We study sample-efficient reinforcement learning (RL) under the general framework of interactive decision making, which includes the Markov decision process, partially observable Markov decision process, and predictive state representation (PSR) as special cases. We propose a novel complexity measure, the generalized eluder coefficient (GEC), which characterizes the fundamental trade-off between exploration and exploitation in online interactive decision making in the context of function approximation. We show that RL problems with low GECs form a remarkably rich class, which subsumes low Bellman eluder dimension problems, bilinear class, low witness rank problems, Partially observable bilinear (PO-bilinear) class, and generalized regular PSR, where generalized regular PSR, a new tractable PSR class identified by us, includes nearly all known tractable partially observable RL models. Furthermore, in terms of algorithm design, we propose a generic posterior sampling algorithm, which can be implemented in both model-free and model-based fashions, under both fully observable and partially observable settings. We prove that the proposed generic posterior sampling algorithm is sample efficient by establishing a sublinear regret upper bound in terms of the GEC. In summary, we provide a new and unified understanding of both fully observable and partially observable RL.

Muon updates matrix parameters via the matrix sign of the gradient and has shown strong empirical gains, yet its dynamics and scaling behavior remain unclear in theory. We study Muon in a linear associative memory model with softmax retrieval and a hierarchical frequency spectrum over query–answer pairs, with and without label noise. In this setting, we show that Gradient Descent (GD) learns frequency components at highly imbalanced rates, leading to slow convergence bottlenecked by low-frequency components. In contrast, the Muon optimizer mitigates this imbalance, leading to faster and more uniform progress. Specifically, in the noiseless case, Muon achieves an exponential speedup over GD; in the noisy case with a power-decay frequency spectrum, we derive Muon's optimization scaling law and demonstrate its superior scaling efficiency over GD. Furthermore, we show that Muon can be interpreted as an implicit matrix preconditioner arising from adaptive task alignment and block-symmetric gradient structure. In contrast, the preconditioner with coordinate-wise sign operator could match Muon under oracle access to unknown task representations, which is infeasible for SignGD in practice. Experiments on synthetic long-tail classification and LLaMA-style pre-training corroborate the theory.

Assortment optimization is a fundamental challenge in modern retail and recommendation systems, where the goal is to select a subset of products that maximizes expected revenue under complex customer choice behaviors. While recent advances in data-driven methods have leveraged historical data to learn and optimize assortments, these approaches typically rely on strong assumptions – namely, the stability of customer preferences and the correctness of the underlying choice models. However, such assumptions frequently break in real-world scenarios due to preference shifts and model misspecification, leading to poor generalization and revenue loss. Motivated by this limitation, we propose a robust framework for data-driven assortment optimization that accounts for potential distributional shifts in customer choice behavior. Our approach models potential preference shift from a nominal choice model that generates data and seeks to maximize worst-case expected revenue. We first establish the computational tractability of robust assortment planning when the nominal model is known, then advance to the data-driven setting, where we design statistically optimal algorithms that minimize the data requirements while maintaining robustness. Our theoretical analysis provides both upper bounds and matching lower bounds on the sample complexity, offering theoretical guarantees for robust generalization. Notably, we uncover and identify the notion of “robust item-wise coverage” as the minimal data requirement to enable sample-efficient robust assortment learning. Our work bridges the gap between robustness and statistical efficiency in assortment learning, contributing new insights and tools for reliable assortment optimization under uncertainty.

Credit assignment is a central challenge in reinforcement learning (RL). Classical actor-critic methods address this challenge through fine-grained advantage estimation based on a learned value function. However, learned value models are often avoided in modern large language model (LLM) RL because conventional discriminative critics are difficult to train reliably. We revisit value modeling and argue that this difficulty is partly due to limited expressiveness. In particular, representation complexity theory suggests that value functions can be hard to approximate under the one-shot prediction paradigm used by existing value models, and our scaling experiments show that such critics do not improve reliably with scale. Motivated by this observation, we propose Generative Actor-Critic (GenAC), which replaces one-shot scalar value prediction with a generative critic that performs chain-of-thought reasoning before producing a value estimate. We further introduce In-Context Conditioning, which helps the critic remain calibrated to the current actor throughout training. GenAC improves value approximation, ranking reliability, and out-of-distribution generalization, and these gains translate into stronger downstream RL performance than both value-based and value-free baselines. Overall, our results suggest that stronger value modeling is a promising direction for improving credit assignment in LLM reinforcement learning.

Multi-armed bandit algorithms are evaluated by regret, yet comparable regret can coexist with different allocations across independent runs. We study the trade-off between worst-case regret ℛ_K,T and instability 𝒮_K,T, defined as the largest standard deviation of a terminal pull count, for K arms and T rounds. We prove the finite-time lower bound ℛ_K,T𝒮_K,T≥ C T^3/2, where C is independent of K and T, under a finite-time regret condition and without the regularity assumptions imposed in the prior asymptotic analysis. We also introduce Stabilized Lower-Envelope UCB (), a new tunable algorithm combining a running lower-envelope index with a decreasing pull-count stabilizer. satisfies ℛ_K,T𝒮_K,T=O(T^3/2log K), with an implicit constant independent of K and T, matching the lower bound exactly in T and within a logarithmic factor in K. To prove the instability bound, we develop a new offline top-prefix representation that removes path dependence from online decisions. Together with single-reward perturbations and the Efron–Stein inequality, this representation controls pull-count variance. Thus, regret and instability depend reciprocally on K, while their product has no polynomial dependence on K. These results resolve the open question raised in the literature concerning the sharp arm-dependent regret–instability frontier.

This paper is concerned with one-bit mean estimation, where each independent sample is represented by a single binary message. We consider distributions on ℝ with mean in [-λ,λ] and absolute k-th central moment at most σ^k, where k>1 is fixed. For this class, previous work attained the optimal sample complexity for general queries using a two-stage protocol. The first stage localizes the mean. The second-stage queries are chosen after localization and refine the estimate around the decoded center. We show that this interaction can be avoided by constructing a randomized fully non-adaptive protocol that fixes all queries before observing the data and matches the optimal adaptive sample complexity. For target accuracy ε and confidence 1-δ, its sample complexity scales as log + (σ/ε)^2log(1/δ), k>2, (σ/ε)^2log(σ/ε)log(1/δ), k=2, (σ/ε)^k/(k-1)log(1/δ), 1.

While deep reinforcement learning has achieved tremendous successes in various applications, most existing works only focus on maximizing the expected value of total return and thus ignore its inherent stochasticity. Such stochasticity is also known as the aleatoric uncertainty and is closely related to the notion of risk. In this work, we make the first attempt to study risk-sensitive deep reinforcement learning under the average reward setting with the variance risk criteria. In particular, we focus on a variance-constrained policy optimization problem where the goal is to find a policy that maximizes the expected value of the long-run average reward, subject to a constraint that the long-run variance of the average reward is upper bounded by a threshold. Utilizing Lagrangian and Fenchel dualities, we transform the original problem into an unconstrained saddle-point policy optimization problem, and propose an actor-critic algorithm that iteratively and efficiently updates the policy, the Lagrange multiplier, and the Fenchel dual variable. When both the value and policy functions are represented by multi-layer overparameterized neural networks, we prove that our actor-critic algorithm generates a sequence of policies that finds a globally optimal policy at a sublinear rate. Further, We provide numerical studies of the proposed method using two real datasets to back up the theoretical results.

Thompson sampling (TS) is widely used for stochastic multi-armed bandits, yet its inferential properties under adaptive data collection are subtle. Classical asymptotic theory for sample means can fail because arm-specific sample sizes are random and coupled with the rewards through the action-selection rule. We study this phenomenon in the K-armed Gaussian bandit and identify optimism as a key mechanism for restoring stability, a sufficient condition for valid asymptotic inference requiring each arm's pull count to concentrate around a deterministic scale. First, we prove that variance-inflated TS is stable for any K ≥ 2, including the challenging regime where multiple arms are optimal. This resolves the open question raised by through extending their results from the two-armed setting to the general K-armed setting. Second, we analyze an alternative optimistic modification that keeps the posterior variance unchanged but adds an explicit mean bonus to posterior mean, and establish the same stability conclusion. In summary, suitably implemented optimism stabilizes Thompson sampling and enables asymptotically valid inference in multi-armed bandits, while incurring only a mild additional regret cost.

This paper investigates a pivotal yet debated component of reinforcement learning (RL) for training large language models (LLMs): controlling entropy (increasing or decreasing it) during RL fine-tuning. The existing literature presents a dichotomy: some studies posit that increasing entropy facilitates exploration, whereas others argue that decreasing entropy enhances performance. To reconcile these conflicting observations, we provide a theoretical framework showing that the effect of entropy is governed by \emph{Entropy Discrepancy}, the distributional divergence between positive and negative samples. Guided by this insight, we derive a principled dynamic scheduling method that adaptively modulates the entropy coefficient, effectively switching between entropy maximization and minimization as training evolves. Extensive experiments confirm the correlation between Entropy Discrepancy and the efficacy of entropy control. Furthermore, our adaptive method yields substantial improvements, boosting Pass@K by 6.7\% on AIME24 and 17.52\% on puzzle tasks compared to vanilla RL, while consistently outperforming recent state-of-the-art reasoning methods.

In the classical Reinforcement Learning from Human Feedback (RLHF) framework, Proximal Policy Optimization (PPO) is employed to learn from sparse, sentence-level rewards---a challenging scenario in traditional deep reinforcement learning. Despite the great successes of PPO in the alignment of state-of-the-art closed-source large language models (LLMs), its open-source implementation is still largely sub-optimal, as widely reported by numerous research studies. To address these issues, we introduce a framework that models RLHF problems as a Markov decision process (MDP), enabling the capture of fine-grained token-wise information. Furthermore, we provide theoretical insights that demonstrate the superiority of our MDP framework over the previous sentence-level bandit formulation. Under this framework, we introduce an algorithm, dubbed as Reinforced Token Optimization (\texttt{RTO}), which learns the token-wise reward function from preference data and performs policy optimization based on this learned token-wise reward signal. Theoretically, \texttt{RTO} is proven to have the capability of finding the near-optimal policy sample-efficiently. For its practical implementation, \texttt{RTO} innovatively integrates Direct Preference Optimization (DPO) and PPO. DPO, originally derived from sparse sentence rewards, surprisingly provides us with a token-wise characterization of response quality, which is seamlessly incorporated into our subsequent PPO training stage. We conduct extensive experiments to evaluate \texttt{RTO} against PPO and other direct preference learning algorithms. The results highlight the effectiveness of RTO, with the algorithm outperforming PPO by 7.5 points on the AlpacaEval 2 benchmark and by 4.1 points on Arena-Hard. Our code and models are available at \href{https://github.com/zkshan2002/RTO}{https://github.com/zkshan2002/RTO}.

Direct Preference Optimization (DPO) has emerged as a promising approach for aligning large language models with human preferences. While prior work mainly extends DPO from the aspect of the objective function, we instead improve DPO from the largely overlooked but critical aspect of data selection. Specifically, we address the issue of parameter shrinkage caused by noisy data by proposing a novel margin-maximization principle for dataset curation in DPO training. To further mitigate the noise in different reward models, we propose a Bayesian Aggregation approach that unifies multiple margin sources (external and implicit) into a single preference probability. Extensive experiments in diverse settings demonstrate the consistently high data efficiency of our approach. Remarkably, by using just 10\% of the Ultrafeedback dataset, our approach achieves 3\% to 8\% improvements across various Llama, Mistral, and Qwen models on the AlpacaEval2 benchmark. Furthermore, our approach seamlessly extends to iterative DPO, yielding a roughly 3\% improvement with 25\% online data, revealing the high redundancy in this presumed high-quality data construction manner. These results highlight the potential of data selection strategies for advancing preference optimization.

We study the fundamental problem of offline assortment optimization under the Multinomial Logit (MNL) model, where sellers must determine the optimal subset of the products to offer based solely on historical customer choice data. While most existing approaches to learning-based assortment optimization focus on the online learning of the optimal assortment through repeated interactions with customers, such exploration can be costly or even impractical in many real-world settings. In this paper, we consider the offline learning paradigm and investigate the minimal data requirements for efficient offline assortment optimization. To this end, we introduce Pessimistic Rank-Breaking (PRB), an algorithm that combines rank-breaking with pessimistic estimation. We prove that PRB is nearly minimax optimal by establishing the tight suboptimality upper bound and a nearly matching lower bound. This further shows that "optimal item coverage" - where each item in the optimal assortment appears sufficiently often in the historical data - is both sufficient and necessary for efficient offline learning. This significantly relaxes the previous requirement of observing the complete optimal assortment in the data. Our results provide fundamental insights into the data requirements for offline assortment optimization under the MNL model.

Large Language Models (LLMs) have demonstrated remarkable capabilities in complex reasoning tasks, yet generating reliable reasoning processes remains a significant challenge. We present a unified probabilistic framework that formalizes LLM reasoning through a novel graphical model incorporating latent thinking processes and evaluation signals. Within this framework, we introduce the Bootstrapping Reinforced Thinking Process (BRiTE) algorithm, which works in two steps. First, it generates high-quality rationales by approximating the optimal thinking process through reinforcement learning, using a novel reward shaping mechanism. Second, it enhances the base LLM by maximizing the joint probability of rationale generation with respect to the model's parameters. Theoretically, we demonstrate BRiTE's convergence at a rate of 1/T with T representing the number of iterations. Empirical evaluations on math and coding benchmarks demonstrate that our approach consistently improves performance across different base models without requiring human-annotated thinking processes. In addition, BRiTE demonstrates superior performance compared to existing algorithms that bootstrap thinking processes use alternative methods such as rejection sampling, and can even match or exceed the results achieved through supervised fine-tuning with human-annotated data.

Information asymmetry is a pervasive feature of multi-agent systems, especially evident in economics and social sciences. In these settings, agents tailor their actions based on private information to maximize their rewards. These strategic behaviors often introduce complexities due to confounding variables. Simultaneously, knowledge transportability poses another significant challenge, arising from the difficulties of conducting experiments in target environments. It requires transferring knowledge from environments where empirical data is more readily available. Against these backdrops, this paper explores a fundamental question in online learning: Can we employ non-i.i.d. actions to learn about confounders even when requiring knowledge transfer? We present a sample-efficient algorithm designed to accurately identify system dynamics under information asymmetry and to navigate the challenges of knowledge transfer effectively in reinforcement learning, framed within an online strategic interaction model. Our method provably achieves learning of an epsilon-optimal policy with a tight sample complexity of (O) over tilde (1/epsilon(2)).

We systematically investigate quantum algorithms and lower bounds for mean estimation given query access to non-identically distributed samples. On the one hand, we give quantum mean estimators with quadratic quantum speed-up given samples from different bounded or sub-Gaussian random variables. On the other hand, we prove that, in general, it is impossible for any quantum algorithm to achieve quadratic speed-up over the number of classical samples needed to estimate the mean μ, where the samples come from different random variables with mean close to μ. Technically, our quantum algorithms reduce bounded and sub-Gaussian random variables to the Bernoulli case, and use an uncomputation trick to overcome the challenge that direct amplitude estimation does not work with non-identical query access. Our quantum query lower bounds are established by simulating non-identical oracles by parallel oracles, and also by an adversarial method with non-identical oracles. Both results pave the way for proving quantum query lower bounds with non-identical oracles in general, which may be of independent interest.

The sim-to-real gap, which represents the disparity between training and testing environments, poses a significant challenge in reinforcement learning (RL). A promising approach to addressing this challenge is distributionally robust RL, often framed as a robust Markov decision process (RMDP). In this framework, the objective is to find a robust policy that achieves good performance under the worst-case scenario among all environments within a pre-specified uncertainty set centered around the training environment. Unlike previous work, which relies on a generative model or a pre-collected offline dataset enjoying good coverage of the deployment environment, we tackle robust RL via interactive data collection, where the learner interacts with the training environment only and refines the policy through trial and error. In this robust RL paradigm, two main challenges emerge: managing distributional robustness while striking a balance between exploration and exploitation during data collection. Initially, we establish that sample-efficient learning without additional assumptions is unattainable owing to the curse of support shift; i.e., the potential disjointedness of the distributional supports between the training and testing environments. To circumvent such a hardness result, we introduce the vanishing minimal value assumption to RMDPs with a total-variation (TV) distance robust set, postulating that the minimal value of the optimal robust value function is zero. We prove that such an assumption effectively eliminates the support shift issue for RMDPs with a TV distance robust set, and present an algorithm with a provable sample complexity guarantee. Our work makes the initial step to uncovering the inherent difficulty of robust RL via interactive data collection and sufficient conditions for designing a sample-efficient algorithm accompanied by sharp sample complexity analysis.

Reinforcement Learning (RL) encompasses diverse paradigms, including model-based RL, policy-based RL, and value-based RL, each tailored to approximate the model, optimal policy, and optimal value function, respectively. This work investigates the potential hierarchy of representation complexity among these RL paradigms. By utilizing computational complexity measures, including time complexity and circuit complexity, we theoretically unveil a potential representation complexity hierarchy within RL. We find that representing the model emerges as the easiest task, followed by the optimal policy, while representing the optimal value function presents the most intricate challenge. Additionally, we reaffirm this hierarchy from the perspective of the expressiveness of Multi-Layer Perceptrons (MLPs), which align more closely with practical deep RL and contribute to a completely new perspective in theoretical studying representation complexity in RL. Finally, we conduct deep RL experiments to validate our theoretical findings.

Preference optimization, particularly through Reinforcement Learning from Human Feedback (RLHF), has achieved significant success in aligning Large Language Models (LLMs) to adhere to human intentions. Unlike offline alignment with a fixed dataset, online feedback collection from humans or AI on model generations typically leads to more capable reward models and better-aligned LLMs through an iterative process. However, achieving a globally accurate reward model requires systematic exploration to generate diverse responses that span the vast space of natural language. Random sampling from standard reward-maximizing LLMs alone is insufficient to fulfill this requirement. To address this issue, we propose a bilevel objective optimistically biased towards potentially high-reward responses to actively explore out-of-distribution regions. By solving the inner-level problem with the reparameterized reward function, the resulting algorithm, named Self-Exploring Language Models (SELM), eliminates the need for a separate RM and iteratively updates the LLM with a straightforward objective. Compared to Direct Preference Optimization (DPO), the SELM objective reduces indiscriminate favor of unseen extrapolations and enhances exploration efficiency. Our experimental results demonstrate that when fine-tuned on Zephyr-7B-SFT and Llama-3-8B-Instruct models, SELM significantly boosts the performance on instruction-following benchmarks such as MT-Bench and AlpacaEval 2.0, as well as various standard academic benchmarks in different settings. Our code and models are available at https://github.com/shenao-zhang/SELM.

Active Constrained Clustering (ACC) is a widely used semi-supervised clustering framework to improve clustering quality through progressive annotation of informative pairwise constraints. However, the application of existing ACC methods to large datasets with numerous classes incurs high computational or query expenses. In this paper, we conduct a theoretical analysis of the inefficiency of sample-based ACC and the rationale behind cluster-based ACC. Moreover, we provide the theoretical guarantee for cluster fusion under a certain purity constraint and a clustering quality constraint with respect to normalized mutual information (NMI). Drawing on these theoretical insights, we introduce a novel Active Probabilistic Clustering (APC) framework designed to scale effectively with large datasets. Compared to previous methods, APC demonstrates superior performance across eight datasets of varying sizes (ranging from 350 to 100,000 samples) in terms of clustering quality, query cost, and computational expense. Specifically, APC accomplishes satisfactory clustering outcomes (e.g., NMI $>0.95$) using 3,920 queries on a dataset with 100,000 samples, while baseline methods yield inferior clustering results (e.g., NMI $\leq0.85$) with 10,000 queries. Concurrently, APC operates at a speed 100x faster than baseline methods.

Offline reinforcement learning (RL) presents a promising approach for learning reinforced policies from offline datasets without the need for costly or unsafe interactions with the environment. However, datasets collected by humans in real-world environments are often noisy and may even be maliciously corrupted, which can significantly degrade the performance of offline RL. In this work, we first investigate the performance of current offline RL algorithms under comprehensive data corruption, including states, actions, rewards, and dynamics. Our extensive experiments reveal that implicit Q-learning (IQL) demonstrates remarkable resilience to data corruption among various offline RL algorithms. Furthermore, we conduct both empirical and theoretical analyses to understand IQL's robust performance, identifying its supervised policy learning scheme as the key factor. Despite its relative robustness, IQL still suffers from heavy-tail targets of Q functions under dynamics corruption. To tackle this challenge, we draw inspiration from robust statistics to employ the Huber loss to handle the heavy-tailedness and utilize quantile estimators to balance penalization for corrupted data and learning stability. By incorporating these simple yet effective modifications into IQL, we propose a more robust offline RL approach named Robust IQL (RIQL). Extensive experiments demonstrate that RIQL exhibits highly robust performance when subjected to diverse data corruption scenarios.