We present for the first time an asymptotic convergence analysis of two time-scale stochastic approximation driven by "controlled" Markov noise. In particular, the faster and slower recursions have non-additive controlled Markov noise components in addition to martingale difference noise. We analyze the asymptotic behavior of our framework by relating it to limiting differential inclusions in both time scales that are defined in terms of the ergodic occupation measures associated with the controlled Markov processes. Using a special case of our results, we present a solution to the off-policy convergence problem for temporal-difference learning with linear function approximation. We compile several aspects of the dynamics of stochastic approximation algorithms with Markov iterate-dependent noise when the iterates are not known to be stable beforehand. We achieve the same by extending the lock-in probability (i.e. the probability of convergence to a specific attractor of the limiting o.d.e. given that the iterates are in its domain of attraction after a sufficiently large number of iterations (say) n_0) framework to such recursions. We use these results to prove almost sure convergence of the iterates to the specified attractor when the iterates satisfy an "asymptotic tightness" condition. This, in turn, is shown to be useful in analyzing the tracking ability of general "adaptive" algorithms. Finally, we obtain the first informative error bounds on function approximation for the policy evaluation algorithm proposed by Basu et al. when the aim is to find the risk-sensitive cost represented using exponential utility. We show that this happens due to the absence of difference term in the earlier bound which is always present in all our bounds when the state space is large.
We give a simple optimistic algorithm for which it is easy to derive regret bounds of $\tilde{O}(\sqrt{t_{\rm mix} SAT})$ after $T$ steps in uniformly ergodic Markov decision processes with $S$ states, $A$ actions, and mixing time parameter $t_{\rm mix}$. These bounds are the first regret bounds in the general, non-episodic setting with an optimal dependence on all given parameters. They could only be improved by using an alternative mixing time parameter.
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…
University of Colorado Boulder（科罗拉多大学博尔德分校）；University of Central Florida（中佛罗里达大学）；Air Force Research Laboratory（空军研究实验室）
The success of reinforcement learning in typical settings is predicated on Markovian assumptions on the reward signal by which an agent learns optimal policies. In recent years, the use of reward machines has relaxed this assumption by enabling a structured representation of non-Markovian rewards. In particular, such representations can be used to augment the state space of the underlying decision process, thereby facilitating non-Markovian reinforcement learning. However, these reward machines cannot capture the semantics of stochastic reward signals. In this paper, we make progress on this f…
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 …
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…
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…
Indian Institute of Technology Bombay（印度理工学院孟买分校）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Indian Institute of Technology Kanpur（印度理工学院坎普尔分校）
To overcome the curses of dimensionality and modeling of Dynamic Programming (DP) methods to solve Markov Decision Process (MDP) problems, Reinforcement Learning (RL) methods are adopted in practice. Contrary to traditional RL algorithms which do not consider the structural properties of the optimal policy, we propose a structure-aware learning algorithm to exploit the ordered multi-threshold structure of the optimal policy, if any. We prove the asymptotic convergence of the proposed algorithm to the optimal policy. Due to the reduction in the policy space, the proposed algorithm provides rema…
This paper presents a theoretical framework for probably approximately correct (PAC) multi-agent reinforcement learning (MARL) algorithms for Markov games. The paper offers an extension to the well-known Nash Q-learning algorithm, using the idea of delayed Q-learning, in order to build a new PAC MARL algorithm for general-sum Markov games. In addition to guiding the design of a provably PAC MARL algorithm, the framework enables checking whether an arbitrary MARL algorithm is PAC. Comparative numerical results demonstrate performance and robustness.
Given $X_1,\cdot ,X_N$ random variables whose joint distribution is given as $μ$ we will use the Martingale Method to show any Lipshitz Function $f$ over these random variables is subgaussian. The Variance parameter however can have a simple expression under certain conditions. For example under the assumption that the random variables follow a Markov Chain and that the function is Lipschitz under a Weighted Hamming Metric. We shall conclude with certain well known techniques from concentration of suprema of random processes with applications in Reinforcement Learning
We consider un-discounted reinforcement learning (RL) in Markov decision processes (MDPs) under drifting non-stationarity, i.e., both the reward and state transition distributions are allowed to evolve over time, as long as their respective total variations, quantified by suitable metrics, do not exceed certain variation budgets. We first develop the Sliding Window Upper-Confidence bound for Reinforcement Learning with Confidence Widening (SWUCRL2-CW) algorithm, and establish its dynamic regret bound when the variation budgets are known. In addition, we propose the Bandit-over-Reinforcement Le…
Software Competence Center Hagenberg（哈根贝格软件能力中心）；Technological University of Dublin（都柏林理工大学）
An open research question in deep reinforcement learning is how to focus the policy learning of key decisions within a sparse domain. This paper emphasizes combining the advantages of inputoutput hidden Markov models and reinforcement learning towards interpretable maintenance decisions. We propose a novel hierarchical-modeling methodology that, at a high level, detects and interprets the root cause of a failure as well as the health degradation of the turbofan engine, while, at a low level, it provides the optimal replacement policy. It outperforms the baseline performance of deep reinforceme…
The combination of Formal Methods with Reinforcement Learning (RL) has recently attracted interest as a way for single-agent RL to learn multiple-task specifications. In this paper we extend this convergence to multi-agent settings and formally define Extended Markov Games as a general mathematical model that allows multiple RL agents to concurrently learn various non-Markovian specifications. To introduce this new model we provide formal definitions and proofs as well as empirical tests of RL algorithms running on this framework. Specifically, we use our model to train two different logic-bas…
Duke University（杜克大学）；University of California San Diego（加州大学圣地亚哥分校）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Probabilistic Computation Tree Logic (PCTL) is frequently used to formally specify control objectives such as probabilistic reachability and safety. In this work, we focus on model checking PCTL specifications statistically on Markov Decision Processes (MDPs) by sampling, e.g., checking whether there exists a feasible policy such that the probability of reaching certain goal states is greater than a threshold. We use reinforcement learning to search for such a feasible policy for PCTL specifications, and then develop a statistical model checking (SMC) method with provable guarantees on its err…
Model-based algorithms -- algorithms that explore the environment through building and utilizing an estimated model -- are widely used in reinforcement learning practice and theoretically shown to achieve optimal sample efficiency for single-agent reinforcement learning in Markov Decision Processes (MDPs). However, for multi-agent reinforcement learning in Markov games, the current best known sample complexity for model-based algorithms is rather suboptimal and compares unfavorably against recent model-free approaches. In this paper, we present a sharp analysis of model-based self-play algorit…
University of Innsbruck（因斯布鲁克大学）；Max Planck Institute of Quantum Optics（马克斯·普朗克量子光学研究所）；Leiden University（莱顿大学）；University of Konstanz（康斯坦茨大学）
In recent years, the interest in leveraging quantum effects for enhancing machine learning tasks has significantly increased. Many algorithms speeding up supervised and unsupervised learning were established. The first framework in which ways to exploit quantum resources specifically for the broader context of reinforcement learning were found is projective simulation. Projective simulation presents an agent-based reinforcement learning approach designed in a manner which may support quantum walk-based speed-ups. Although classical variants of projective simulation have been benchmarked agains…
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…
University of Twente（特文特大学）；University of Colorado Boulder（科罗拉多大学博尔德分校）；University of Liverpool（利物浦大学）
We study reinforcement learning for the optimal control of Branching Markov Decision Processes (BMDPs), a natural extension of (multitype) Branching Markov Chains (BMCs). The state of a (discrete-time) BMCs is a collection of entities of various types that, while spawning other entities, generate a payoff. In comparison with BMCs, where the evolution of a each entity of the same type follows the same probabilistic pattern, BMDPs allow an external controller to pick from a range of options. This permits us to study the best/worst behaviour of the system. We generalise model-free reinforcement l…
We consider the problem of finding Nash equilibrium for two-player turn-based zero-sum games. Inspired by the AlphaGo Zero (AGZ) algorithm, we develop a Reinforcement Learning based approach. Specifically, we propose Explore-Improve-Supervise (EIS) method that combines "exploration", "policy improvement"' and "supervised learning" to find the value function and policy associated with Nash equilibrium. We identify sufficient conditions for convergence and correctness for such an approach. For a concrete instance of EIS where random policy is used for "exploration", Monte-Carlo Tree Search is us…
The Hong Kong University of Science Technology（香港科技大学）；Peking University（北京大学）；University of Virginia（弗吉尼亚大学）
Offline reinforcement learning (RL) aims at learning an optimal strategy using a pre-collected dataset without further interactions with the environment. While various algorithms have been proposed for offline RL in the previous literature, the minimax optimality has only been (nearly) established for tabular Markov decision processes (MDPs). In this paper, we focus on offline RL with linear function approximation and propose a new pessimism-based algorithm for offline linear MDP. At the core of our algorithm is the uncertainty decomposition via a reference function, which is new in the litera…
RISE AI Research Institutes of Sweden（瑞典RISE人工智能研究院）
In this paper, we consider reinforcement learning of Markov Decision Processes (MDP) with peak constraints, where an agent chooses a policy to optimize an objective and at the same time satisfy additional constraints. The agent has to take actions based on the observed states, reward outputs, and constraint-outputs, without any knowledge about the dynamics, reward functions, and/or the knowledge of the constraint-functions. We introduce a game theoretic approach to construct reinforcement learning algorithms where the agent maximizes an unconstrained objective that depends on the simulated act…
National Research Council of Canada（加拿大国家研究委员会）；University of Victoria（维多利亚大学）；University of Waterloo（滑铁卢大学）；Vector Institute for Artificial Intelligence（Vector人工智能研究所）
Reinforcement learning (RL) has been demonstrated to have great potential in many applications of scientific discovery and design. Recent work includes, for example, the design of new structures and compositions of molecules for therapeutic drugs. Much of the existing work related to the application of RL to scientific domains, however, assumes that the available state representation obeys the Markov property. For reasons associated with time, cost, sensor accuracy, and gaps in scientific knowledge, many scientific design and discovery problems do not satisfy the Markov property. Thus, somethi…
The Ohio State University（俄亥俄州立大学）；Southern University of Science and Technology（南方科技大学）
Despite the wide applications of Adam in reinforcement learning (RL), the theoretical convergence of Adam-type RL algorithms has not been established. This paper provides the first such convergence analysis for two fundamental RL algorithms of policy gradient (PG) and temporal difference (TD) learning that incorporate AMSGrad updates (a standard alternative of Adam in theoretical analysis), referred to as PG-AMSGrad and TD-AMSGrad, respectively. Moreover, our analysis focuses on Markovian sampling for both algorithms. We show that under general nonlinear function approximation, PG-AMSGrad with…
We consider an agent who is involved in a Markov decision process and receives a vector of outcomes every round. Her objective is to maximize a global concave reward function on the average vectorial outcome. The problem models applications such as multi-objective optimization, maximum entropy exploration, and constrained optimization in Markovian environments. In our general setting where a stationary policy could have multiple recurrent classes, the agent faces a subtle yet consequential trade-off in alternating among different actions for balancing the vectorial outcomes. In particular, sta…
In applications of offline reinforcement learning to observational data, such as in healthcare or education, a general concern is that observed actions might be affected by unobserved factors, inducing confounding and biasing estimates derived under the assumption of a perfect Markov decision process (MDP) model. Here we tackle this by considering off-policy evaluation in a partially observed MDP (POMDP). Specifically, we consider estimating the value of a given target policy in a POMDP given trajectories with only partial state observations generated by a different and unknown policy that may…
Northwestern University（西北大学）；Yale University（耶鲁大学）；University of California, Berkeley（加州大学伯克利分校）
We study a Markov matching market involving a planner and a set of strategic agents on the two sides of the market. At each step, the agents are presented with a dynamical context, where the contexts determine the utilities. The planner controls the transition of the contexts to maximize the cumulative social welfare, while the agents aim to find a myopic stable matching at each step. Such a setting captures a range of applications including ridesharing platforms. We formalize the problem by proposing a reinforcement learning framework that integrates optimistic value iteration with maximum we…
Safe reinforcement learning has been a promising approach for optimizing the policy of an agent that operates in safety-critical applications. In this paper, we propose an algorithm, SNO-MDP, that explores and optimizes Markov decision processes under unknown safety constraints. Specifically, we take a stepwise approach for optimizing safety and cumulative reward. In our method, the agent first learns safety constraints by expanding the safe region, and then optimizes the cumulative reward in the certified safe region. We provide theoretical guarantees on both the satisfaction of the safety co…
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.
Tsinghua University（清华大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Reinforcement learning (RL) algorithms can be used to provide personalized services, which rely on users' private and sensitive data. To protect the users' privacy, privacy-preserving RL algorithms are in demand. In this paper, we study RL with linear function approximation and local differential privacy (LDP) guarantees. We propose a novel $(\varepsilon, δ)$-LDP algorithm for learning a class of Markov decision processes (MDPs) dubbed linear mixture MDPs, and obtains an $\tilde{\mathcal{O}}( d^{5/4}H^{7/4}T^{3/4}\left(\log(1/δ)\right)^{1/4}\sqrt{1/\varepsilon})$ regret, where $d$ is the dimen…
DRL-STAF: A Deep Reinforcement Learning Framework for State-Aware Forecasting of Complex Multivariate Hidden Markov Processes
Department of Industrial Engineering, Tsinghua University, Beijing 100084, China（清华大学工业工程系）；Department of Industrial and Systems Engineering, University of Iowa, Iowa City, IA 52242, USA（爱荷华大学工业与系统工程系）
Princeton University（普林斯顿大学）；DeepMind（深度思维）；University of Alberta（阿尔伯塔大学）
This paper considers the challenging tasks of Multi-Agent Reinforcement Learning (MARL) under partial observability, where each agent only sees her own individual observations and actions that reveal incomplete information about the underlying state of system. This paper studies these tasks under the general model of multiplayer general-sum Partially Observable Markov Games (POMGs), which is significantly larger than the standard model of Imperfect Information Extensive-Form Games (IIEFGs). We identify a rich subclass of POMGs -- weakly revealing POMGs -- in which sample-efficient learning is …
Reinforcement learning (RL) is a technique to learn the control policy for an agent that interacts with a stochastic environment. In any given state, the agent takes some action, and the environment determines the probability distribution over the next state as well as gives the agent some reward. Most RL algorithms typically assume that the environment satisfies Markov assumptions (i.e. the probability distribution over the next state depends only on the current state). In this paper, we propose a model-based RL technique for a system that has non-Markovian dynamics. Such environments are com…
Sparebank 1 SR-Bank；Center for Artificial Intelligence Research（人工智能研究中心）；University of Agder（阿格德大学）
The proliferation of artificial intelligence is increasingly dependent on model understanding. Understanding demands both an interpretation - a human reasoning about a model's behavior - and an explanation - a symbolic representation of the functioning of the model. Notwithstanding the imperative of transparency for safety, trust, and acceptance, the opacity of state-of-the-art reinforcement learning algorithms conceals the rudiments of their learned strategies. We have developed a policy regularization method that asserts the global intrinsic affinities of learned strategies. These affinities…
Huawei UK R&D（华为英国研发中心）；University of Cambridge（剑桥大学）；University College London（伦敦大学学院）
We consider a context-dependent Reinforcement Learning (RL) setting, which is characterized by: a) an unknown finite number of not directly observable contexts; b) abrupt (discontinuous) context changes occurring during an episode; and c) Markovian context evolution. We argue that this challenging case is often met in applications and we tackle it using a Bayesian approach and variational inference. We adapt a sticky Hierarchical Dirichlet Process (HDP) prior for model learning, which is arguably best-suited for Markov process modeling. We then derive a context distillation procedure, which id…
Using a martingale concentration inequality, concentration bounds `from time $n_0$ on' are derived for stochastic approximation algorithms with contractive maps and both martingale difference and Markov noises. These are applied to reinforcement learning algorithms, in particular to asynchronous Q-learning and TD(0).
University of Toronto（多伦多大学）；Vector Institute for Artificial Intelligence（矢量人工智能研究所）
Many real-world reinforcement learning (RL) problems necessitate learning complex, temporally extended behavior that may only receive reward signal when the behavior is completed. If the reward-worthy behavior is known, it can be specified in terms of a non-Markovian reward function - a function that depends on aspects of the state-action history, rather than just the current state and action. Such reward functions yield sparse rewards, necessitating an inordinate number of experiences to find a policy that captures the reward-worthy pattern of behavior. Recent work has leveraged Knowledge Rep…
