At the working heart of policy iteration algorithms commonly used and studied in the discounted setting of reinforcement learning, the policy evaluation step estimates the value of states with samples from a Markov reward process induced by following a Markov policy in a Markov decision process. We propose a simple and efficient estimator called loop estimator that exploits the regenerative structure of Markov reward processes without explicitly estimating a full model. Our method enjoys a space complexity of $O(1)$ when estimating the value of a single positive recurrent state $s$ unlike TD with $O(S)$ or model-based methods with $O\left(S^2\right)$. Moreover, the regenerative structure enables us to show, without relying on the generative model approach, that the estimator has an instance-dependent convergence rate of $\widetilde{O}\left(\sqrt{τ_s/T}\right)$ over steps $T$ on a single sample path, where $τ_s$ is the maximal expected hitting time to state $s$. In preliminary numerical experiments, the loop estimator outperforms model-free methods, such as TD(k), and is competitive with the model-based estimator.
Although in recent years reinforcement learning has become very popular the number of successful applications to different kinds of operations research problems is rather scarce. Reinforcement learning is based on the well-studied dynamic programming technique and thus also aims at finding the best stationary policy for a given Markov Decision Process, but in contrast does not require any model knowledge. The policy is assessed solely on consecutive states (or state-action pairs), which are observed while an agent explores the solution space. The contributions of this paper are manifold. First…
Nara Institute of Science and Technology（奈良先端科学技术大学院大学）
This paper proposes a new reinforcement learning with hyperbolic discounting. Combining a new temporal difference error with the hyperbolic discounting in recursive manner and reward-punishment framework, a new scheme to learn the optimal policy is derived. In simulations, it is found that the proposal outperforms the standard reinforcement learning, although the performance depends on the design of reward and punishment. In addition, the averages of discount factors w.r.t. reward and punishment are different from each other, like a sign effect in animal behaviors.
UM-SJTU Joint Institute, Shanghai Jiao Tong University（上海交通大学密西根学院）；Department of Automation, Shanghai Jiao Tong University（上海交通大学自动化系）
As the operations of autonomous systems generally affect simultaneously several users, it is crucial that their designs account for fairness considerations. In contrast to standard (deep) reinforcement learning (RL), we investigate the problem of learning a policy that treats its users equitably. In this paper, we formulate this novel RL problem, in which an objective function, which encodes a notion of fairness that we formally define, is optimized. For this problem, we provide a theoretical discussion where we examine the case of discounted rewards and that of average rewards. During this an…
INRIA Lille - Nord Europe（法国国立信息学及其应用研究院里尔-北欧中心）；Adobe Research（奥多比研究院）；INRIA Lille（法国国立信息学及其应用研究院里尔中心）
In many sequential decision-making problems we may want to manage risk by minimizing some measure of variability in rewards in addition to maximizing a standard criterion. Variance related risk measures are among the most common risk-sensitive criteria in finance and operations research. However, optimizing many such criteria is known to be a hard problem. In this paper, we consider both discounted and average reward Markov decision processes. For each formulation, we first define a measure of variability for a policy, which in turn gives us a set of risk-sensitive criteria to optimize. For ea…
In this work, we provide theoretical guarantees for reward decomposition in deterministic MDPs. Reward decomposition is a special case of Hierarchical Reinforcement Learning, that allows one to learn many policies in parallel and combine them into a composite solution. Our approach builds on mapping this problem into a Reward Discounted Traveling Salesman Problem, and then deriving approximate solutions for it. In particular, we focus on approximate solutions that are local, i.e., solutions that only observe information about the current state. Local policies are easy to implement and do not r…
School of Information Technology and Electrical Engineering（信息技术与电气工程学院）；University of Queensland（昆士兰大学）
In reinforcement learning (RL), the goal is to obtain an optimal policy, for which the optimality criterion is fundamentally important. Two major optimality criteria are average and discounted rewards. While the latter is more popular, it is problematic to apply in environments without an inherent notion of discounting. This motivates us to revisit a) the progression of optimality criteria in dynamic programming, b) justification for and complication of an artificial discount factor, and c) benefits of directly maximizing the average reward criterion, which is discounting-free. Our contributio…
Peking University（北京大学）；Microsoft Research Asia（微软亚洲研究院）
In reinforcement learning, Return, which is the weighted accumulated future rewards, and Value, which is the expected return, serve as the objective that guides the learning of the policy. In classic RL, return is defined as the exponentially discounted sum of future rewards. One key insight is that there could be many feasible ways to define the form of the return function (and thus the value), from which the same optimal policy can be derived, yet these different forms might render dramatically different speeds of learning this policy. In this paper, we research how to modify the form of the…
University at Buffalo, The State University of New York（纽约州立大学布法罗分校）；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）
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 fun…
Tsinghua University（清华大学）；University College London（伦敦大学学院）；Sun Yat-sen University（中山大学）
Most of reinforcement learning algorithms optimize the discounted criterion which is beneficial to accelerate the convergence and reduce the variance of estimates. Although the discounted criterion is appropriate for certain tasks such as financial related problems, many engineering problems treat future rewards equally and prefer a long-run average criterion. In this paper, we study the reinforcement learning problem with the long-run average criterion. Firstly, we develop a unified trust region theory with discounted and average criteria and derive a novel performance bound within the trust …
Fudan University（复旦大学）；Columbia University（哥伦比亚大学）；University of California, Los Angeles（加州大学洛杉矶分校）
In this paper, we propose AsyncQVI, an asynchronous-parallel Q-value iteration for discounted Markov decision processes whose transition and reward can only be sampled through a generative model. Given such a problem with $|\mathcal{S}|$ states, $|\mathcal{A}|$ actions, and a discounted factor $γ\in(0,1)$, AsyncQVI uses memory of size $\mathcal{O}(|\mathcal{S}|)$ and returns an $\varepsilon$-optimal policy with probability at least $1-δ$ using $$\tilde{\mathcal{O}}\big(\frac{|\mathcal{S}||\mathcal{A}|}{(1-γ)^5\varepsilon^2}\log(\frac{1}δ)\big)$$ samples. AsyncQVI is also the first asynchronous…
We study the computational complexity of the infinite-horizon discounted-reward Markov Decision Problem (MDP) with a finite state space $|\mathcal{S}|$ and a finite action space $|\mathcal{A}|$. We show that any randomized algorithm needs a running time at least $Ω(|\mathcal{S}|^2|\mathcal{A}|)$ to compute an $ε$-optimal policy with high probability. We consider two variants of the MDP where the input is given in specific data structures, including arrays of cumulative probabilities and binary trees of transition probabilities. For these cases, we show that the complexity lower bound reduces t…
In this paper, we present an online reinforcement learning algorithm, called Renewal Monte Carlo (RMC), for infinite horizon Markov decision processes with a designated start state. RMC is a Monte Carlo algorithm and retains the advantages of Monte Carlo methods including low bias, simplicity, and ease of implementation while, at the same time, circumvents their key drawbacks of high variance and delayed (end of episode) updates. The key ideas behind RMC are as follows. First, under any reasonable policy, the reward process is ergodic. So, by renewal theory, the performance of a policy is equa…
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…
University of Alberta（阿尔伯塔大学）；Amii（阿尔伯塔机器智能研究所）；DeepMind
We extend the options framework for temporal abstraction in reinforcement learning from discounted Markov decision processes (MDPs) to average-reward MDPs. Our contributions include general convergent off-policy inter-option learning algorithms, intra-option algorithms for learning values and models, as well as sample-based planning variants of our learning algorithms. Our algorithms and convergence proofs extend those recently developed by Wan, Naik, and Sutton. We also extend the notion of option-interrupting behavior from the discounted to the average-reward formulation. We show the efficac…
Improved Sample Complexity Analysis of Natural Policy Gradient Algorithm with General Parameterization for Infinite Horizon Discounted Reward Markov Decision Processes
Consider an agent interacting with an environment in cycles. In every interaction cycle the agent is rewarded for its performance. We compare the average reward U from cycle 1 to m (average value) with the future discounted reward V from cycle k to infinity (discounted value). We consider essentially arbitrary (non-geometric) discount sequences and arbitrary reward sequences (non-MDP environments). We show that asymptotically U for m->infinity and V for k->infinity are equal, provided both limits exist. Further, if the effective horizon grows linearly with k or faster, then existence of the li…
In classical Q-learning, the objective is to maximize the sum of discounted rewards through iteratively using the Bellman equation as an update, in an attempt to estimate the action value function of the optimal policy. Conventionally, the loss function is defined as the temporal difference between the action value and the expected (discounted) reward, however it focuses solely on the future, leading to overestimation errors. We extend the well-established Q-learning techniques by introducing the hindsight factor, an additional loss term that takes into account how the model progresses, by int…
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Amazon（亚马逊）；Sony AI（索尼AI）
In continuing tasks, average-reward reinforcement learning may be a more appropriate problem formulation than the more common discounted reward formulation. As usual, learning an optimal policy in this setting typically requires a large amount of training experiences. Reward shaping is a common approach for incorporating domain knowledge into reinforcement learning in order to speed up convergence to an optimal policy. However, to the best of our knowledge, the theoretical properties of reward shaping have thus far only been established in the discounted setting. This paper presents the first …
In reinforcement learning, the discount factor $γ$ controls the agent's effective planning horizon. Traditionally, this parameter was considered part of the MDP; however, as deep reinforcement learning algorithms tend to become unstable when the effective planning horizon is long, recent works refer to $γ$ as a hyper-parameter -- thus changing the underlying MDP and potentially leading the agent towards sub-optimal behavior on the original task. In this work, we introduce \emph{reward tweaking}. Reward tweaking learns a surrogate reward function $\tilde r$ for the discounted setting that induc…
We study the reinforcement learning problem for discounted Markov Decision Processes (MDPs) under the tabular setting. We propose a model-based algorithm named UCBVI-$γ$, which is based on the \emph{optimism in the face of uncertainty principle} and the Bernstein-type bonus. We show that UCBVI-$γ$ achieves an $\tilde{O}\big({\sqrt{SAT}}/{(1-γ)^{1.5}}\big)$ regret, where $S$ is the number of states, $A$ is the number of actions, $γ$ is the discount factor and $T$ is the number of steps. In addition, we construct a class of hard MDPs and show that for any algorithm, the expected regret is at lea…
We motivate and analyse a new Tree Search algorithm, GPTS, based on recent theoretical advances in the use of Gaussian Processes for Bandit problems. We consider tree paths as arms and we assume the target/reward function is drawn from a GP distribution. The posterior mean and variance, after observing data, are used to define confidence intervals for the function values, and we sequentially play arms with highest upper confidence bounds. We give an efficient implementation of GPTS and we adapt previous regret bounds by determining the decay rate of the eigenvalues of the kernel matrix on the …
Australian National University（澳大利亚国立大学）；ETH Zürich（苏黎世联邦理工学院）；NICTA（国家信息与通信技术研究院）
We study upper and lower bounds on the sample-complexity of learning near-optimal behaviour in finite-state discounted Markov Decision Processes (MDPs). For the upper bound we make the assumption that each action leads to at most two possible next-states and prove a new bound for a UCRL-style algorithm on the number of time-steps when it is not Probably Approximately Correct (PAC). The new lower bound strengthens previous work by being both more general (it applies to all policies) and tighter. The upper and lower bounds match up to logarithmic factors.
IMT School for Advanced Studies Lucca（卢卡IMT高等研究院）；Gran Sasso Science Institute - GSSI（格兰萨索科学研究院（GSSI））
In this paper, we consider solving discounted Markov Decision Processes (MDPs) under the constraint that the resulting policy is stabilizing. In practice MDPs are solved based on some form of policy approximation. We will leverage recent results proposing to use Model Predictive Control (MPC) as a structured policy in the context of Reinforcement Learning to make it possible to introduce stability requirements directly inside the MPC-based policy. This will restrict the solution of the MDP to stabilizing policies by construction. The stability theory for MPC is most mature for the undiscounted…
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…
Masaryk University（马萨里克大学）；Institute of Science and Technology Austria（奥地利科学技术研究所）
Markov decision processes (MDPs) are the defacto frame-work for sequential decision making in the presence ofstochastic uncertainty. A classical optimization criterion forMDPs is to maximize the expected discounted-sum pay-off, which ignores low probability catastrophic events withhighly negative impact on the system. On the other hand,risk-averse policies require the probability of undesirableevents to be below a given threshold, but they do not accountfor optimization of the expected payoff. We consider MDPswith discounted-sum payoff with failure states which repre-sent catastrophic outcomes…
There exist a number of reinforcement learning algorithms which learnby climbing the gradient of expected reward. Their long-runconvergence has been proved, even in partially observableenvironments with non-deterministic actions, and without the need fora system model. However, the variance of the gradient estimator hasbeen found to be a significant practical problem. Recent approacheshave discounted future rewards, introducing a bias-variance trade-offinto the gradient estimate. We incorporate a reward baseline into thelearning system, and show that it affects variance without introducingfurt…
Scale-free adaptive planning for deterministic dynamics & discounted rewards
University of California, Berkeley, USA（加州大学伯克利分校）；Noah's Ark Lab, Huawei Technologies, London, UK（华为技术伦敦诺亚实验室）；Adobe Research, San Jose, USA（Adobe研究实验室）；SequeL team, INRIA Lille - Nord Europe, France（INRIA里尔-北欧洲SequeL团队）
