Constrained Markov Decision Processes (CMDPs) formalize sequential decision-making problems whose objective is to minimize a cost function while satisfying constraints on various cost functions. In this paper, we consider the setting of episodic fixed-horizon CMDPs. We propose an online algorithm which leverages the linear programming formulation of finite-horizon CMDP for repeated optimistic planning to provide a probably approximately correct (PAC) guarantee on the number of episodes needed to ensure an $ε$-optimal policy, i.e., with resulting objective value within $ε$ of the optimal value and satisfying the constraints within $ε$-tolerance, with probability at least $1-δ$. The number of episodes needed is shown to be of the order $\tilde{\mathcal{O}}\big(\frac{|S||A|C^{2}H^{2}}{ε^{2}}\log\frac{1}δ\big)$, where $C$ is the upper bound on the number of possible successor states for a state-action pair. Therefore, if $C \ll |S|$, the number of episodes needed have a linear dependence on the state and action space sizes $|S|$ and $|A|$, respectively, and quadratic dependence on the time horizon $H$.
In this paper, we propose new problem-independent lower bounds on the sample complexity and regret in episodic MDPs, with a particular focus on the non-stationary case in which the transition kernel is allowed to change in each stage of the episode. Our main contribution is a novel lower bound of $Ω((H^3SA/ε^2)\log(1/δ))$ on the sample complexity of an $(\varepsilon,δ)$-PAC algorithm for best policy identification in a non-stationary MDP. This lower bound relies on a construction of "hard MDPs" which is different from the ones previously used in the literature. Using this same class of MDPs, w…
We study reinforcement learning in non-episodic factored Markov decision processes (FMDPs). We propose two near-optimal and oracle-efficient algorithms for FMDPs. Assuming oracle access to an FMDP planner, they enjoy a Bayesian and a frequentist regret bound respectively, both of which reduce to the near-optimal bound $\widetilde{O}(DS\sqrt{AT})$ for standard non-factored MDPs. We propose a tighter connectivity measure, factored span, for FMDPs and prove a lower bound that depends on the factored span rather than the diameter $D$. In order to decrease the gap between lower and upper bounds, we…
This work studies the problem of learning episodic Markov Decision Processes with known transition and bandit feedback. We develop the first algorithm with a ``best-of-both-worlds'' guarantee: it achieves $\mathcal{O}(log T)$ regret when the losses are stochastic, and simultaneously enjoys worst-case robustness with $\tilde{\mathcal{O}}(\sqrt{T})$ regret even when the losses are adversarial, where $T$ is the number of episodes. More generally, it achieves $\tilde{\mathcal{O}}(\sqrt{C})$ regret in an intermediate setting where the losses are corrupted by a total amount of $C$. Our algorithm is …
We investigate online Markov Decision Processes (MDPs) with adversarially changing loss functions and known transitions. We choose dynamic regret as the performance measure, defined as the performance difference between the learner and any sequence of feasible changing policies. The measure is strictly stronger than the standard static regret that benchmarks the learner's performance with a fixed compared policy. We consider three foundational models of online MDPs, including episodic loop-free Stochastic Shortest Path (SSP), episodic SSP, and infinite-horizon MDPs. For these three models, we …
On the Convergence of Monte Carlo UCB for Random-Length Episodic MDPs
SFSC of AI and DL, NYU Shanghai（纽约大学上海人工智能与深度学习科学与技术中心）；New York University（纽约大学）；New York University Shanghai（纽约大学上海分校）；New York University Abu Dhabi（纽约大学阿布扎比分校）
Recently, there has been significant progress in understanding reinforcement learning in discounted infinite-horizon Markov decision processes (MDPs) by deriving tight sample complexity bounds. However, in many real-world applications, an interactive learning agent operates for a fixed or bounded period of time, for example tutoring students for exams or handling customer service requests. Such scenarios can often be better treated as episodic fixed-horizon MDPs, for which only looser bounds on the sample complexity exist. A natural notion of sample complexity in this setting is the number of …
University of Southern California（南加利福尼亚大学）；Tsinghua University（清华大学）
We consider the best-of-both-worlds problem for learning an episodic Markov Decision Process through $T$ episodes, with the goal of achieving $\widetilde{\mathcal{O}}(\sqrt{T})$ regret when the losses are adversarial and simultaneously $\mathcal{O}(\text{polylog}(T))$ regret when the losses are (almost) stochastic. Recent work by [Jin and Luo, 2020] achieves this goal when the fixed transition is known, and leaves the case of unknown transition as a major open question. In this work, we resolve this open problem by using the same Follow-the-Regularized-Leader ($\text{FTRL}$) framework together…
Reinforcement learning (RL) has traditionally been understood from an episodic perspective; the concept of non-episodic RL, where there is no restart and therefore no reliable recovery, remains elusive. A fundamental question in non-episodic RL is how to measure the performance of a learner and derive algorithms to maximize such performance. Conventional wisdom is to maximize the difference between the average reward received by the learner and the maximal long-term average reward. In this paper, we argue that if the total time budget is relatively limited compared to the complexity of the env…
Thompson Sampling is one of the most effective methods for contextual bandits and has been generalized to posterior sampling for certain MDP settings. However, existing posterior sampling methods for reinforcement learning are limited by being model-based or lack worst-case theoretical guarantees beyond linear MDPs. This paper proposes a new model-free formulation of posterior sampling that applies to more general episodic reinforcement learning problems with theoretical guarantees. We introduce novel proof techniques to show that under suitable conditions, the worst-case regret of our posteri…
Meta AI；UMPA, ENS Lyon（里昂高等师范学院纯数学与应用数学实验室）；Univ. Lille（里尔大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；Centrale Lille（里尔中央理工学院）
In probably approximately correct (PAC) reinforcement learning (RL), an agent is required to identify an $ε$-optimal policy with probability $1-δ$. While minimax optimal algorithms exist for this problem, its instance-dependent complexity remains elusive in episodic Markov decision processes (MDPs). In this paper, we propose the first nearly matching (up to a horizon squared factor and logarithmic terms) upper and lower bounds on the sample complexity of PAC RL in deterministic episodic MDPs with finite state and action spaces. In particular, our bounds feature a new notion of sub-optimality g…
Google Research（谷歌研究院）；Johns Hopkins University（约翰斯·霍普金斯大学）；Courant Institute（库朗研究所）
We provide improved gap-dependent regret bounds for reinforcement learning in finite episodic Markov decision processes. Compared to prior work, our bounds depend on alternative definitions of gaps. These definitions are based on the insight that, in order to achieve a favorable regret, an algorithm does not need to learn how to behave optimally in states that are not reached by an optimal policy. We prove tighter upper regret bounds for optimistic algorithms and accompany them with new information-theoretic lower bounds for a large class of MDPs. Our results show that optimistic algorithms ca…
We study online learning of finite Markov decision process (MDP) problems when a side information vector is available. The problem is motivated by applications such as clinical trials, recommendation systems, etc. Such applications have an episodic structure, where each episode corresponds to a patient/customer. Our objective is to compete with the optimal dynamic policy that can take side information into account. We propose a computationally efficient algorithm and show that its regret is at most $O(\sqrt{T})$, where $T$ is the number of rounds. To best of our knowledge, this is the first re…
Statistical performance bounds for reinforcement learning (RL) algorithms can be critical for high-stakes applications like healthcare. This paper introduces a new framework for theoretically measuring the performance of such algorithms called Uniform-PAC, which is a strengthening of the classical Probably Approximately Correct (PAC) framework. In contrast to the PAC framework, the uniform version may be used to derive high probability regret guarantees and so forms a bridge between the two setups that has been missing in the literature. We demonstrate the benefits of the new framework for fin…
In this paper, we investigate the problem of \textit{episodic reinforcement learning} with quantum oracles for state evolution. To this end, we propose an \textit{Upper Confidence Bound} (UCB) based quantum algorithmic framework to facilitate learning of a finite-horizon MDP. Our quantum algorithm achieves an exponential improvement in regret as compared to the classical counterparts, achieving a regret of $\Tilde{\mathcal{O}}(1)$ as compared to $\Tilde{\mathcal{O}}(\sqrt{K})$ \footnote{$\Tilde{\mathcal{O}}(\cdot)$ hides logarithmic terms.}, $K$ being the number of training episodes. In order …
A Markov Decision Process (MDP) is a popular model for reinforcement learning. However, its commonly used assumption of stationary dynamics and rewards is too stringent and fails to hold in adversarial, nonstationary, or multi-agent problems. We study an episodic setting where the parameters of an MDP can differ across episodes. We learn a reliable policy of this potentially adversarial MDP by developing an Adversarial Reinforcement Learning (ARL) algorithm that reduces our MDP to a sequence of \emph{adversarial} bandit problems. ARL achieves $O(\sqrt{SATH^3})$ regret, which is optimal with re…
When designing algorithms for finite-time-horizon episodic reinforcement learning problems, a common approach is to introduce a fictitious discount factor and use stationary policies for approximations. Empirically, it has been shown that the fictitious discount factor helps reduce variance, and stationary policies serve to save the per-iteration computational cost. Theoretically, however, there is no existing work on convergence analysis for algorithms with this fictitious discount recipe. This paper takes the first step towards analyzing these algorithms. It focuses on two vanilla policy gra…
Solving general Markov decision processes (MDPs) is a computationally hard problem. Solving finite-horizon MDPs, on the other hand, is highly tractable with well known polynomial-time algorithms. What drives this extreme disparity, and do problems exist that lie between these diametrically opposed complexities? In this paper we identify and analyse a sub-class of stochastic shortest path problems (SSPs) for general state-action spaces whose dynamics satisfy a particular drift condition. This construction generalises the traditional, temporal notion of a horizon via decreasing reachability: a p…
Indian Institute of Science（印度科学理工学院）；Wayne State University（韦恩州立大学）
We study regret minimization in finite horizon tabular Markov decision processes (MDPs) under the constraints of differential privacy (DP). This is motivated by the widespread applications of reinforcement learning (RL) in real-world sequential decision making problems, where protecting users' sensitive and private information is becoming paramount. We consider two variants of DP -- joint DP (JDP), where a centralized agent is responsible for protecting users' sensitive data and local DP (LDP), where information needs to be protected directly on the user side. We first propose two general fram…
Model-Free Adaptive Optimal Control of Episodic Fixed-Horizon Manufacturing Processes using Reinforcement Learning
Institute Intelligent Systems Research Group, Karlsruhe University of Applied Sciences（智能系统研究组，卡尔斯鲁厄应用科学大学）；Institute for Applied Materials (IAM-CMS), Karlsruhe Institute of Technology（应用材料研究所（IAM-CMS），卡尔斯鲁厄理工学院）
In this paper, we propose and study opportunistic reinforcement learning - a new variant of reinforcement learning problems where the regret of selecting a suboptimal action varies under an external environmental condition known as the variation factor. When the variation factor is low, so is the regret of selecting a suboptimal action and vice versa. Our intuition is to exploit more when the variation factor is high, and explore more when the variation factor is low. We demonstrate the benefit of this novel framework for finite-horizon episodic MDPs by designing and evaluating OppUCRL2 and Op…
Technical University of Munich（慕尼黑工业大学）；Microsoft Research, NYC（微软研究院纽约）
We study meta-learning in Markov Decision Processes (MDP) with linear transition models in the undiscounted episodic setting. Under a task sharedness metric based on model proximity we study task families characterized by a distribution over models specified by a bias term and a variance component. We then propose BUC-MatrixRL, a version of the UC-Matrix RL algorithm, and show it can meaningfully leverage a set of sampled training tasks to quickly solve a test task sampled from the same task distribution by learning an estimator of the bias parameter of the task distribution. The analysis leve…
DeepMind；University College London（伦敦大学学院）；University of Illinois（伊利诺伊大学）；University of Alberta（阿尔伯塔大学）
We consider the problem of local planning in fixed-horizon and discounted Markov Decision Processes (MDPs) with linear function approximation and a generative model under the assumption that the optimal action-value function lies in the span of a feature map that is available to the planner. Previous work has left open the question of whether there exist sound planners that need only poly(H,d) queries regardless of the MDP, where H is the horizon and d is the dimensionality of the features. We answer this question in the negative: we show that any sound planner must query at least $\min(\exp(Ω…
Model-free reinforcement learning is known to be memory and computation efficient and more amendable to large scale problems. In this paper, two model-free algorithms are introduced for learning infinite-horizon average-reward Markov Decision Processes (MDPs). The first algorithm reduces the problem to the discounted-reward version and achieves $\mathcal{O}(T^{2/3})$ regret after $T$ steps, under the minimal assumption of weakly communicating MDPs. To our knowledge, this is the first model-free algorithm for general MDPs in this setting. The second algorithm makes use of recent advances in ada…
UT Austin（德克萨斯大学奥斯汀分校）；Tsinghua University（清华大学）；Google Brain（谷歌大脑）；University of Washington（华盛顿大学）；UT Austin & Amazon Search（德克萨斯大学奥斯汀分校 & 亚马逊搜索）
We revisit offline reinforcement learning on episodic time-homogeneous Markov Decision Processes (MDP). For tabular MDP with $S$ states and $A$ actions, or linear MDP with anchor points and feature dimension $d$, given the collected $K$ episodes data with minimum visiting probability of (anchor) state-action pairs $d_m$, we obtain nearly horizon $H$-free sample complexity bounds for offline reinforcement learning when the total reward is upper bounded by $1$. Specifically: 1. For offline policy evaluation, we obtain an $\tilde{O}\left(\sqrt{\frac{1}{Kd_m}} \right)$ error bound for the plug-in …
Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院）；Key Laboratory of Machine Perception, MOE, School of EECS, Peking University（北京大学信息科学技术学院教育部机器感知重点实验室）；Center for Data Science, Peking University（北京大学数据科学中心）；Beijing Institute of Big Data Research（北京大数据研究院）
A fundamental question in reinforcement learning is whether model-free algorithms are sample efficient. Recently, Jin et al. \cite{jin2018q} proposed a Q-learning algorithm with UCB exploration policy, and proved it has nearly optimal regret bound for finite-horizon episodic MDP. In this paper, we adapt Q-learning with UCB-exploration bonus to infinite-horizon MDP with discounted rewards \emph{without} accessing a generative model. We show that the \textit{sample complexity of exploration} of our algorithm is bounded by $\tilde{O}({\frac{SA}{ε^2(1-γ)^7}})$. This improves the previously best kn…
Recently, model-free reinforcement learning has attracted research attention due to its simplicity, memory and computation efficiency, and the flexibility to combine with function approximation. In this paper, we propose Exploration Enhanced Q-learning (EE-QL), a model-free algorithm for infinite-horizon average-reward Markov Decision Processes (MDPs) that achieves regret bound of $O(\sqrt{T})$ for the general class of weakly communicating MDPs, where $T$ is the number of interactions. EE-QL assumes that an online concentrating approximation of the optimal average reward is available. This is …
We study reinforcement learning in an infinite-horizon average-reward setting with linear function approximation, where the transition probability function of the underlying Markov Decision Process (MDP) admits a linear form over a feature mapping of the current state, action, and next state. We propose a new algorithm UCRL2-VTR, which can be seen as an extension of the UCRL2 algorithm with linear function approximation. We show that UCRL2-VTR with Bernstein-type bonus can achieve a regret of $\tilde{O}(d\sqrt{DT})$, where $d$ is the dimension of the feature mapping, $T$ is the horizon, and $\…
EDF R&D（法国电力公司研发部）；FIME（FIME实验室）；University of California, Berkeley（加州大学伯克利分校）；University of Oxford（牛津大学）
We study finite-time horizon continuous-time linear-quadratic reinforcement learning problems in an episodic setting, where both the state and control coefficients are unknown to the controller. We first propose a least-squares algorithm based on continuous-time observations and controls, and establish a logarithmic regret bound of order $O((\ln M)(\ln\ln M))$, with $M$ being the number of learning episodes. The analysis consists of two parts: perturbation analysis, which exploits the regularity and robustness of the associated Riccati differential equation; and parameter estimation error, whi…
Recent studies have shown that episodic reinforcement learning (RL) is not more difficult than contextual bandits, even with a long planning horizon and unknown state transitions. However, these results are limited to either tabular Markov decision processes (MDPs) or computationally inefficient algorithms for linear mixture MDPs. In this paper, we propose the first computationally efficient horizon-free algorithm for linear mixture MDPs, which achieves the optimal $\tilde O(d\sqrt{K} +d^2)$ regret up to logarithmic factors. Our algorithm adapts a weighted least square estimator for the unknow…
Learning Markov decision processes (MDPs) in the presence of the adversary is a challenging problem in reinforcement learning (RL). In this paper, we study RL in episodic MDPs with adversarial reward and full information feedback, where the unknown transition probability function is a linear function of a given feature mapping, and the reward function can change arbitrarily episode by episode. We propose an optimistic policy optimization algorithm POWERS and show that it can achieve $\tilde{O}(dH\sqrt{T})$ regret, where $H$ is the length of the episode, $T$ is the number of interactions with t…
Inria Lille（里尔法国国家信息与自动化研究所）；Facebook AI Research（脸书人工智能研究院）
We derive a novel asymptotic problem-dependent lower-bound for regret minimization in finite-horizon tabular Markov Decision Processes (MDPs). While, similar to prior work (e.g., for ergodic MDPs), the lower-bound is the solution to an optimization problem, our derivation reveals the need for an additional constraint on the visitation distribution over state-action pairs that explicitly accounts for the dynamics of the MDP. We provide a characterization of our lower-bound through a series of examples illustrating how different MDPs may have significantly different complexity. 1) We first consi…
We develop several new algorithms for learning Markov Decision Processes in an infinite-horizon average-reward setting with linear function approximation. Using the optimism principle and assuming that the MDP has a linear structure, we first propose a computationally inefficient algorithm with optimal $\widetilde{O}(\sqrt{T})$ regret and another computationally efficient variant with $\widetilde{O}(T^{3/4})$ regret, where $T$ is the number of interactions. Next, taking inspiration from adversarial linear bandits, we develop yet another efficient algorithm with $\widetilde{O}(\sqrt{T})$ regret…
Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）
We consider online learning in episodic loop-free Markov decision processes (MDPs), where the loss function can change arbitrarily between episodes, and the transition function is not known to the learner. We show $\tilde{O}(L|X|\sqrt{|A|T})$ regret bound, where $T$ is the number of episodes, $X$ is the state space, $A$ is the action space, and $L$ is the length of each episode. Our online algorithm is implemented using entropic regularization methodology, which allows to extend the original adversarial MDP model to handle convex performance criteria (different ways to aggregate the losses of …
We study regret minimization in non-episodic factored Markov decision processes (FMDPs), where all existing algorithms make the strong assumption that the factored structure of the FMDP is known to the learner in advance. In this paper, we provide the first algorithm that learns the structure of the FMDP while minimizing the regret. Our algorithm is based on the optimism in face of uncertainty principle, combined with a simple statistical method for structure learning, and can be implemented efficiently given oracle-access to an FMDP planner. Moreover, we give a variant of our algorithm that r…
Univ. of Michigan（密歇根大学）；Microsoft Research（微软研究院）
We consider a reinforcement learning (RL) setting in which the agent interacts with a sequence of episodic MDPs. At the start of each episode the agent has access to some side-information or context that determines the dynamics of the MDP for that episode. Our setting is motivated by applications in healthcare where baseline measurements of a patient at the start of a treatment episode form the context that may provide information about how the patient might respond to treatment decisions. We propose algorithms for learning in such Contextual Markov Decision Processes (CMDPs) under an assumpti…
Universitat Pompeu Fabra（庞培法布拉大学）；Barcelona Graduate School of Economics（巴塞罗那经济学研究生院）；Imperial College London（帝国理工学院）
The principle of optimism in the face of uncertainty underpins many theoretically successful reinforcement learning algorithms. In this paper we provide a general framework for designing, analyzing and implementing such algorithms in the episodic reinforcement learning problem. This framework is built upon Lagrangian duality, and demonstrates that every model-optimistic algorithm that constructs an optimistic MDP has an equivalent representation as a value-optimistic dynamic programming algorithm. Typically, it was thought that these two classes of algorithms were distinct, with model-optimist…
Key Laboratory of Machine Perception, MOE（教育部机器感知重点实验室）；School of EECS, Peking University（北京大学电子工程与计算机科学学院）；Amazon（亚马逊公司）；Center for Data Science, Peking University（北京大学数据科学中心）
Reinforcement learning (RL) in episodic, factored Markov decision processes (FMDPs) is studied. We propose an algorithm called FMDP-BF, which leverages the factorization structure of FMDP. The regret of FMDP-BF is shown to be exponentially smaller than that of optimal algorithms designed for non-factored MDPs, and improves on the best previous result for FMDPs~\citep{osband2014near} by a factored of $\sqrt{H|\mathcal{S}_i|}$, where $|\mathcal{S}_i|$ is the cardinality of the factored state subspace and $H$ is the planning horizon. To show the optimality of our bounds, we also provide a lower b…
Oxford Robotics Institute, University of Oxford（牛津大学牛津机器人研究所）
Previous work on planning as active inference addresses finite horizon problems and solutions valid for online planning. We propose solving the general Stochastic Shortest-Path Markov Decision Process (SSP MDP) as probabilistic inference. Furthermore, we discuss online and offline methods for planning under uncertainty. In an SSP MDP, the horizon is indefinite and unknown a priori. SSP MDPs generalize finite and infinite horizon MDPs and are widely used in the artificial intelligence community. Additionally, we highlight some of the differences between solving an MDP using dynamic programming …
University of Southern California（南加州大学）；Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）；Amazon（亚马逊）
The standard assumption in reinforcement learning (RL) is that agents observe feedback for their actions immediately. However, in practice feedback is often observed in delay. This paper studies online learning in episodic Markov decision process (MDP) with unknown transitions, adversarially changing costs, and unrestricted delayed bandit feedback. More precisely, the feedback for the agent in episode $k$ is revealed only in the end of episode $k + d^k$, where the delay $d^k$ can be changing over episodes and chosen by an oblivious adversary. We present the first algorithms that achieve near-o…
Princeton University（普林斯顿大学）；University of Southern California（南加利福尼亚大学）；Massachusetts Institute of Technology（麻省理工学院）
We consider the problem of learning in episodic finite-horizon Markov decision processes with an unknown transition function, bandit feedback, and adversarial losses. We propose an efficient algorithm that achieves $\mathcal{\tilde{O}}(L|X|\sqrt{|A|T})$ regret with high probability, where $L$ is the horizon, $|X|$ is the number of states, $|A|$ is the number of actions, and $T$ is the number of episodes. To the best of our knowledge, our algorithm is the first to ensure $\mathcal{\tilde{O}}(\sqrt{T})$ regret in this challenging setting; in fact it achieves the same regret bound as (Rosenberg &…
We consider the recently proposed reinforcement learning (RL) framework of Contextual Markov Decision Processes (CMDP), where the agent interacts with a (potentially adversarial) sequence of episodic tabular MDPs. In addition, a context vector determining the MDP parameters is available to the agent at the start of each episode, thereby allowing it to learn a context-dependent near-optimal policy. In this paper, we propose a no-regret online RL algorithm in the setting where the MDP parameters are obtained from the context using generalized linear mappings (GLMs). We propose and analyze optimi…
