimensional Knapsack Problem (MDKP), and the Maximum Independent Set (MIS). We demonstrate that MDKP and TSP, with their inequality-based or degree-constrained structures, allow for slack-free reformulations, leading to significant qubit savings without compromising performance. In contrast, MIS does not inherently benefit from slack elimination but still gains in feasibility and objective quality from principled Lagrangian updates. We benchmark these methods across classically hard instances, analyzing trade-offs in qubit usage, feasibility, and optimality gaps. Our results highlight the flexibility of Lagrangian formulations as a scalable alternative to naive QUBO penalization, even when qubit savings are not always achievable. This work provides practical insights for deploying constraint-aware quantum optimization pipelines, with applications in logistics, network design, and resource allocation.

Decision-making under distribution shift is a central challenge in reinforcement learning (RL), where training and deployment environments differ. We study this problem through the lens of robust Markov decision processes (RMDPs), which optimize performance against adversarial transition dynamics. Our focus is the online setting, where the agent has only limited interaction with the environment, making sample efficiency and exploration especially critical. Policy optimization, despite its success in standard RL, remains theoretically and empirically underexplored in robust RL. To bridge this gap, we propose Distributionally Robust Regularized Policy Optimization algorithm (DR-RPO), a model-free online policy optimization method that learns robust policies with sublinear regret. To enable tractable optimization within the softmax policy class, DR-RPO incorporates reference-policy regularization, yielding RMDP variants that are doubly constrained in both transitions and policies. To scale to large state-action spaces, we adopt the d-rectangular linear MDP formulation and combine linear function approximation with an upper confidence bonus for optimistic exploration. We provide theoretical guarantees showing that policy optimization can achieve polynomial suboptimality bounds and sample efficiency in robust RL, matching the performance of value-based approaches. Finally, empirical results across diverse domains corroborate our theory and demonstrate the robustness of DR-RPO.

We solve a challenging yet practically useful variant of 3D Bin Packing Problem (3D-BPP). In our problem, the agent has limited information about the items to be packed into the bin, and an item must be packed immediately after its arrival without buffering or readjusting. The item's placement also subjects to the constraints of collision avoidance and physical stability. We formulate this online 3D-BPP as a constrained Markov decision process. To solve the problem, we propose an effective and easy-to-implement constrained deep reinforcement learning (DRL) method under the actor-critic framework. In particular, we introduce a feasibility predictor to predict the feasibility mask for the placement actions and use it to modulate the action probabilities output by the actor during training. Such supervisions and transformations to DRL facilitate the agent to learn feasible policies efficiently. Our method can also be generalized e.g., with the ability to handle lookahead or items with different orientations. We have conducted extensive evaluation showing that the learned policy significantly outperforms the state-of-the-art methods. A user study suggests that our method attains a human-level performance.

We introduce a framework for automatically defining and learning deep generative models with problem-specific structure. We tackle problem domains that are more traditionally solved by algorithms such as sorting, constraint satisfaction for Sudoku, and matrix factorization. Concretely, we train diffusion models with an architecture tailored to the problem specification. This problem specification should contain a graphical model describing relationships between variables, and often benefits from explicit representation of subcomputations. Permutation invariances can also be exploited. Across a diverse set of experiments we improve the scaling relationship between problem dimension and our model's performance, in terms of both training time and final accuracy. Our code can be found at https://github.com/plai-group/gsdm.

Recent improvements in conditional generative modeling have made it possible to generate high-quality images from language descriptions alone. We investigate whether these methods can directly address the problem of sequential decision-making. We view decision-making not through the lens of reinforcement learning (RL), but rather through conditional generative modeling. To our surprise, we find that our formulation leads to policies that can outperform existing offline RL approaches across standard benchmarks. By modeling a policy as a return-conditional diffusion model, we illustrate how we may circumvent the need for dynamic programming and subsequently eliminate many of the complexities that come with traditional offline RL. We further demonstrate the advantages of modeling policies as conditional diffusion models by considering two other conditioning variables: constraints and skills. Conditioning on a single constraint or skill during training leads to behaviors at test-time that can satisfy several constraints together or demonstrate a composition of skills. Our results illustrate that conditional generative modeling is a powerful tool for decision-making.

In online reinforcement learning (online RL), balancing exploration and exploitation is crucial for finding an optimal policy in a sample-efficient way. To achieve this, existing sample-efficient online RL algorithms typically consist of three components: estimation, planning, and exploration. However, in order to cope with general function approximators, most of them involve impractical algorithmic components to incentivize exploration, such as optimization within data-dependent level-sets or complicated sampling procedures. To address this challenge, we propose an easy-to-implement RL framework called Maximize to Explore (MEX), which only needs to optimize unconstrainedly a single objective that integrates the estimation and planning components while balancing exploration and exploitation automatically. Theoretically, we prove that MEX achieves a sublinear regret with general function approximations for Markov decision processes (MDP) and is further extendable to two-player zero-sum Markov games (MG). Meanwhile, we adapt deep RL baselines to design practical versions of MEX, in both model-free and model-based manners, which can outperform baselines by a stable margin in various MuJoCo environments with sparse rewards. Compared with existing sample-efficient online RL algorithms with general function approximations, MEX achieves similar sample efficiency while enjoying a lower computational cost and is more compatible with modern deep RL methods.

The application of Reinforcement Learning (RL) in real world environments can be expensive or risky due to sub-optimal policies during training. In Offline RL, this problem is avoided since interactions with an environment are prohibited. Policies are learned from a given dataset, which solely determines their performance. Despite this fact, how dataset characteristics influence Offline RL algorithms is still hardly investigated. The dataset characteristics are determined by the behavioral policy that samples this dataset. Therefore, we define characteristics of behavioral policies as exploratory for yielding high expected information in their interaction with the Markov Decision Process (MDP) and as exploitative for having high expected return. We implement two corresponding empirical measures for the datasets sampled by the behavioral policy in deterministic MDPs. The first empirical measure SACo is defined by the normalized unique state-action pairs and captures exploration. The second empirical measure TQ is defined by the normalized average trajectory return and captures exploitation. Empirical evaluations show the effectiveness of TQ and SACo. In large-scale experiments using our proposed measures, we show that the unconstrained off-policy Deep Q-Network family requires datasets with high SACo to find a good policy. Furthermore, experiments show that policy constraint algorithms perform well on datasets with high TQ and SACo. Finally, the experiments show, that purely dataset-constrained Behavioral Cloning performs competitively to the best Offline RL algorithms for datasets with high TQ.

We study the autonomous exploration (AX) problem proposed by Lim & Auer (2012). In this setting, the objective is to discover a set of epsilon-optimal policies reaching a set S_L^{rightarrow} of incrementally L-controllable states. We introduce a novel layered decomposition of the set of incrementally L-controllable states that is based on the iterative application of a state-expansion operator. We leverage these results to design Layered Autonomous Exploration (LAE), a novel algorithm for AX that attains a sample complexity of mathcal{O}(LS^{rightarrow}_{L(1+epsilon)}Gamma_{L(1+epsilon)} A ln^{12}(S^{rightarrow}_{L(1+epsilon)})/epsilon^2), where S^{rightarrow}_{L(1+epsilon)} is the number of states that are incrementally L(1+epsilon)-controllable, A is the number of actions, and Gamma_{L(1+epsilon)} is the branching factor of the transitions over such states. LAE improves over the algorithm of Tarbouriech et al. (2020a) by a factor of L^2 and it is the first algorithm for AX that works in a countably-infinite state space. Moreover, we show that, under a certain identifiability assumption, LAE achieves minimax-optimal sample complexity of mathcal{O}(LS^{rightarrow}_{L}Aln^{12}(S^{rightarrow}_{L})/epsilon^2), outperforming existing algorithms and matching for the first time the lower bound proved by Cai et al. (2022) up to logarithmic factors.

In multi-task reinforcement learning (RL) under Markov decision processes (MDPs), the presence of shared latent structures among multiple MDPs has been shown to yield significant benefits to the sample efficiency compared to single-task RL. In this paper, we investigate whether such a benefit can extend to more general sequential decision making problems, such as partially observable MDPs (POMDPs) and more general predictive state representations (PSRs). The main challenge here is that the large and complex model space makes it hard to identify what types of common latent structure of multi-task PSRs can reduce the model complexity and improve sample efficiency. To this end, we posit a joint model class for tasks and use the notion of eta-bracketing number to quantify its complexity; this number also serves as a general metric to capture the similarity of tasks and thus determines the benefit of multi-task over single-task RL. We first study upstream multi-task learning over PSRs, in which all tasks share the same observation and action spaces. We propose a provably efficient algorithm UMT-PSR for finding near-optimal policies for all PSRs, and demonstrate that the advantage of multi-task learning manifests if the joint model class of PSRs has a smaller eta-bracketing number compared to that of individual single-task learning. We also provide several example multi-task PSRs with small eta-bracketing numbers, which reap the benefits of multi-task learning. We further investigate downstream learning, in which the agent needs to learn a new target task that shares some commonalities with the upstream tasks via a similarity constraint. By exploiting the learned PSRs from the upstream, we develop a sample-efficient algorithm that provably finds a near-optimal policy.

Policy Optimization (PO) is one of the most popular methods in Reinforcement Learning (RL). Thus, theoretical guarantees for PO algorithms have become especially important to the RL community. In this paper, we study PO in adversarial MDPs with a challenge that arises in almost every real-world application -- delayed bandit feedback. We give the first near-optimal regret bounds for PO in tabular MDPs, and may even surpass state-of-the-art (which uses less efficient methods). Our novel Delay-Adapted PO (DAPO) is easy to implement and to generalize, allowing us to extend our algorithm to: (i) infinite state space under the assumption of linear Q-function, proving the first regret bounds for delayed feedback with function approximation. (ii) deep RL, demonstrating its effectiveness in experiments on MuJoCo domains.

We consider the problem of decentralized multi-agent reinforcement learning in Markov games. A fundamental question is whether there exist algorithms that, when adopted by all agents and run independently in a decentralized fashion, lead to no-regret for each player, analogous to celebrated convergence results in normal-form games. While recent work has shown that such algorithms exist for restricted settings (notably, when regret is defined with respect to deviations to Markovian policies), the question of whether independent no-regret learning can be achieved in the standard Markov game framework was open. We provide a decisive negative resolution this problem, both from a computational and statistical perspective. We show that: - Under the widely-believed assumption that PPAD-hard problems cannot be solved in polynomial time, there is no polynomial-time algorithm that attains no-regret in general-sum Markov games when executed independently by all players, even when the game is known to the algorithm designer and the number of players is a small constant. - When the game is unknown, no algorithm, regardless of computational efficiency, can achieve no-regret without observing a number of episodes that is exponential in the number of players. Perhaps surprisingly, our lower bounds hold even for seemingly easier setting in which all agents are controlled by a a centralized algorithm. They are proven via lower bounds for a simpler problem we refer to as SparseCCE, in which the goal is to compute a coarse correlated equilibrium that is sparse in the sense that it can be represented as a mixture of a small number of product policies. The crux of our approach is a novel application of aggregation techniques from online learning, whereby we show that any algorithm for the SparseCCE problem can be used to compute approximate Nash equilibria for non-zero sum normal-form games.

Reinforcement Learning algorithms that learn from human feedback (RLHF) need to be efficient in terms of statistical complexity, computational complexity, and query complexity. In this work, we consider the RLHF setting where the feedback is given in the format of preferences over pairs of trajectories. In the linear MDP model, using randomization in algorithm design, we present an algorithm that is sample efficient (i.e., has near-optimal worst-case regret bounds) and has polynomial running time (i.e., computational complexity is polynomial with respect to relevant parameters). Our algorithm further minimizes the query complexity through a novel randomized active learning procedure. In particular, our algorithm demonstrates a near-optimal tradeoff between the regret bound and the query complexity. To extend the results to more general nonlinear function approximation, we design a model-based randomized algorithm inspired by the idea of Thompson sampling. Our algorithm minimizes Bayesian regret bound and query complexity, again achieving a near-optimal tradeoff between these two quantities. Computation-wise, similar to the prior Thompson sampling algorithms under the regular RL setting, the main computation primitives of our algorithm are Bayesian supervised learning oracles which have been heavily investigated on the empirical side when applying Thompson sampling algorithms to RL benchmark problems.

We present an algorithm for learning mixtures of Markov chains and Markov decision processes (MDPs) from short unlabeled trajectories. Specifically, our method handles mixtures of Markov chains with optional control input by going through a multi-step process, involving (1) a subspace estimation step, (2) spectral clustering of trajectories using "pairwise distance estimators," along with refinement using the EM algorithm, (3) a model estimation step, and (4) a classification step for predicting labels of new trajectories. We provide end-to-end performance guarantees, where we only explicitly require the length of trajectories to be linear in the number of states and the number of trajectories to be linear in a mixing time parameter. Experimental results support these guarantees, where we attain 96.6% average accuracy on a mixture of two MDPs in gridworld, outperforming the EM algorithm with random initialization (73.2% average accuracy).

Preference-based Reinforcement Learning (PbRL) is a paradigm in which an RL agent learns to optimize a task using pair-wise preference-based feedback over trajectories, rather than explicit reward signals. While PbRL has demonstrated practical success in fine-tuning language models, existing theoretical work focuses on regret minimization and fails to capture most of the practical frameworks. In this study, we fill in such a gap between theoretical PbRL and practical algorithms by proposing a theoretical reward-agnostic PbRL framework where exploratory trajectories that enable accurate learning of hidden reward functions are acquired before collecting any human feedback. Theoretical analysis demonstrates that our algorithm requires less human feedback for learning the optimal policy under preference-based models with linear parameterization and unknown transitions, compared to the existing theoretical literature. Specifically, our framework can incorporate linear and low-rank MDPs with efficient sample complexity. Additionally, we investigate reward-agnostic RL with action-based comparison feedback and introduce an efficient querying algorithm tailored to this scenario.

Machine learning approaches relying on such criteria as adversarial robustness or multi-agent settings have raised the need for solving game-theoretic equilibrium problems. Of particular relevance to these applications are methods targeting finite-sum structure, which generically arises in empirical variants of learning problems in these contexts. Further, methods with computable approximation errors are highly desirable, as they provide verifiable exit criteria. Motivated by these applications, we study finite-sum monotone inclusion problems, which model broad classes of equilibrium problems. Our main contributions are variants of the classical Halpern iteration that employ variance reduction to obtain improved complexity guarantees in which n component operators in the finite sum are ``on average'' either cocoercive or Lipschitz continuous and monotone, with parameter L. The resulting oracle complexity of our methods, which provide guarantees for the last iterate and for a (computable) operator norm residual, is mathcal{O}( n + nLvarepsilon^{-1}), which improves upon existing methods by a factor up to n. This constitutes the first variance reduction-type result for general finite-sum monotone inclusions and for more specific problems such as convex-concave optimization when operator norm residual is the optimality measure. We further argue that, up to poly-logarithmic factors, this complexity is unimprovable in the monotone Lipschitz setting; i.e., the provided result is near-optimal.

Diffusion language models, as a promising alternative to traditional autoregressive (AR) models, enable faster generation and richer conditioning on bidirectional context. However, they suffer from a key discrepancy between training and inference: during inference, MDLMs progressively reveal the structure of the generated sequence by producing fewer and fewer masked tokens, whereas this structure is ignored in training as tokens are masked at random. Although this discrepancy between training and inference can lead to suboptimal performance, it has been largely overlooked by previous works, leaving closing this gap between the two stages an open problem. To address this, we frame the problem of learning effective denoising trajectories as a sequential decision-making problem and use the resulting framework to apply reinforcement learning. We propose a novel Masked Diffusion Policy Optimization (MDPO) to exploit the Markov property diffusion possesses and explicitly train the model under the same progressive refining schedule used at inference. MDPO matches the performance of the previous state-of-the-art (SOTA) method with 60x fewer gradient updates, while achieving average improvements of 9.6% on MATH500 and 54.2% on Countdown over SOTA when trained within the same number of weight updates. Additionally, we improve the remasking strategy of MDLMs as a plug-in inference replacement to overcome the limitation that the model cannot refine tokens flexibly. This simple yet effective training-free strategy, what we refer to as RCR, consistently improves performance and yields additional gains when combined with MDPO. Our findings establish great potential for investigating the discrepancy between pre-training and inference of MDLMs. Code: https://github.com/autonomousvision/mdpo. Project Page: https://cli212.github.io/MDPO/.

Mixed Integer Linear Programming (MILP) is a fundamental tool for modeling combinatorial optimization problems. Recently, a growing body of research has used machine learning to accelerate MILP solving. Despite the increasing popularity of this approach, there is a lack of a common repository that provides distributions of similar MILP instances across different domains, at different hardness levels, with standardized test sets. In this paper, we introduce Distributional MIPLIB, a multi-domain library of problem distributions for advancing ML-guided MILP methods. We curate MILP distributions from existing work in this area as well as real-world problems that have not been used, and classify them into different hardness levels. It will facilitate research in this area by enabling comprehensive evaluation on diverse and realistic domains. We empirically illustrate the benefits of using Distributional MIPLIB as a research vehicle in two ways. We evaluate the performance of ML-guided variable branching on previously unused distributions to identify potential areas for improvement. Moreover, we propose to learn branching policies from a mix of distributions, demonstrating that mixed distributions achieve better performance compared to homogeneous distributions when there is limited data and generalize well to larger instances. The dataset is publicly available at https://sites.google.com/usc.edu/distributional-miplib/home.

Diffusion-based generative models (DBGMs) perturb data to a target noise distribution and reverse this process to generate samples. The choice of noising process, or inference diffusion process, affects both likelihoods and sample quality. For example, extending the inference process with auxiliary variables leads to improved sample quality. While there are many such multivariate diffusions to explore, each new one requires significant model-specific analysis, hindering rapid prototyping and evaluation. In this work, we study Multivariate Diffusion Models (MDMs). For any number of auxiliary variables, we provide a recipe for maximizing a lower-bound on the MDMs likelihood without requiring any model-specific analysis. We then demonstrate how to parameterize the diffusion for a specified target noise distribution; these two points together enable optimizing the inference diffusion process. Optimizing the diffusion expands easy experimentation from just a few well-known processes to an automatic search over all linear diffusions. To demonstrate these ideas, we introduce two new specific diffusions as well as learn a diffusion process on the MNIST, CIFAR10, and ImageNet32 datasets. We show learned MDMs match or surpass bits-per-dims (BPDs) relative to fixed choices of diffusions for a given dataset and model architecture.

In this paper we address the solution of the popular Wordle puzzle, using new reinforcement learning methods, which apply more generally to adaptive control of dynamic systems and to classes of Partially Observable Markov Decision Process (POMDP) problems. These methods are based on approximation in value space and the rollout approach, admit a straightforward implementation, and provide improved performance over various heuristic approaches. For the Wordle puzzle, they yield on-line solution strategies that are very close to optimal at relatively modest computational cost. Our methods are viable for more complex versions of Wordle and related search problems, for which an optimal strategy would be impossible to compute. They are also applicable to a wide range of adaptive sequential decision problems that involve an unknown or frequently changing environment whose parameters are estimated on-line.

The problem of designing NLP solvers for math word problems (MWP) has seen sustained research activity and steady gains in the test accuracy. Since existing solvers achieve high performance on the benchmark datasets for elementary level MWPs containing one-unknown arithmetic word problems, such problems are often considered "solved" with the bulk of research attention moving to more complex MWPs. In this paper, we restrict our attention to English MWPs taught in grades four and lower. We provide strong evidence that the existing MWP solvers rely on shallow heuristics to achieve high performance on the benchmark datasets. To this end, we show that MWP solvers that do not have access to the question asked in the MWP can still solve a large fraction of MWPs. Similarly, models that treat MWPs as bag-of-words can also achieve surprisingly high accuracy. Further, we introduce a challenge dataset, SVAMP, created by applying carefully chosen variations over examples sampled from existing datasets. The best accuracy achieved by state-of-the-art models is substantially lower on SVAMP, thus showing that much remains to be done even for the simplest of the MWPs.

We address the problem of finite-horizon control of a discrete-time linear system, where the initial state distribution follows a Gaussian mixture model, the terminal state must follow a specified Gaussian distribution, and the state and control inputs must obey chance constraints. We show that, throughout the time horizon, the state and control distributions are fully characterized by Gaussian mixtures. We then formulate the cost, distributional terminal constraint, and affine/2-norm chance constraints on the state and control, as convex functions of the decision variables. This is leveraged to formulate the chance-constrained path planning problem as a single convex optimization problem. A numerical example demonstrates the effectiveness of the proposed method.

In this paper, we propose a policy gradient method for confounded partially observable Markov decision processes (POMDPs) with continuous state and observation spaces in the offline setting. We first establish a novel identification result to non-parametrically estimate any history-dependent policy gradient under POMDPs using the offline data. The identification enables us to solve a sequence of conditional moment restrictions and adopt the min-max learning procedure with general function approximation for estimating the policy gradient. We then provide a finite-sample non-asymptotic bound for estimating the gradient uniformly over a pre-specified policy class in terms of the sample size, length of horizon, concentratability coefficient and the measure of ill-posedness in solving the conditional moment restrictions. Lastly, by deploying the proposed gradient estimation in the gradient ascent algorithm, we show the global convergence of the proposed algorithm in finding the history-dependent optimal policy under some technical conditions. To the best of our knowledge, this is the first work studying the policy gradient method for POMDPs under the offline setting.

In this paper, we consider the minimum submodular cost allocation (MSCA) problem. The input of MSCA is k non-negative submodular functions f_1,ldots,f_k on the ground set N given by evaluation oracles, and the goal is to partition N into k (possibly empty) sets X_1,ldots,X_k so that sum_{i=1}^k f_i(X_i) is minimized. In this paper, we focus on the case when f_1,ldots,f_k are monotone (denoted by Mono-MSCA). We provide a natural LP-relaxation for Mono-MSCA, which is equivalent to the convex program relaxation introduced by Chekuri and Ene. We show that the integrality gap of the LP-relaxation is at most k/2, which yields a k/2-approximation algorithm for Mono-MSCA. We also show that the integrality gap of the LP-relaxation is at least k/2-epsilon for any constant epsilon>0 when k is fixed.

Two main challenges in Reinforcement Learning (RL) are designing appropriate reward functions and ensuring the safety of the learned policy. To address these challenges, we present a theoretical framework for Inverse Reinforcement Learning (IRL) in constrained Markov decision processes. From a convex-analytic perspective, we extend prior results on reward identifiability and generalizability to both the constrained setting and a more general class of regularizations. In particular, we show that identifiability up to potential shaping (Cao et al., 2021) is a consequence of entropy regularization and may generally no longer hold for other regularizations or in the presence of safety constraints. We also show that to ensure generalizability to new transition laws and constraints, the true reward must be identified up to a constant. Additionally, we derive a finite sample guarantee for the suboptimality of the learned rewards, and validate our results in a gridworld environment.

Policy optimization methods are popular reinforcement learning algorithms in practice. Recent works have built theoretical foundation for them by proving T regret bounds even when the losses are adversarial. Such bounds are tight in the worst case but often overly pessimistic. In this work, we show that in tabular Markov decision processes (MDPs), by properly designing the regularizer, the exploration bonus and the learning rates, one can achieve a more favorable polylog(T) regret when the losses are stochastic, without sacrificing the worst-case guarantee in the adversarial regime. To our knowledge, this is also the first time a gap-dependent polylog(T) regret bound is shown for policy optimization. Specifically, we achieve this by leveraging a Tsallis entropy or a Shannon entropy regularizer in the policy update. Then we show that under known transitions, we can further obtain a first-order regret bound in the adversarial regime by leveraging the log-barrier regularizer.

Meta-reinforcement learning (meta-RL) is a promising framework for tackling challenging domains requiring efficient exploration. Existing meta-RL algorithms are characterized by low sample efficiency, and mostly focus on low-dimensional task distributions. In parallel, model-based RL methods have been successful in solving partially observable MDPs, of which meta-RL is a special case. In this work, we leverage this success and propose a new model-based approach to meta-RL, based on elements from existing state-of-the-art model-based and meta-RL methods. We demonstrate the effectiveness of our approach on common meta-RL benchmark domains, attaining greater return with better sample efficiency (up to 15times) while requiring very little hyperparameter tuning. In addition, we validate our approach on a slate of more challenging, higher-dimensional domains, taking a step towards real-world generalizing agents.

Off-policy evaluation (OPE) is a method for estimating the return of a target policy using some pre-collected observational data generated by a potentially different behavior policy. In some cases, there may be unmeasured variables that can confound the action-reward or action-next-state relationships, rendering many existing OPE approaches ineffective. This paper develops an instrumental variable (IV)-based method for consistent OPE in confounded Markov decision processes (MDPs). Similar to single-stage decision making, we show that IV enables us to correctly identify the target policy's value in infinite horizon settings as well. Furthermore, we propose an efficient and robust value estimator and illustrate its effectiveness through extensive simulations and analysis of real data from a world-leading short-video platform.

Mixed integer linear programs are commonly solved by Branch and Bound algorithms. A key factor of the efficiency of the most successful commercial solvers is their fine-tuned heuristics. In this paper, we leverage patterns in real-world instances to learn from scratch a new branching strategy optimised for a given problem and compare it with a commercial solver. We propose FMSTS, a novel Reinforcement Learning approach specifically designed for this task. The strength of our method lies in the consistency between a local value function and a global metric of interest. In addition, we provide insights for adapting known RL techniques to the Branch and Bound setting, and present a new neural network architecture inspired from the literature. To our knowledge, it is the first time Reinforcement Learning has been used to fully optimise the branching strategy. Computational experiments show that our method is appropriate and able to generalise well to new instances.

Partially Observable Markov Decision Processes (POMDPs) are used to model environments where the full state cannot be perceived by an agent. As such the agent needs to reason taking into account the past observations and actions. However, simply remembering the full history is generally intractable due to the exponential growth in the history space. Maintaining a probability distribution that models the belief over what the true state is can be used as a sufficient statistic of the history, but its computation requires access to the model of the environment and is often intractable. While SOTA algorithms use Recurrent Neural Networks to compress the observation-action history aiming to learn a sufficient statistic, they lack guarantees of success and can lead to sub-optimal policies. To overcome this, we propose the Wasserstein Belief Updater, an RL algorithm that learns a latent model of the POMDP and an approximation of the belief update. Our approach comes with theoretical guarantees on the quality of our approximation ensuring that our outputted beliefs allow for learning the optimal value function.

We study offline multi-agent reinforcement learning (RL) in Markov games, where the goal is to learn an approximate equilibrium -- such as Nash equilibrium and (Coarse) Correlated Equilibrium -- from an offline dataset pre-collected from the game. Existing works consider relatively restricted tabular or linear models and handle each equilibria separately. In this work, we provide the first framework for sample-efficient offline learning in Markov games under general function approximation, handling all 3 equilibria in a unified manner. By using Bellman-consistent pessimism, we obtain interval estimation for policies' returns, and use both the upper and the lower bounds to obtain a relaxation on the gap of a candidate policy, which becomes our optimization objective. Our results generalize prior works and provide several additional insights. Importantly, we require a data coverage condition that improves over the recently proposed "unilateral concentrability". Our condition allows selective coverage of deviation policies that optimally trade-off between their greediness (as approximate best responses) and coverage, and we show scenarios where this leads to significantly better guarantees. As a new connection, we also show how our algorithmic framework can subsume seemingly different solution concepts designed for the special case of two-player zero-sum games.

Research on applications of Reinforcement Learning (RL) to Large Language Models (LLMs) has mostly been focused on single-turn problems, such as mathematical reasoning or single-shot code generation. While these problems can be viewed as token-level multi-turn MDPs, this view corresponds to a degenerate case of multi-turn interaction where the environment provides no feedback. This contrasts with many real-world domains, such as software engineering (SWE), which require rich multi-turn interactions with a stateful environment that responds to each action with a non-trivial observation. To bridge this gap, we demonstrate the successful application of RL to this general regime. Using a modified Decoupled Advantage Policy Optimization (DAPO) algorithm, we train an agent based on Qwen2.5-72B-Instruct to solve real-world software engineering tasks. Our approach increases the agent's success rate on the SWE-bench Verified benchmark from a 20% rejection fine-tuned baseline to 39%, without relying on any teacher models. On SWE-rebench, our agent matches or outperforms leading open-weight models such as DeepSeek-V3-0324 and Qwen3-235B-A22B using an identical scaffolding, offering a viable path toward building more capable autonomous agents for complex real-world problems based on open models.

The Multi-Agent Pathfinding (MAPF) problem involves finding a set of conflict-free paths for a group of agents confined to a graph. In typical MAPF scenarios, the graph and the agents' starting and ending vertices are known beforehand, allowing the use of centralized planning algorithms. However, in this study, we focus on the decentralized MAPF setting, where the agents may observe the other agents only locally and are restricted in communications with each other. Specifically, we investigate the lifelong variant of MAPF, where new goals are continually assigned to the agents upon completion of previous ones. Drawing inspiration from the successful AlphaZero approach, we propose a decentralized multi-agent Monte Carlo Tree Search (MCTS) method for MAPF tasks. Our approach utilizes the agent's observations to recreate the intrinsic Markov decision process, which is then used for planning with a tailored for multi-agent tasks version of neural MCTS. The experimental results show that our approach outperforms state-of-the-art learnable MAPF solvers. The source code is available at https://github.com/AIRI-Institute/mats-lp.

We address the problem of code generation from multi-turn execution feedback. Existing methods either generate code without feedback or use complex, hierarchical reinforcement learning to optimize multi-turn rewards. We propose a simple yet scalable approach, muCode, that solves multi-turn code generation using only single-step rewards. Our key insight is that code generation is a one-step recoverable MDP, where the correct code can be recovered from any intermediate code state in a single turn. muCode iteratively trains both a generator to provide code solutions conditioned on multi-turn execution feedback and a verifier to score the newly generated code. Experimental evaluations show that our approach achieves significant improvements over the state-of-the-art baselines. We provide analysis of the design choices of the reward models and policy, and show the efficacy of muCode at utilizing the execution feedback. Our code is available at https://github.com/portal-cornell/muCode.

Mirror descent (MD), a well-known first-order method in constrained convex optimization, has recently been shown as an important tool to analyze trust-region algorithms in reinforcement learning (RL). However, there remains a considerable gap between such theoretically analyzed algorithms and the ones used in practice. Inspired by this, we propose an efficient RL algorithm, called {\em mirror descent policy optimization} (MDPO). MDPO iteratively updates the policy by {\em approximately} solving a trust-region problem, whose objective function consists of two terms: a linearization of the standard RL objective and a proximity term that restricts two consecutive policies to be close to each other. Each update performs this approximation by taking multiple gradient steps on this objective function. We derive {\em on-policy} and {\em off-policy} variants of MDPO, while emphasizing important design choices motivated by the existing theory of MD in RL. We highlight the connections between on-policy MDPO and two popular trust-region RL algorithms: TRPO and PPO, and show that explicitly enforcing the trust-region constraint is in fact {\em not} a necessity for high performance gains in TRPO. We then show how the popular soft actor-critic (SAC) algorithm can be derived by slight modifications of off-policy MDPO. Overall, MDPO is derived from the MD principles, offers a unified approach to viewing a number of popular RL algorithms, and performs better than or on-par with TRPO, PPO, and SAC in a number of continuous control tasks. Code is available at https://github.com/manantomar/Mirror-Descent-Policy-Optimization.

We propose Monte Carlo Permutation Search (MCPS), a general-purpose Monte Carlo Tree Search (MCTS) algorithm that improves upon the GRAVE algorithm. MCPS is relevant when deep reinforcement learning is not an option, or when the computing power available before play is not substantial, such as in General Game Playing, for example. The principle of MCPS is to include in the exploration term of a node the statistics on all the playouts that contain all the moves on the path from the root to the node. We extensively test MCPS on a variety of games: board games, wargame, investment game, video game and multi-player games. MCPS has better results than GRAVE in all the two-player games. It has equivalent results for multi-player games because these games are inherently balanced even when players have different strengths. We also show that using abstract codes for moves instead of exact codes can be beneficial to both MCPS and GRAVE, as they improve the permutation statistics and the AMAF statistics. We also provide a mathematical derivation of the formulas used for weighting the three sources of statistics. These formulas are an improvement on the GRAVE formula since they no longer use the bias hyperparameter of GRAVE. Moreover, MCPS is not sensitive to the ref hyperparameter.

Offline reinforcement learning (RL) aims to learn optimal policies from offline datasets, where the parameterization of policies is crucial but often overlooked. Recently, Diffsuion-QL significantly boosts the performance of offline RL by representing a policy with a diffusion model, whose success relies on a parametrized Markov Chain with hundreds of steps for sampling. However, Diffusion-QL suffers from two critical limitations. 1) It is computationally inefficient to forward and backward through the whole Markov chain during training. 2) It is incompatible with maximum likelihood-based RL algorithms (e.g., policy gradient methods) as the likelihood of diffusion models is intractable. Therefore, we propose efficient diffusion policy (EDP) to overcome these two challenges. EDP approximately constructs actions from corrupted ones at training to avoid running the sampling chain. We conduct extensive experiments on the D4RL benchmark. The results show that EDP can reduce the diffusion policy training time from 5 days to 5 hours on gym-locomotion tasks. Moreover, we show that EDP is compatible with various offline RL algorithms (TD3, CRR, and IQL) and achieves new state-of-the-art on D4RL by large margins over previous methods. Our code is available at https://github.com/sail-sg/edp.

We study the problem of planning under model uncertainty in an online meta-reinforcement learning (RL) setting where an agent is presented with a sequence of related tasks with limited interactions per task. The agent can use its experience in each task and across tasks to estimate both the transition model and the distribution over tasks. We propose an algorithm to meta-learn the underlying structure across tasks, utilize it to plan in each task, and upper-bound the regret of the planning loss. Our bound suggests that the average regret over tasks decreases as the number of tasks increases and as the tasks are more similar. In the classical single-task setting, it is known that the planning horizon should depend on the estimated model's accuracy, that is, on the number of samples within task. We generalize this finding to meta-RL and study this dependence of planning horizons on the number of tasks. Based on our theoretical findings, we derive heuristics for selecting slowly increasing discount factors, and we validate its significance empirically.

We incorporate discrete and continuous time Markov processes as building blocks into probabilistic graphical models with latent and observed variables. We introduce the automatic Backward Filtering Forward Guiding (BFFG) paradigm (Mider et al., 2021) for programmable inference on latent states and model parameters. Our starting point is a generative model, a forward description of the probabilistic process dynamics. We backpropagate the information provided by observations through the model to transform the generative (forward) model into a pre-conditional model guided by the data. It approximates the actual conditional model with known likelihood-ratio between the two. The backward filter and the forward change of measure are suitable to be incorporated into a probabilistic programming context because they can be formulated as a set of transformation rules. The guided generative model can be incorporated in different approaches to efficiently sample latent states and parameters conditional on observations. We show applicability in a variety of settings, including Markov chains with discrete state space, interacting particle systems, state space models, branching diffusions and Gamma processes.

In this study, we introduce an innovative deep learning framework that employs a transformer model to address the challenges of mixed-integer programs, specifically focusing on the Capacitated Lot Sizing Problem (CLSP). Our approach, to our knowledge, is the first to utilize transformers to predict the binary variables of a mixed-integer programming (MIP) problem. Specifically, our approach harnesses the encoder decoder transformer's ability to process sequential data, making it well-suited for predicting binary variables indicating production setup decisions in each period of the CLSP. This problem is inherently dynamic, and we need to handle sequential decision making under constraints. We present an efficient algorithm in which CLSP solutions are learned through a transformer neural network. The proposed post-processed transformer algorithm surpasses the state-of-the-art solver, CPLEX and Long Short-Term Memory (LSTM) in solution time, optimal gap, and percent infeasibility over 240K benchmark CLSP instances tested. After the ML model is trained, conducting inference on the model, reduces the MIP into a linear program (LP). This transforms the ML-based algorithm, combined with an LP solver, into a polynomial-time approximation algorithm to solve a well-known NP-Hard problem, with almost perfect solution quality.

Goal-conditioned Hierarchical Reinforcement Learning (HRL) is a promising approach for scaling up reinforcement learning (RL) techniques. However, it often suffers from training inefficiency as the action space of the high-level, i.e., the goal space, is large. Searching in a large goal space poses difficulty for both high-level subgoal generation and low-level policy learning. In this paper, we show that this problem can be effectively alleviated by restricting the high-level action space from the whole goal space to a k-step adjacent region of the current state using an adjacency constraint. We theoretically prove that in a deterministic Markov Decision Process (MDP), the proposed adjacency constraint preserves the optimal hierarchical policy, while in a stochastic MDP the adjacency constraint induces a bounded state-value suboptimality determined by the MDP's transition structure. We further show that this constraint can be practically implemented by training an adjacency network that can discriminate between adjacent and non-adjacent subgoals. Experimental results on discrete and continuous control tasks including challenging simulated robot locomotion and manipulation tasks show that incorporating the adjacency constraint significantly boosts the performance of state-of-the-art goal-conditioned HRL approaches.

Masked diffusion models (MDMs) have emerged as a popular research topic for generative modeling of discrete data, thanks to their superior performance over other discrete diffusion models, and are rivaling the auto-regressive models (ARMs) for language modeling tasks. The recent effort in simplifying the masked diffusion framework further leads to alignment with continuous-space diffusion models and more principled training and sampling recipes. In this paper, however, we reveal that both training and sampling of MDMs are theoretically free from the time variable, arguably the key signature of diffusion models, and are instead equivalent to masked models. The connection on the sampling aspect is drawn by our proposed first-hitting sampler (FHS). Specifically, we show that the FHS is theoretically equivalent to MDMs' original generation process while significantly alleviating the time-consuming categorical sampling and achieving a 20times speedup. In addition, our investigation raises doubts about whether MDMs can truly beat ARMs. We identify, for the first time, an underlying numerical issue, even with the commonly used 32-bit floating-point precision, which results in inaccurate categorical sampling. We show that the numerical issue lowers the effective temperature both theoretically and empirically, and the resulting decrease in token diversity makes previous evaluations, which assess the generation quality solely through the incomplete generative perplexity metric, somewhat unfair.

Having explored an environment, intelligent agents should be able to transfer their knowledge to most downstream tasks within that environment. Referred to as "zero-shot learning," this ability remains elusive for general-purpose reinforcement learning algorithms. While recent works have attempted to produce zero-shot RL agents, they make assumptions about the nature of the tasks or the structure of the MDP. We present Proto Successor Measure: the basis set for all possible solutions of Reinforcement Learning in a dynamical system. We provably show that any possible policy can be represented using an affine combination of these policy independent basis functions. Given a reward function at test time, we simply need to find the right set of linear weights to combine these basis corresponding to the optimal policy. We derive a practical algorithm to learn these basis functions using only interaction data from the environment and show that our approach can produce the optimal policy at test time for any given reward function without additional environmental interactions. Project page: https://agarwalsiddhant10.github.io/projects/psm.html.

We study the representation complexity of model-based and model-free reinforcement learning (RL) in the context of circuit complexity. We prove theoretically that there exists a broad class of MDPs such that their underlying transition and reward functions can be represented by constant depth circuits with polynomial size, while the optimal Q-function suffers an exponential circuit complexity in constant-depth circuits. By drawing attention to the approximation errors and building connections to complexity theory, our theory provides unique insights into why model-based algorithms usually enjoy better sample complexity than model-free algorithms from a novel representation complexity perspective: in some cases, the ground-truth rule (model) of the environment is simple to represent, while other quantities, such as Q-function, appear complex. We empirically corroborate our theory by comparing the approximation error of the transition kernel, reward function, and optimal Q-function in various Mujoco environments, which demonstrates that the approximation errors of the transition kernel and reward function are consistently lower than those of the optimal Q-function. To the best of our knowledge, this work is the first to study the circuit complexity of RL, which also provides a rigorous framework for future research.

Policy optimization methods with function approximation are widely used in multi-agent reinforcement learning. However, it remains elusive how to design such algorithms with statistical guarantees. Leveraging a multi-agent performance difference lemma that characterizes the landscape of multi-agent policy optimization, we find that the localized action value function serves as an ideal descent direction for each local policy. Motivated by the observation, we present a multi-agent PPO algorithm in which the local policy of each agent is updated similarly to vanilla PPO. We prove that with standard regularity conditions on the Markov game and problem-dependent quantities, our algorithm converges to the globally optimal policy at a sublinear rate. We extend our algorithm to the off-policy setting and introduce pessimism to policy evaluation, which aligns with experiments. To our knowledge, this is the first provably convergent multi-agent PPO algorithm in cooperative Markov games.

We propose algorithms for approximate filtering and smoothing in high-dimensional Factorial hidden Markov models. The approximation involves discarding, in a principled way, likelihood factors according to a notion of locality in a factor graph associated with the emission distribution. This allows the exponential-in-dimension cost of exact filtering and smoothing to be avoided. We prove that the approximation accuracy, measured in a local total variation norm, is "dimension-free" in the sense that as the overall dimension of the model increases the error bounds we derive do not necessarily degrade. A key step in the analysis is to quantify the error introduced by localizing the likelihood function in a Bayes' rule update. The factorial structure of the likelihood function which we exploit arises naturally when data have known spatial or network structure. We demonstrate the new algorithms on synthetic examples and a London Underground passenger flow problem, where the factor graph is effectively given by the train network.

We propose Hydra-MDP, a novel paradigm employing multiple teachers in a teacher-student model. This approach uses knowledge distillation from both human and rule-based teachers to train the student model, which features a multi-head decoder to learn diverse trajectory candidates tailored to various evaluation metrics. With the knowledge of rule-based teachers, Hydra-MDP learns how the environment influences the planning in an end-to-end manner instead of resorting to non-differentiable post-processing. This method achieves the 1^{st} place in the Navsim challenge, demonstrating significant improvements in generalization across diverse driving environments and conditions. More details by visiting https://github.com/NVlabs/Hydra-MDP.

Inspired by human conscious planning, we propose Skipper, a model-based reinforcement learning framework utilizing spatio-temporal abstractions to generalize better in novel situations. It automatically decomposes the given task into smaller, more manageable subtasks, and thus enables sparse decision-making and focused computation on the relevant parts of the environment. The decomposition relies on the extraction of an abstracted proxy problem represented as a directed graph, in which vertices and edges are learned end-to-end from hindsight. Our theoretical analyses provide performance guarantees under appropriate assumptions and establish where our approach is expected to be helpful. Generalization-focused experiments validate Skipper's significant advantage in zero-shot generalization, compared to some existing state-of-the-art hierarchical planning methods.

In an effort to better understand the different ways in which the discount factor affects the optimization process in reinforcement learning, we designed a set of experiments to study each effect in isolation. Our analysis reveals that the common perception that poor performance of low discount factors is caused by (too) small action-gaps requires revision. We propose an alternative hypothesis that identifies the size-difference of the action-gap across the state-space as the primary cause. We then introduce a new method that enables more homogeneous action-gaps by mapping value estimates to a logarithmic space. We prove convergence for this method under standard assumptions and demonstrate empirically that it indeed enables lower discount factors for approximate reinforcement-learning methods. This in turn allows tackling a class of reinforcement-learning problems that are challenging to solve with traditional methods.

Reinforcement Learning From Human Feedback (RLHF) has been a critical to the success of the latest generation of generative AI models. In response to the complex nature of the classical RLHF pipeline, direct alignment algorithms such as Direct Preference Optimization (DPO) have emerged as an alternative approach. Although DPO solves the same objective as the standard RLHF setup, there is a mismatch between the two approaches. Standard RLHF deploys reinforcement learning in a specific token-level MDP, while DPO is derived as a bandit problem in which the whole response of the model is treated as a single arm. In this work we rectify this difference, first we theoretically show that we can derive DPO in the token-level MDP as a general inverse Q-learning algorithm, which satisfies the Bellman equation. Using our theoretical results, we provide three concrete empirical insights. First, we show that because of its token level interpretation, DPO is able to perform some type of credit assignment. Next, we prove that under the token level formulation, classical search-based algorithms, such as MCTS, which have recently been applied to the language generation space, are equivalent to likelihood-based search on a DPO policy. Empirically we show that a simple beam search yields meaningful improvement over the base DPO policy. Finally, we show how the choice of reference policy causes implicit rewards to decline during training. We conclude by discussing applications of our work, including information elicitation in multi-tun dialogue, reasoning, agentic applications and end-to-end training of multi-model systems.

When trying to solve a computational problem, we are often faced with a choice between algorithms that are guaranteed to return the right answer but differ in their runtime distributions (e.g., SAT solvers, sorting algorithms). This paper aims to lay theoretical foundations for such choices by formalizing preferences over runtime distributions. It might seem that we should simply prefer the algorithm that minimizes expected runtime. However, such preferences would be driven by exactly how slow our algorithm is on bad inputs, whereas in practice we are typically willing to cut off occasional, sufficiently long runs before they finish. We propose a principled alternative, taking a utility-theoretic approach to characterize the scoring functions that describe preferences over algorithms. These functions depend on the way our value for solving our problem decreases with time and on the distribution from which captimes are drawn. We describe examples of realistic utility functions and show how to leverage a maximum-entropy approach for modeling underspecified captime distributions. Finally, we show how to efficiently estimate an algorithm's expected utility from runtime samples.

While originally developed for continuous control problems, Proximal Policy Optimization (PPO) has emerged as the work-horse of a variety of reinforcement learning (RL) applications including the fine-tuning of generative models. Unfortunately, PPO requires multiple heuristics to enable stable convergence (e.g. value networks, clipping) and is notorious for its sensitivity to the precise implementation of these components. In response, we take a step back and ask what a minimalist RL algorithm for the era of generative models would look like. We propose REBEL, an algorithm that cleanly reduces the problem of policy optimization to regressing the relative rewards via a direct policy parameterization between two completions to a prompt, enabling strikingly lightweight implementation. In theory, we prove that fundamental RL algorithms like Natural Policy Gradient can be seen as variants of REBEL, which allows us to match the strongest known theoretical guarantees in terms of convergence and sample complexity in the RL literature. REBEL can also cleanly incorporate offline data and handle the intransitive preferences we frequently see in practice. Empirically, we find that REBEL provides a unified approach to language modeling and image generation with stronger or similar performance as PPO and DPO, all while being simpler to implement and more computationally tractable than PPO.

Reinforcement Learning (RL) plays a crucial role in aligning large language models (LLMs) with human preferences and improving their ability to perform complex tasks. However, current approaches either require significant computational resources due to the use of multiple models and extensive online sampling for training (e.g., PPO) or are framed as bandit problems (e.g., DPO, DRO), which often struggle with multi-step reasoning tasks, such as math problem-solving and complex reasoning that involve long chains of thought. To overcome these limitations, we introduce Direct Q-function Optimization (DQO), which formulates the response generation process as a Markov Decision Process (MDP) and utilizes the soft actor-critic (SAC) framework to optimize a Q-function directly parameterized by the language model. The MDP formulation of DQO offers structural advantages over bandit-based methods, enabling more effective process supervision. Experimental results on two math problem-solving datasets, GSM8K and MATH, demonstrate that DQO outperforms previous methods, establishing it as a promising offline reinforcement learning approach for aligning language models.

We introduce distributional dynamic programming (DP) methods for optimizing statistical functionals of the return distribution, with standard reinforcement learning as a special case. Previous distributional DP methods could optimize the same class of expected utilities as classic DP. To go beyond expected utilities, we combine distributional DP with stock augmentation, a technique previously introduced for classic DP in the context of risk-sensitive RL, where the MDP state is augmented with a statistic of the rewards obtained so far (since the first time step). We find that a number of recently studied problems can be formulated as stock-augmented return distribution optimization, and we show that we can use distributional DP to solve them. We analyze distributional value and policy iteration, with bounds and a study of what objectives these distributional DP methods can or cannot optimize. We describe a number of applications outlining how to use distributional DP to solve different stock-augmented return distribution optimization problems, for example maximizing conditional value-at-risk, and homeostatic regulation. To highlight the practical potential of stock-augmented return distribution optimization and distributional DP, we combine the core ideas of distributional value iteration with the deep RL agent DQN, and empirically evaluate it for solving instances of the applications discussed.

Constraint Programming (CP) is a declarative programming paradigm that allows for modeling and solving combinatorial optimization problems, such as the Job-Shop Scheduling Problem (JSSP). While CP solvers manage to find optimal or near-optimal solutions for small instances, they do not scale well to large ones, i.e., they require long computation times or yield low-quality solutions. Therefore, real-world scheduling applications often resort to fast, handcrafted, priority-based dispatching heuristics to find a good initial solution and then refine it using optimization methods. This paper proposes a novel end-to-end approach to solving scheduling problems by means of CP and Reinforcement Learning (RL). In contrast to previous RL methods, tailored for a given problem by including procedural simulation algorithms, complex feature engineering, or handcrafted reward functions, our neural-network architecture and training algorithm merely require a generic CP encoding of some scheduling problem along with a set of small instances. Our approach leverages existing CP solvers to train an agent learning a Priority Dispatching Rule (PDR) that generalizes well to large instances, even from separate datasets. We evaluate our method on seven JSSP datasets from the literature, showing its ability to find higher-quality solutions for very large instances than obtained by static PDRs and by a CP solver within the same time limit.