价值迭代（VI）是一种基础动态编程方法，对于最佳控制和强化学习的学习和计划很重要。 VI分批进行，其中必须完成对每个状态值的更新，然后才能开始下一批更新。如果状态空间较大，完成单批次的昂贵，那么对于许多应用来说，VI不切实际。异步VI通过一次，就地和任意顺序一次更新一个状态来帮助解决大型状态空间问题。但是，异步VI仍然需要在整个动作空间上最大化，这使得对具有较大动作空间的域不切实际。为了解决这个问题，我们提出了双重同步价值迭代（DAVI），这是一种新算法，将异步从各州到州和行动的概念推广。更具体地说，DAVI在可以使用用户定义的大小的采样子集上最大化。使用采样来减少计算的这种简单方法使VI具有类似吸引人的理论属性，而无需等待每个更新中的整个动作空间进行全面扫描。在本文中，我们显示了DAVI收敛到最佳值函数，概率是，以接近几何的速率与概率1-delta收敛，并在计算时间中返回近乎最佳的策略，该策略几乎与先前建立的对VI结合的限制。我们还从经验上证明了Davi在几个实验中的有效性。

我们研究了在随机最短路径（SSP）设置中的学习问题，其中代理试图最小化在达到目标状态之前累积的预期成本。我们设计了一种新型基于模型的算法EB-SSP，仔细地偏离了经验转变，并通过探索奖励来赋予经验成本，以诱导乐观的SSP问题，其相关价值迭代方案被保证收敛。我们证明了EB-SSP实现了Minimax后悔率$ \ tilde {o}（b _ {\ star} \ sqrt {sak}）$，其中$ k $是剧集的数量，$ s $是状态的数量， $ a $是行动的数量，而B _ {\ star} $绑定了从任何状态的最佳策略的预期累积成本，从而缩小了下限的差距。有趣的是，EB-SSP在没有参数的同时获得此结果，即，它不需要任何先前的$ B _ {\ star} $的知识，也不需要$ t _ {\ star} $，它绑定了预期的时间 ​​- 任何州的最佳政策的目标。此外，我们说明了各种情况（例如，当$ t _ {\ star} $的订单准确估计可用时，遗憾地仅包含对$ t _ {\ star} $的对数依赖性，因此产生超出有限范围MDP设置的第一个（几乎）的免地相会遗憾。

一种简单自然的增强学习算法（RL）是蒙特卡洛探索开始（MCES），通过平均蒙特卡洛回报来估算Q功能，并通过选择最大化Q当前估计的行动来改进策略。 -功能。探索是通过“探索开始”来执行的，即每个情节以随机选择的状态和动作开始，然后遵循当前的策略到终端状态。在Sutton＆Barto（2018）的RL经典书中，据说建立MCES算法的收敛是RL中最重要的剩余理论问题之一。但是，MCE的收敛问题证明是非常细微的。 Bertsekas＆Tsitsiklis（1996）提供了一个反例，表明MCES算法不一定会收敛。 TSITSIKLIS（2002）进一步表明，如果修改了原始MCES算法，以使Q-功能估计值以所有状态行动对以相同的速率更新，并且折现因子严格少于一个，则MCES算法收敛。在本文中，我们通过Sutton＆Barto（1998）中给出的原始，更有效的MCES算法取得进展政策。这样的MDP包括大量的环境，例如所有确定性环境和所有具有时间步长的情节环境或作为状态的任何单调变化的值。与以前使用随机近似的证据不同，我们引入了一种新型的感应方法，该方法非常简单，仅利用大量的强规律。

We incorporate statistical confidence intervals in both the multi-armed bandit and the reinforcement learning problems. In the bandit problem we show that given n arms, it suffices to pull the arms a total of O (n/ε 2 ) log(1/δ) times to find an ε-optimal arm with probability of at least 1 − δ. This bound matches the lower bound of Mannor and Tsitsiklis (2004) up to constants. We also devise action elimination procedures in reinforcement learning algorithms. We describe a framework that is based on learning the confidence interval around the value function or the Q-function and eliminating actions that are not optimal (with high probability). We provide a model-based and a model-free variants of the elimination method. We further derive stopping conditions guaranteeing that the learned policy is approximately optimal with high probability. Simulations demonstrate a considerable speedup and added robustness over ε-greedy Q-learning. * . Preliminary and partial results from this work appeared as extended abstracts in COLT 2002 and ICML 2003.

我们研究了马尔可夫潜在游戏（MPG）中多机构增强学习（RL）问题的策略梯度方法的全球非反应收敛属性。要学习MPG的NASH平衡，在该MPG中，状态空间的大小和/或玩家数量可能非常大，我们建议使用TANDEM所有玩家运行的新的独立政策梯度算法。当梯度评估中没有不确定性时，我们表明我们的算法找到了$ \ epsilon $ -NASH平衡，$ o（1/\ epsilon^2）$迭代复杂性并不明确取决于状态空间大小。如果没有确切的梯度，我们建立$ O（1/\ epsilon^5）$样品复杂度在潜在的无限大型状态空间中，用于利用函数近似的基于样本的算法。此外，我们确定了一类独立的政策梯度算法，这些算法都可以融合零和马尔可夫游戏和马尔可夫合作游戏，并与玩家不喜欢玩的游戏类型。最后，我们提供了计算实验来证实理论发展的优点和有效性。

Epsilon-Greedy，SoftMax或Gaussian噪声等近视探索政策在某些强化学习任务中无法有效探索，但是在许多其他方面，它们的表现都很好。实际上，实际上，由于简单性，它们通常被选为最佳选择。但是，对于哪些任务执行此类政策成功？我们可以为他们的有利表现提供理论保证吗？尽管这些政策具有显着的实际重要性，但这些关键问题几乎没有得到研究。本文介绍了对此类政策的理论分析，并为通过近视探索提供了对增强学习的首次遗憾和样本复杂性。我们的结果适用于具有有限的Bellman Eluder维度的情节MDP中的基于价值功能的算法。我们提出了一种新的复杂度度量，称为近视探索差距，用Alpha表示，该差距捕获了MDP的结构属性，勘探策略和给定的值函数类别。我们表明，近视探索的样品复杂性与该数量的倒数1 / alpha^2二次地量表。我们通过具体的例子进一步证明，由于相应的动态和奖励结构，在近视探索成功的几项任务中，近视探索差距确实是有利的。

This work considers the sample complexity of obtaining an $\varepsilon$-optimal policy in an average reward Markov Decision Process (AMDP), given access to a generative model (simulator). When the ground-truth MDP is weakly communicating, we prove an upper bound of $\widetilde O(H \varepsilon^{-3} \ln \frac{1}{\delta})$ samples per state-action pair, where $H := sp(h^*)$ is the span of bias of any optimal policy, $\varepsilon$ is the accuracy and $\delta$ is the failure probability. This bound improves the best-known mixing-time-based approaches in [Jin & Sidford 2021], which assume the mixing-time of every deterministic policy is bounded. The core of our analysis is a proper reduction bound from AMDP problems to discounted MDP (DMDP) problems, which may be of independent interests since it allows the application of DMDP algorithms for AMDP in other settings. We complement our upper bound by proving a minimax lower bound of $\Omega(|\mathcal S| |\mathcal A| H \varepsilon^{-2} \ln \frac{1}{\delta})$ total samples, showing that a linear dependent on $H$ is necessary and that our upper bound matches the lower bound in all parameters of $(|\mathcal S|, |\mathcal A|, H, \ln \frac{1}{\delta})$ up to some logarithmic factors.

我们在$ \ Gamma $ -diScounted MDP中使用Polyak-Ruppert平均（A.K.A.，平均Q-Leaning）进行同步Q学习。我们为平均迭代$ \ bar {\ boldsymbol {q}}建立渐近常态。此外，我们展示$ \ bar {\ boldsymbol {q}} _ t $实际上是一个常规的渐近线性（RAL）估计值，用于最佳q-value函数$ \ boldsymbol {q} ^ * $与最有效的影响功能。它意味着平均Q学习迭代在所有RAL估算器之间具有最小的渐近方差。此外，我们为$ \ ell _ {\ infty} $错误$ \ mathbb {e} \ | \ | \ bar {\ boldsymbol {q}} _ t- \ boldsymbol {q} ^ *} ^ *} _ {\ idty} $，显示它与实例相关的下限以及最佳最低限度复杂性下限。作为一个副产品，我们发现Bellman噪音具有var-gaussian坐标，具有方差$ \ mathcal {o}（（1- \ gamma）^ {-1}）$而不是现行$ \ mathcal {o}（（1- \ Gamma）^ { - 2}）$根据标准界限奖励假设。子高斯结果有可能提高许多R1算法的样本复杂性。简而言之，我们的理论分析显示平均Q倾斜在统计上有效。

Value-function approximation methods that operate in batch mode have foundational importance to reinforcement learning (RL). Finite sample guarantees for these methods often crucially rely on two types of assumptions: (1) mild distribution shift, and (2) representation conditions that are stronger than realizability. However, the necessity ("why do we need them?") and the naturalness ("when do they hold?") of such assumptions have largely eluded the literature. In this paper, we revisit these assumptions and provide theoretical results towards answering the above questions, and make steps towards a deeper understanding of value-function approximation.

在标准数据分析框架中，首先收集数据（全部一次），然后进行数据分析。此外，通常认为数据生成过程是外源性的。当数据分析师对数据的生成方式没有影响时，这种方法是自然的。但是，数字技术的进步使公司促进了从数据中学习并同时做出决策。随着这些决定生成新数据，数据分析师（业务经理或算法）也成为数据生成器。这种相互作用会产生一种新型的偏见 - 增强偏见 - 加剧了静态数据分析中的内生性问题。因果推理技术应该被纳入加强学习中以解决此类问题。

In robust Markov decision processes (MDPs), the uncertainty in the transition kernel is addressed by finding a policy that optimizes the worst-case performance over an uncertainty set of MDPs. While much of the literature has focused on discounted MDPs, robust average-reward MDPs remain largely unexplored. In this paper, we focus on robust average-reward MDPs, where the goal is to find a policy that optimizes the worst-case average reward over an uncertainty set. We first take an approach that approximates average-reward MDPs using discounted MDPs. We prove that the robust discounted value function converges to the robust average-reward as the discount factor $\gamma$ goes to $1$, and moreover, when $\gamma$ is large, any optimal policy of the robust discounted MDP is also an optimal policy of the robust average-reward. We further design a robust dynamic programming approach, and theoretically characterize its convergence to the optimum. Then, we investigate robust average-reward MDPs directly without using discounted MDPs as an intermediate step. We derive the robust Bellman equation for robust average-reward MDPs, prove that the optimal policy can be derived from its solution, and further design a robust relative value iteration algorithm that provably finds its solution, or equivalently, the optimal robust policy.

We consider infinite horizon Markov decision processes (MDPs) with fast-slow structure, meaning that certain parts of the state space move "fast" (and in a sense, are more influential) while other parts transition more "slowly." Such structure is common in real-world problems where sequential decisions need to be made at high frequencies, yet information that varies at a slower timescale also influences the optimal policy. Examples include: (1) service allocation for a multi-class queue with (slowly varying) stochastic costs, (2) a restless multi-armed bandit with an environmental state, and (3) energy demand response, where both day-ahead and real-time prices play a role in the firm's revenue. Models that fully capture these problems often result in MDPs with large state spaces and large effective time horizons (due to frequent decisions), rendering them computationally intractable. We propose an approximate dynamic programming algorithmic framework based on the idea of "freezing" the slow states, solving a set of simpler finite-horizon MDPs (the lower-level MDPs), and applying value iteration (VI) to an auxiliary MDP that transitions on a slower timescale (the upper-level MDP). We also extend the technique to a function approximation setting, where a feature-based linear architecture is used. On the theoretical side, we analyze the regret incurred by each variant of our frozen-state approach. Finally, we give empirical evidence that the frozen-state approach generates effective policies using just a fraction of the computational cost, while illustrating that simply omitting slow states from the decision modeling is often not a viable heuristic.

与表征解决马尔可夫决策过程（MDP）样品复杂性的进步相反，解决约束MDP（CMDP）的最佳统计复杂性仍然未知。我们通过在折扣CMDP中学习近乎最佳策略的样本复杂性上的最小上限和下限来解决这个问题，并访问生成模型（模拟器）。特别是，我们设计了一种基于模型的算法，该算法解决了两个设置：（i）允许违反小小的约束的可行性，以及（ii）严格的可行性，其中需要输出策略来满足约束。对于（i），我们证明我们的算法通过制作$ \ tilde {o} \ left（\ frac {s a \ log（1/\ delta）来返回带有概率$ 1- \ delta $的$ \ epsilon $ - 优势策略} {（1- \ gamma）^3 \ epsilon^2} \ right）$ QUERIES $ QUERIES与生成模型相匹配，因此与无约束的MDP的样品复杂性匹配。对于（ii），我们表明该算法的样本复杂性是由$ \ tilde {o} \ left（\ frac {s a a \ log，\ log（1/\ delta）} {（1 - \ gamma）^5 \，\ epsilon^2 \ zeta^2} \ right）$，其中$ \ zeta $是与问题相关的slater常数，其特征是可行区域的大小。最后，我们证明了严格的可行性设置的匹配较低限制，因此获得了折扣CMDP的第一个最小值最佳界限。我们的结果表明，在允许违反小小的约束时，学习CMDP与MDP一样容易，但是当我们要求零约束违规时，本质上更加困难。

我们介绍了一种改进政策改进的方法，该方法在基于价值的强化学习（RL）的贪婪方法与基于模型的RL的典型计划方法之间进行了插值。新方法建立在几何视野模型（GHM，也称为伽马模型）的概念上，该模型对给定策略的折现状态验证分布进行了建模。我们表明，我们可以通过仔细的基本策略GHM的仔细组成，而无需任何其他学习，可以评估任何非马尔科夫策略，以固定的概率在一组基本马尔可夫策略之间切换。然后，我们可以将广义政策改进（GPI）应用于此类非马尔科夫政策的收集，以获得新的马尔可夫政策，通常将其表现优于其先驱。我们对这种方法提供了彻底的理论分析，开发了转移和标准RL的应用，并在经验上证明了其对标准GPI的有效性，对充满挑战的深度RL连续控制任务。我们还提供了GHM培训方法的分析，证明了关于先前提出的方法的新型收敛结果，并显示了如何在深度RL设置中稳定训练这些模型。

In this paper we develop a theoretical analysis of the performance of sampling-based fitted value iteration (FVI) to solve infinite state-space, discounted-reward Markovian decision processes (MDPs) under the assumption that a generative model of the environment is available. Our main results come in the form of finite-time bounds on the performance of two versions of sampling-based FVI. The convergence rate results obtained allow us to show that both versions of FVI are well behaving in the sense that by using a sufficiently large number of samples for a large class of MDPs, arbitrary good performance can be achieved with high probability. An important feature of our proof technique is that it permits the study of weighted L p -norm performance bounds. As a result, our technique applies to a large class of function-approximation methods (e.g., neural networks, adaptive regression trees, kernel machines, locally weighted learning), and our bounds scale well with the effective horizon of the MDP. The bounds show a dependence on the stochastic stability properties of the MDP: they scale with the discounted-average concentrability of the future-state distributions. They also depend on a new measure of the approximation power of the function space, the inherent Bellman residual, which reflects how well the function space is "aligned" with the dynamics and rewards of the MDP. The conditions of the main result, as well as the concepts introduced in the analysis, are extensively discussed and compared to previous theoretical results. Numerical experiments are used to substantiate the theoretical findings.

我们介绍了一种普遍的策略，可实现有效的多目标勘探。它依赖于adagoal，一种基于简单约束优化问题的新的目标选择方案，其自适应地针对目标状态，这既不是太困难也不是根据代理目前的知识达到的。我们展示了Adagoal如何用于解决学习$ \ epsilon $ -optimal的目标条件的政策，以便在$ L $ S_0 $ S_0 $奖励中获得的每一个目标状态，以便在$ S_0 $中获取。免费马尔可夫决策过程。在标准的表格外壳中，我们的算法需要$ \ tilde {o}（l ^ 3 s a \ epsilon ^ { - 2}）$探索步骤，这几乎很少最佳。我们还容易在线性混合Markov决策过程中实例化Adagoal，其产生具有线性函数近似的第一目标导向的PAC保证。除了强大的理论保证之外，迈克纳队以现有方法的高级别算法结构为锚定，为目标条件的深度加固学习。

逆增强学习（IRL）是从专家演示中推断奖励功能的强大范式。许多IRL算法都需要已知的过渡模型，有时甚至是已知的专家政策，或者至少需要访问生成模型。但是，对于许多现实世界应用，这些假设太强了，在这些应用程序中，只能通过顺序相互作用访问环境。我们提出了一种新颖的IRL算法：逆增强学习（ACEIRL）的积极探索，该探索积极探索未知的环境和专家政策，以快速学习专家的奖励功能并确定良好的政策。 Aceirl使用以前的观察来构建置信区间，以捕获合理的奖励功能，并找到关注环境最有用区域的勘探政策。 Aceirl是使用样品复杂性界限的第一种活动IRL的方法，不需要环境的生成模型。在最坏情况下，Aceirl与活性IRL的样品复杂性与生成模型匹配。此外，我们建立了一个与问题相关的结合，该结合将Aceirl的样品复杂性与给定IRL问题的次级隔离间隙联系起来。我们在模拟中对Aceirl进行了经验评估，发现它的表现明显优于更幼稚的探索策略。

在强化学习中，蒙特卡洛算法通过平均偶发回报来更新Q功能。在Monte Carlo UCB（MC-UCB）算法中，在每个状态下采取的动作是最大化Q函数加上UCB勘探项的动作，该术语偏向于选择频率较低的动作的选择。尽管在为MC-UCB建立遗憾界限方面已经进行了重要的工作，但大多数工作都集中在该问题的有限培训版本上，每个情节都在不断数量的步骤后终止。对于此类有限的Horizo​​n问题，最佳策略既取决于当前状态和情节中的时间。但是，对于许多自然的情节问题，例如GO，CHESS和机器人任务等游戏，该情节是随机的，最佳政策是静止的。对于此类环境，MC-UCB中的Q功能是否会收敛到最佳Q函数，这是一个空旷的问题。我们猜想，与Q学习不同，它并不是所有MDP的收敛。尽管如此，我们表明，对于大型MDP，其中包括二十一点和确定性MDP等随机MDP，例如GO，MC-UCB中的Q功能几乎可以肯定地收敛到最佳Q函数。该结果的直接推论是，它几乎肯定会为所有有限的Horizo​​n MDP收敛。我们还提供了数值实验，为MC-UCB提供了进一步的见解。

We study a multi-agent reinforcement learning (MARL) problem where the agents interact over a given network. The goal of the agents is to cooperatively maximize the average of their entropy-regularized long-term rewards. To overcome the curse of dimensionality and to reduce communication, we propose a Localized Policy Iteration (LPI) algorithm that provably learns a near-globally-optimal policy using only local information. In particular, we show that, despite restricting each agent's attention to only its $\kappa$-hop neighborhood, the agents are able to learn a policy with an optimality gap that decays polynomially in $\kappa$. In addition, we show the finite-sample convergence of LPI to the global optimal policy, which explicitly captures the trade-off between optimality and computational complexity in choosing $\kappa$. Numerical simulations demonstrate the effectiveness of LPI.

Partially observable Markov decision processes (POMDPs) provide a flexible representation for real-world decision and control problems. However, POMDPs are notoriously difficult to solve, especially when the state and observation spaces are continuous or hybrid, which is often the case for physical systems. While recent online sampling-based POMDP algorithms that plan with observation likelihood weighting have shown practical effectiveness, a general theory characterizing the approximation error of the particle filtering techniques that these algorithms use has not previously been proposed. Our main contribution is bounding the error between any POMDP and its corresponding finite sample particle belief MDP (PB-MDP) approximation. This fundamental bridge between PB-MDPs and POMDPs allows us to adapt any sampling-based MDP algorithm to a POMDP by solving the corresponding particle belief MDP, thereby extending the convergence guarantees of the MDP algorithm to the POMDP. Practically, this is implemented by using the particle filter belief transition model as the generative model for the MDP solver. While this requires access to the observation density model from the POMDP, it only increases the transition sampling complexity of the MDP solver by a factor of $\mathcal{O}(C)$, where $C$ is the number of particles. Thus, when combined with sparse sampling MDP algorithms, this approach can yield algorithms for POMDPs that have no direct theoretical dependence on the size of the state and observation spaces. In addition to our theoretical contribution, we perform five numerical experiments on benchmark POMDPs to demonstrate that a simple MDP algorithm adapted using PB-MDP approximation, Sparse-PFT, achieves performance competitive with other leading continuous observation POMDP solvers.