Large language models have demonstrated impressive performance on challenging mathematical reasoning tasks, which has triggered the discussion of whether the performance is achieved by true reasoning capability or memorization. To investigate this question, prior work has constructed mathematical benchmarks when questions undergo simple perturbations – modifications that still preserve the underlying reasoning patterns of the solutions. However, no work has explored hard perturbations, which fundamentally change the nature of the problem so that the original solution steps do not apply. To bridge the gap, we construct MATH-P-Simple and MATH-P-Hard via simple perturbation and hard perturbation, respectively. Each consists of 279 perturbed math problems derived from level-5 (hardest) problems in the MATH dataset (Hendrycks et al., 2021). We observe significant performance drops on MATH-P-Hard across various models, including o1-mini (-16.49%) and gemini-2.0-flash-thinking (-12.9%). We also raise concerns about a novel form of memorization where models blindly apply learned problem-solving skills without assessing their applicability to modified contexts. This issue is amplified when using original problems for in-context learning. We call for research efforts to address this challenge, which is critical for developing more robust and reliable reasoning models. The project is available at https://math-perturb.github.io/.

The problem of two-player zero-sum Markov games has recently attracted increasing interests in theoretical studies of multi-agent reinforcement learning (RL). In particular, for finite-horizon episodic Markov decision processes (MDPs), it has been shown that model-based algorithms can find an $\epsilon$-optimal Nash Equilibrium (NE) with the sample complexity of $O(H^3SAB/\epsilon^2)$, which is optimal in the dependence of the horizon $H$ and the number of states $S$ (where $A$ and $B$ denote the number of actions of the two players, respectively). However, none of the existing model-free algorithms can achieve such an optimality. In this work, we propose a model-free stage-based algorithm and show that it achieves the same sample complexity as the best model-based algorithm, and hence for the first time demonstrate that model-free algorithms can enjoy the same optimality in the $H$ dependence as model-based algorithms. The main improvement of the dependency on $H$ arises by leveraging the popular variance reduction technique based on the reference-advantage decomposition previously used only for single-agent RL. However, such a technique relies on a critical monotonicity property of the value function, which does not hold in Markov games due to the update of the policy via the coarse correlated equilibrium (CCE) oracle. Thus, to extend such a technique to Markov games, our algorithm features a key novel design of updating the reference value functions as the pair of optimistic and pessimistic value functions whose value difference is the smallest in the history in order to achieve the desired improvement in the sample efficiency.

Function approximation has experienced significant success in the field of reinforcement learning (RL). Despite a handful of progress on developing theory for Nonstationary RL with function approximation under structural assumptions, existing work for nonstationary RL with general function approximation is still limited. In this work, we propose a UCB-type of algorithm LSVI-Nonstationary following the popular least-square-value-iteration (LSVI) framework. LSVI-Nonstationary features the restart mechanism and a new design of bonus term to handle nonstationarity, and performs no worse than the existing confidence-set based algorithm SW-OPEA in [1], which has been shown to outperform the existing algorithms for nonstationary linear and tabular MDPs in the small variation budget setting.

Transformer-based autoregressive sampling has been the major bottleneck for slowing down large language model inferences. One effective way to accelerate inference is Speculative Decoding, which employs a small model to sample a sequence of draft tokens and a large model to validate. Given its empirical effectiveness, the theoretical understanding of Speculative Decoding is falling behind. This paper tackles this gap by conceptualizing the decoding problem via markov chain abstraction and studying the key properties, output quality and inference acceleration, from a theoretical perspective. Our analysis covers the theoretical limits of speculative decoding, batch algorithms, and output quality-inference acceleration tradeoffs. Our results reveal the fundamental connections between different components of LLMs via total variation distances and show how they jointly affect the efficiency of decoding algorithms.

Aligning foundation models is essential for their safe and trustworthy deployment. However, traditional fine-tuning methods are computationally intensive and require updating billions of model parameters. A promising alternative, alignment via decoding, adjusts the response distribution directly without model updates to maximize a target reward $r$, thus providing a lightweight and adaptable framework for alignment. However, principled decoding methods rely on oracle access to an optimal Q-function ($Q^*$), which is often unavailable in practice. Hence, prior SoTA methods either approximate this $Q^*$ using $Q^{\pi_{\text{sft}}}$ (derived from the reference $\texttt{SFT}$ model) or rely on short-term rewards, resulting in sub-optimal decoding performance. In this work, we propose $\texttt{Transfer Q}^*$, which implicitly estimates the optimal value function for a target reward $r$ through a baseline model $\rho_{\texttt{BL}}$ aligned with a baseline reward $r_{\texttt{BL}}$ (which can be different from the target reward $r$). Theoretical analyses of $\texttt{Transfer Q}^*$ provide a rigorous characterization of its optimality, deriving an upper bound on the sub-optimality gap and identifying a hyperparameter to control the deviation from the pre-trained reference $\texttt{SFT}$ model based on user needs. Our approach significantly reduces the sub-optimality gap observed in prior SoTA methods and demonstrates superior empirical performance across key metrics such as coherence, diversity, and quality in extensive tests on several synthetic and real datasets.

Mobile devices such as smartphones, laptops, and tablets can often connect to multiple access networks (e.g., Wi-Fi, LTE, and 5G) simultaneously.Recent advancements facilitate seamless integration of these connections below the transport layer, enhancing the experience for apps that lack inherent multi-path support.This optimization hinges on dynamically determining the traffic distribution across networks for each device, a process referred to as multi-access traffic splitting.This paper introduces NetworkGym, a high-fidelity network environment simulator that facilitates generating multiple network traffic flows and multi-access traffic splitting.This simulator facilitates training and evaluating different RL-based solutions for the multi-access traffic splitting problem.Our initial explorations demonstrate that the majority of existing state-of-the-art offline RL algorithms (e.g. CQL) fail to outperform certain hand-crafted heuristic policies on average.This illustrates the urgent need to evaluate offline RL algorithms against a broader range of benchmarks, rather than relying solely on popular ones such as D4RL.We also propose an extension to the TD3+BC algorithm, named Pessimistic TD3 (PTD3), and demonstrate that it outperforms many state-of-the-art offline RL algorithms.PTD3's behavioral constraint mechanism, which relies on value-function pessimism, is theoretically motivated and relatively simple to implement.We open source our code and offline datasets at github.com/hmomin/networkgym.

We study offline multitask representation learning in reinforcement learning (RL), where a learner is provided with an offline dataset from different tasks that share a common representation and is asked to learn the shared representation. We theoretically investigate offline multitask low-rank RL, and propose a new algorithm called MORL for offline multitask representation learning. Furthermore, we examine downstream RL in reward-free, offline and online scenarios, where a new task is introduced to the agent that shares the same representation as the upstream offline tasks. Our theoretical results demonstrate the benefits of using the learned representation from the upstream offline task instead of directly learning the representation of the low-rank model.

The safe and effective deployment of Large Language Models (LLMs) involves a critical step called alignment, which ensures that the model's responses are in accordance with human preferences. Prevalent alignment techniques, such as DPO, PPO and their variants, align LLMs by changing the pre-trained model weights during a phase called post-training. While predominant, these post-training methods add substantial complexity before LLMs can be deployed. Inference-time alignment methods avoid the complex post-training step and instead bias the generation towards responses that are aligned with human preferences. The best-known inference-time alignment method, called Best-of-N, is as effective as the state-of-the-art post-training procedures. Unfortunately, Best-of-N requires vastly more resources at inference time than standard decoding strategies, which makes it computationally not viable. In this work, we introduce Speculative Rejection, a computationally-viable inference-time alignment algorithm. It generates high-scoring responses according to a given reward model, like Best-of-N does, while being between 16 to 32 times more computationally efficient.

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.

Behavior constrained policy optimization has been demonstrated to be a successful paradigm for tackling Offline Reinforcement Learning. By exploiting historical transitions, a policy is trained to maximize a learned value function while constrained by the behavior policy to avoid a significant distributional shift. In this paper, we propose our closed-form policy improvement operators. We make a novel observation that the behavior constraint naturally motivates the use of first-order Taylor approximation, leading to a linear approximation of the policy objective. Additionally, as practical datasets are usually collected by heterogeneous policies, we model the behavior policies as a Gaussian Mixture and overcome the induced optimization difficulties by leveraging the LogSumExp’s lower bound and Jensen’s Inequality, giving rise to a closed-form policy improvement operator. We instantiate both one-step and iterative offline RL algorithms with our novel policy improvement operators and empirically demonstrate their effectiveness over state-of-the-art algorithms on the standard D4RL benchmark. Our code is available at https://cfpi-icml23.github.io/.

Recent studies in reinforcement learning (RL) have made significant progress by leveraging function approximation to alleviate the sample complexity hurdle for better performance. Despite the success, existing provably efficient algorithms typically rely on the accessibility of immediate feedback upon taking actions. The failure to account for the impact of delay in observations can significantly degrade the performance of real-world systems due to the regret blow-up. In this work, we tackle the challenge of delayed feedback in RL with linear function approximation by employing posterior sampling, which has been shown to empirically outperform the popular UCB algorithms in a wide range of regimes. We first introduce \textit{Delayed-PSVI}, an optimistic value-based algorithm that effectively explores the value function space via noise perturbation with posterior sampling. We provide the first analysis for posterior sampling algorithms with delayed feedback in RL and show our algorithm achieves $\widetilde{O}(\sqrt{d^3H^3 T} + d^2H^2 \mathbb{E}[\tau])$ worst-case regret in the presence of unknown stochastic delays. Here $\mathbb{E}[\tau]$ is the expected delay. To further improve its computational efficiency and to expand its applicability in high-dimensional RL problems, we incorporate a gradient-based approximate sampling scheme via Langevin dynamics for \textit{Delayed-LPSVI}, which maintains the same order-optimal regret guarantee with $\widetilde{O}(dHK)$ computational cost. Empirical evaluations are performed to demonstrate the statistical and computational efficacy of our algorithms.

We study linear bandits when the underlying reward function is not linear. Existing work relies on a uniform misspecification parameter $\epsilon$ that measures the sup-norm error of the best linear approximation. This results in an unavoidable linear regret whenever $\epsilon > 0$. We describe a more natural model of misspecification which only requires the approximation error at each input $x$ to be proportional to the suboptimality gap at $x$. It captures the intuition that, for optimization problems, near-optimal regions should matter more and we can tolerate larger approximation errors in suboptimal regions. Quite surprisingly, we show that the classical LinUCB algorithm — designed for the realizable case — is automatically robust against such gap-adjusted misspecification. It achieves a near-optimal $\sqrt{T}$ regret for problems that the best-known regret is almost linear in time horizon $T$. Technically, our proof relies on a novel self-bounding argument that bounds the part of the regret due to misspecification by the regret itself.

We study the problem of reinforcement learning (RL) with low (policy) switching cost {—} a problem well-motivated by real-life RL applications in which deployments of new policies are costly and the number of policy updates must be low. In this paper, we propose a new algorithm based on stage-wise exploration and adaptive policy elimination that achieves a regret of $\widetilde{O}(\sqrt{H^4S^2AT})$ while requiring a switching cost of $O(HSA \log\log T)$. This is an exponential improvement over the best-known switching cost $O(H^2SA\log T)$ among existing methods with $\widetilde{O}(\mathrm{poly}(H,S,A)\sqrt{T})$ regret. In the above, $S,A$ denotes the number of states and actions in an $H$-horizon episodic Markov Decision Process model with unknown transitions, and $T$ is the number of steps. As a byproduct of our new techniques, we also derive a reward-free exploration algorithm with a switching cost of $O(HSA)$. Furthermore, we prove a pair of information-theoretical lower bounds which say that (1) Any no-regret algorithm must have a switching cost of $\Omega(HSA)$; (2) Any $\widetilde{O}(\sqrt{T})$ regret algorithm must incur a switching cost of $\Omega(HSA\log\log T)$. Both our algorithms are thus optimal in their switching costs.

Goal-oriented Reinforcement Learning, where the agent needs to reach the goal state while simultaneously minimizing the cost, has received significant attention in real-world applications. Its theoretical formulation, stochastic shortest path (SSP), has been intensively researched in the online setting. Nevertheless, it remains understudied when such an online interaction is prohibited and only historical data is provided. In this paper, we consider the offline stochastic shortest path problem when the state space and the action space are finite. We design the simple value iteration-based algorithms for tackling both offline policy evaluation (OPE) and offline policy learning tasks. Notably, our analysis of these simple algorithms yields strong instance-dependent bounds which can imply worst-case bounds that are near-minimax optimal. We hope our study could help illuminate the fundamental statistical limits of the offline SSP problem and motivate further studies beyond the scope of current consideration.

The problem of \emph{Offline Policy Evaluation} (OPE) in Reinforcement Learning (RL) is a critical step towards applying RL in real life applications. Existing work on OPE mostly focus on evaluating a \emph{fixed} target policy $\pi$, which does not provide useful bounds for offline policy learning as $\pi$ will then be data-dependent. We address this problem by \emph{simultaneously} evaluating all policies in a policy class $\Pi$ — uniform convergence in OPE — and obtain nearly optimal error bounds for a number of global / local policy classes. Our results imply that the model-based planning achieves an optimal episode complexity of $\widetilde{O}(H^3/d_m\epsilon^2)$ in identifying an $\epsilon$-optimal policy under the \emph{time-inhomogeneous episodic} MDP model ($H$ is the planning horizon, $d_m$ is a quantity that reflects the exploration of the logging policy $\mu$). To the best of our knowledge, this is the first time the optimal rate is shown to be possible for the offline RL setting and the paper is the first that systematically investigates the uniform convergence in OPE.

We consider the problem of offline reinforcement learning (RL) --- a well-motivated setting of RL that aims at policy optimization using only historical data. Despite its wide applicability, theoretical understandings of offline RL, such as its optimal sample complexity, remain largely open even in basic settings such as \emph{tabular} Markov Decision Processes (MDPs). In this paper, we propose \emph{Off-Policy Double Variance Reduction} (OPDVR), a new variance reduction-based algorithm for offline RL. Our main result shows that OPDVR provably identifies an $\epsilon$-optimal policy with $\widetilde{O}(H^2/d_m\epsilon^2)$ episodes of offline data in the finite-horizon \emph{stationary transition} setting, where $H$ is the horizon length and $d_m$ is the minimal marginal state-action distribution induced by the behavior policy. This improves over the best-known upper bound by a factor of $H$. Moreover, we establish an information-theoretic lower bound of $\Omega(H^2/d_m\epsilon^2)$ which certifies that OPDVR is optimal up to logarithmic factors. Lastly, we show that OPDVR also achieves rate-optimal sample complexity under alternative settings such as the finite-horizon MDPs with non-stationary transitions and the infinite horizon MDPs with discounted rewards.

We study the \emph{offline reinforcement learning} (offline RL) problem, where the goal is to learn a reward-maximizing policy in an unknown \emph{Markov Decision Process} (MDP) using the data coming from a policy $\mu$. In particular, we consider the sample complexity problems of offline RL for the finite horizon MDPs. Prior works derive the information-theoretical lower bounds based on different data-coverage assumptions and their upper bounds are expressed by the covering coefficients which lack the explicit characterization of system quantities. In this work, we analyze the \emph{Adaptive Pessimistic Value Iteration} (APVI) algorithm and derive the suboptimality upper bound that nearly matches\[O\left(\sum_{h=1}^H\sum_{s_h,a_h}d^{\pi^\star}_h(s_h,a_h)\sqrt{\frac{\mathrm{Var}_{P_{s_h,a_h}}{(V^\star_{h+1}+r_h)}}{d^\mu_h(s_h,a_h)}}\sqrt{\frac{1}{n}}\right).\]We also prove an information-theoretical lower bound to show this quantity is required under the weak assumption that $d^\mu_h(s_h,a_h)>0$ if $d^{\pi^\star}_h(s_h,a_h)>0$. Here $\pi^\star$ is a optimal policy, $\mu$ is the behavior policy and $d(s_h,a_h)$ is the marginal state-action probability. We call this adaptive bound the \emph{intrinsic offline reinforcement learning bound} since it directly implies all the existing optimal results: minimax rate under uniform data-coverage assumption, horizon-free setting, single policy concentrability, and the tight problem-dependent results. Later, we extend the result to the \emph{assumption-free} regime (where we make no assumption on $\mu$) and obtain the assumption-free intrinsic bound. Due to its generic form, we believe the intrinsic bound could help illuminate what makes a specific problem hard and reveal the fundamental challenges in offline RL.

This work studies the statistical limits of uniform convergence for offline policy evaluation (OPE) problems with model-based methods (for episodic MDP) and provides a unified framework towards optimal learning for several well-motivated offline tasks. Uniform OPE $\sup_\Pi|Q^\pi-\hat{Q}^\pi|<\epsilon$ is a stronger measure than the point-wise OPE and ensures offline learning when $\Pi$ contains all policies (the global class). In this paper, we establish an $\Omega(H^2 S/d_m\epsilon^2)$ lower bound (over model-based family) for the global uniform OPE and our main result establishes an upper bound of $\tilde{O}(H^2/d_m\epsilon^2)$ for the \emph{local} uniform convergence that applies to all \emph{near-empirically optimal} policies for the MDPs with \emph{stationary} transition. Here $d_m$ is the minimal marginal state-action probability. Critically, the highlight in achieving the optimal rate $\tilde{O}(H^2/d_m\epsilon^2)$ is our design of \emph{singleton absorbing MDP}, which is a new sharp analysis tool that works with the model-based approach. We generalize such a model-based framework to the new settings: offline task-agnostic and the offline reward-free with optimal complexity $\tilde{O}(H^2\log(K)/d_m\epsilon^2)$ ($K$ is the number of tasks) and $\tilde{O}(H^2S/d_m\epsilon^2)$ respectively. These results provide a unified solution for simultaneously solving different offline RL problems.

We consider the problem of off-policy evaluation for reinforcement learning, where the goal is to estimate the expected reward of a target policy $\pi$ using offline data collected by running a logging policy $\mu$. Standard importance-sampling based approaches for this problem suffer from a variance that scales exponentially with time horizon $H$, which motivates a splurge of recent interest in alternatives that break the "Curse of Horizon" (Liu et al. 2018, Xie et al. 2019). In particular, it was shown that a marginalized importance sampling (MIS) approach can be used to achieve an estimation error of order $O(H^3/ n)$ in mean square error (MSE) under an episodic Markov Decision Process model with finite states and potentially infinite actions. The MSE bound however is still a factor of $H$ away from a Cramer-Rao lower bound of order $\Omega(H^2/n)$. In this paper, we prove that with a simple modification to the MIS estimator, we can asymptotically attain the Cramer-Rao lower bound, provided that the action space is finite. We also provide a general method for constructing MIS estimators with high-probability error bounds.