We consider reinforcement learning (RL) in Markov Decision Processes in which an agent repeatedly interacts with an environment that is modeled by a controlled Markov process. At each time step $t$, it earns a reward, and also incurs a cost-vector consisting of $M$ costs. We design model-based RL algorithms that maximize the cumulative reward earned over a time horizon of $T$ time-steps, while simultaneously ensuring that the average values of the $M$ cost expenditures are bounded by agent-specified thresholds $c^{ub}_i,i=1,2,\ldots,M$.
In order to measure the performance of a reinforcement learning algorithm that satisfies the average cost constraints, we define an $M+1$ dimensional regret vector that is composed of its reward regret, and $M$ cost regrets. The reward regret measures the sub-optimality in the cumulative reward, while the $i$-th component of the cost regret vector is the difference between its $i$-th cumulative cost expense and the expected cost expenditures $Tc^{ub}_i$.
We prove that the expected value of the regret vector of UCRL-CMDP, is upper-bounded as $\tilde{O}\left(T^{2\slash 3}\right)$, where $T$ is the time horizon. We further show how to reduce the regret of a desired subset of the $M$ costs, at the expense of increasing the regrets of rewards and the remaining costs. To the best of our knowledge, ours is the only work that considers non-episodic RL under average cost constraints, and derive algorithms that can~\emph{tune the regret vector} according to the agent's requirements on its cost regrets.
London School of Economics and Political Science（伦敦政治经济学院）；North Carolina State University（北卡罗来纳州立大学）；Amazon（亚马逊公司（Amazon））
The Markov assumption (MA) is fundamental to the empirical validity of reinforcement learning. In this paper, we propose a novel Forward-Backward Learning procedure to test MA in sequential decision making. The proposed test does not assume any parametric form on the joint distribution of the observed data and plays an important role for identifying the optimal policy in high-order Markov decision processes and partially observable MDPs. We apply our test to both synthetic datasets and a real data example from mobile health studies to illustrate its usefulness.
We study minimax optimal reinforcement learning in episodic factored Markov decision processes (FMDPs), which are MDPs with conditionally independent transition components. Assuming the factorization is known, we propose two model-based algorithms. The first one achieves minimax optimal regret guarantees for a rich class of factored structures, while the second one enjoys better computational complexity with a slightly worse regret. A key new ingredient of our algorithms is the design of a bonus term to guide exploration. We complement our algorithms by presenting several structure-dependent l…
We develop algorithms with low regret for learning episodic Markov decision processes based on kernel approximation techniques. The algorithms are based on both the Upper Confidence Bound (UCB) as well as Posterior or Thompson Sampling (PSRL) philosophies, and work in the general setting of continuous state and action spaces when the true unknown transition dynamics are assumed to have smoothness induced by an appropriate Reproducing Kernel Hilbert Space (RKHS).
In this short note we derive a relationship between the Bregman divergence from the current policy to the optimal policy and the suboptimality of the current value function in a regularized Markov decision process. This result has implications for multi-task reinforcement learning, offline reinforcement learning, and regret analysis under function approximation, among others.
Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院）
We consider the adversarial Markov Decision Process (MDP) problem, where the rewards for the MDP can be adversarially chosen, and the transition function can be either known or unknown. In both settings, Follow-the-PerturbedLeader (FPL) based algorithms have been proposed in previous literature. However, the established regret bounds for FPL based algorithms are worse than algorithms based on mirrordescent. We improve the analysis of FPL based algorithms in both settings, matching the current best regret bounds using faster and simpler algorithms.
We consider reinforcement learning in changing Markov Decision Processes where both the state-transition probabilities and the reward functions may vary over time. For this problem setting, we propose an algorithm using a sliding window approach and provide performance guarantees for the regret evaluated against the optimal non-stationary policy. We also characterize the optimal window size suitable for our algorithm. These results are complemented by a sample complexity bound on the number of sub-optimal steps taken by the algorithm. Finally, we present some experimental results to support ou…
Technion（以色列理工学院）；National University of Singapore（新加坡国立大学）
We present an efficient robust value iteration for \texttt{s}-rectangular robust Markov Decision Processes (MDPs) with a time complexity comparable to standard (non-robust) MDPs which is significantly faster than any existing method. We do so by deriving the optimal robust Bellman operator in concrete forms using our $L_p$ water filling lemma. We unveil the exact form of the optimal policies, which turn out to be novel threshold policies with the probability of playing an action proportional to its advantage.
We present two elegant solutions for modeling continuous-time dynamics, in a novel model-based reinforcement learning (RL) framework for semi-Markov decision processes (SMDPs), using neural ordinary differential equations (ODEs). Our models accurately characterize continuous-time dynamics and enable us to develop high-performing policies using a small amount of data. We also develop a model-based approach for optimizing time schedules to reduce interaction rates with the environment while maintaining the near-optimal performance, which is not possible for model-free methods. We experimentally …
Our work aims at developing reinforcement learning algorithms that do not rely on the Markov assumption. We consider the class of Non-Markov Decision Processes where histories can be abstracted into a finite set of states while preserving the dynamics. We call it a Markov abstraction since it induces a Markov Decision Process over a set of states that encode the non-Markov dynamics. This phenomenon underlies the recently introduced Regular Decision Processes (as well as POMDPs where only a finite number of belief states is reachable). In all such kinds of decision process, an agent that uses a…
Tokyo Institute of Technology（东京工业大学）；The University of Tokyo（东京大学）
The online Markov decision process (MDP) is a generalization of the classical Markov decision process that incorporates changing reward functions. In this paper, we propose practical online MDP algorithms with policy iteration and theoretically establish a sublinear regret bound. A notable advantage of the proposed algorithm is that it can be easily combined with function approximation, and thus large and possibly continuous state spaces can be efficiently handled. Through experiments, we demonstrate the usefulness of the proposed algorithm.
Decision makers, such as doctors and judges, make crucial decisions such as recommending treatments to patients, and granting bails to defendants on a daily basis. Such decisions typically involve weighting the potential benefits of taking an action against the costs involved. In this work, we aim to automate this task of learning \emph{cost-effective, interpretable and actionable treatment regimes}. We formulate this as a problem of learning a decision list -- a sequence of if-then-else rules -- which maps characteristics of subjects (eg., diagnostic test results of patients) to treatments. W…
Sharif University of Technology（谢里夫理工大学）；University of Birmingham（伯明翰大学）
Partially observable Markov decision processes have been widely used to provide models for real-world decision making problems. In this paper, we will provide a method in which a slightly different version of them called Mixed observability Markov decision process, MOMDP, is going to join with our problem. Basically, we aim at offering a behavioural model for interaction of intelligent agents with musical pitch environment and we will show that how MOMDP can shed some light on building up a decision making model for musical pitch conveniently.
We show that combinations of optimal (stationary) policies in unichain Markov decision processes are optimal. That is, let M be a unichain Markov decision process with state space S, action space A and policies π_j^*: S -> A (1\leq j\leq n) with optimal average infinite horizon reward. Then any combination πof these policies, where for each state i in S there is a j such that π(i)=π_j^*(i), is optimal as well. Furthermore, we prove that any mixture of optimal policies, where at each visit in a state i an arbitrary action π_j^*(i) of an optimal policy is chosen, yields optimal average reward, t…
University of Cambridge（剑桥大学）；CREST—ENSAE（法国国家统计与经济研究所-国立统计与经济管理学院）；HEC Paris（巴黎高等商学院）；University of Bristol（布里斯托大学）
We consider the inverse reinforcement learning problem, that is, the problem of learning from, and then predicting or mimicking a controller based on state/action data. We propose a statistical model for such data, derived from the structure of a Markov decision process. Adopting a Bayesian approach to inference, we show how latent variables of the model can be estimated, and how predictions about actions can be made, in a unified framework. A new Markov chain Monte Carlo (MCMC) sampler is devised for simulation from the posterior distribution. This step includes a parameter expansion step, wh…
Off-policy evaluation (OPE) in reinforcement learning allows one to evaluate novel decision policies without needing to conduct exploration, which is often costly or otherwise infeasible. We consider for the first time the semiparametric efficiency limits of OPE in Markov decision processes (MDPs), where actions, rewards, and states are memoryless. We show existing OPE estimators may fail to be efficient in this setting. We develop a new estimator based on cross-fold estimation of $q$-functions and marginalized density ratios, which we term double reinforcement learning (DRL). We show that DRL…
Standard Markov decision process (MDP) and reinforcement learning algorithms optimize the policy with respect to the expected gain. We propose an algorithm which enables to optimize an alternative objective: the probability that the gain is greater than a given value. The algorithm can be seen as an extension of the value iteration algorithm. We also show how the proposed algorithm could be generalized to use neural networks, similarly to the deep Q learning extension of Q learning.
George Washington University（乔治华盛顿大学）；Harvard University（哈佛大学）
We study the offline data-driven sequential decision making problem in the framework of Markov decision process (MDP). In order to enhance the generalizability and adaptivity of the learned policy, we propose to evaluate each policy by a set of the average rewards with respect to distributions centered at the policy induced stationary distribution. Given a pre-collected dataset of multiple trajectories generated by some behavior policy, our goal is to learn a robust policy in a pre-specified policy class that can maximize the smallest value of this set. Leveraging the theory of semi-parametric…
Technion（以色列理工学院）；Massachusetts Institute of Technology（麻省理工学院）
We consider finite horizon Markov decision processes under performance measures that involve both the mean and the variance of the cumulative reward. We show that either randomized or history-based policies can improve performance. We prove that the complexity of computing a policy that maximizes the mean reward under a variance constraint is NP-hard for some cases, and strongly NP-hard for others. We finally offer pseudopolynomial exact and approximation algorithms.
This work provides a novel interpretation of Markov Decision Processes (MDP) from the online optimization viewpoint. In such an online optimization context, the policy of the MDP is viewed as the decision variable while the corresponding value function is treated as payoff feedback from the environment. Based on this interpretation, we construct a Blackwell game induced by MDP, which bridges the gap among regret minimization, Blackwell approachability theory, and learning theory for MDP. Specifically, from the approachability theory, we propose 1) Blackwell value iteration for offline planning…
Reinforcement learning has solid foundations, but becomes inefficient in partially observed (non-Markovian) environments. Thus, a learning agent -born with a representation and a policy- might wish to investigate to what extent the Markov property holds. We propose a learning architecture that utilizes combinatorial policy optimization to overcome non-Markovity and to develop efficient behaviors, which are easy to inherit, tests the Markov property of the behavioral states, and corrects against non-Markovity by running a deterministic factored Finite State Model, which can be learned. We illus…
Harvard Medical School（哈佛医学院）；Massachusetts Institute of Technology（麻省理工学院）
Control applications often feature tasks with similar, but not identical, dynamics. We introduce the Hidden Parameter Markov Decision Process (HiP-MDP), a framework that parametrizes a family of related dynamical systems with a low-dimensional set of latent factors, and introduce a semiparametric regression approach for learning its structure from data. In the control setting, we show that a learned HiP-MDP rapidly identifies the dynamics of a new task instance, allowing an agent to flexibly adapt to task variations.
Markov decision processes (MDP) are useful to model concurrent process optimisation problems, but verifying them with numerical methods is often intractable. Existing approximative approaches do not scale well and are limited to memoryless schedulers. Here we present the basis of scalable verification for MDPSs, using an O(1) memory representation of history-dependent schedulers. We thus facilitate scalable learning techniques and the use of massively parallel verification.
Mirror Descent Policy Optimisation for Robust Constrained Markov Decision Processes
Centre for Frontier AI Research, Agency for Science, Technology and Research (A*STAR)（前沿人工智能研究中心，科技研究局（A*STAR））；Singapore Institute of High Performance Computing, Agency for Science, Technology and Research (A*STAR)（新加坡高性能计算研究所，科技研究局（A*STAR））；College of Computing and Data Science, Nanyang Technological University, Singapore（计算与数据科学学院，南洋理工大学，新加坡）
Remembering to Be Fair: Non-Markovian Fairness in Sequential Decision Making
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；Schwartz Reisman Institute for Technology and Society（施瓦茨·赖斯曼技术与社会研究所）；University of Waterloo（滑铁卢大学）
We propose a learning-based approach for the sparse Gaussian Elimination. There are many hard combinatorial optimization problems in modern sparse solver. These NP-hard problems could be handled in the framework of Markov Decision Process, especially the Q-Learning technique. We proposed some Q-Learning algorithms for the main modules of sparse solver: minimum degree ordering, task scheduling and adaptive pivoting. Finally, we recast the sparse solver into the framework of Q-Learning. Our study is the first step to connect these two classical mathematical models: Gaussian Elimination and Marko…
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 …
In this paper, we are interested in optimal decisions in a partially observable Markov universe. Our viewpoint departs from the dynamic programming viewpoint: we are directly approximating an optimal strategic tree depending on the observation. This approximation is made by means of a parameterized probabilistic law. In this paper, a particular family of hidden Markov models, with input and output, is considered as a learning framework. A method for optimizing the parameters of these HMMs is proposed and applied. This optimization method is based on the cross-entropic principle.
