In this paper, we present a link between preference-based and multiobjective sequential decision-making. While transforming a multiobjective problem to a preference-based one is quite natural, the other direction is a bit less obvious. We present how this transformation (from preference-based to multiobjective) can be done under the classic condition that preferences over histories can be represented by additively decomposable utilities and that the decision criterion to evaluate policies in a state is based on expectation. This link yields a new source of multiobjective sequential decision-making problems (i.e., when reward values are unknown) and justifies the use of solving methods developed in one setting in the other one.
A preference-based MDP (PBMDP) is an MDP where possibly no reward function is given. Instead, one assumes that a preference relation @math is defined over histories. In the case where the dynamics of the system is not known, this setting is referred to as preference-based reinforcement learning (PBRL) @cite_21 @cite_30 @cite_34 @cite_6 . Due to this ordinal preferential information, it is not possible to directly use the same decision criterion based on expectation like in the standard or multiobjective cases. Most approaches in PBRL @cite_21 @cite_34 @cite_6 relies on comparing policies with probabilistic dominance , which is defined as follows: where @math denotes the probability that policy @math generates a history preferred or equivalent to that generated by policy @math . Probabilistic dominance is related to Condorcet methods (where a candidate is preferred to another if more voters prefers the former than the latter) in social choice theory. This is why the optimal policy for probabilistic dominance is often called a Condorcet winner .
@cite_30 @cite_21 @cite_6 @cite_34
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Consider the problem of approximating the optimal policy of a Markov decision process (MDP) by sampling state transitions. In contrast to existing reinforcement learning methods that are based on successive approximations to the nonlinear Bellman equation, we propose a Primal-Dual @math Learning method in light of the linear duality between the value and policy. The @math learning method is model-free and makes primal-dual updates to the policy and value vectors as new data are revealed. For infinite-horizon undiscounted Markov decision process with finite state space @math and finite action space @math , the @math learning method finds an @math -optimal policy using the following number of sample transitions @math where @math is an upper bound of mixing times across all policies and @math is a parameter characterizing the range of stationary distributions across policies. The @math learning method also applies to the computational problem of MDP where the transition probabilities and rewards are explicitly given as the input. In the case where each state transition can be sampled in @math time, the @math learning method gives a sublinear-time algorithm for solving the averaged-reward MDP. Specifying a numeric reward function for reinforcement learning typically requires a lot of hand-tuning from a human expert. In contrast, preference-based reinforcement learning (PBRL) utilizes only pairwise comparisons between trajectories as a feedback signal, which are often more intuitive to specify. Currently available approaches to PBRL for control problems with continuous state action spaces require a known or estimated model, which is often not available and hard to learn. In this paper, we integrate preference-based estimation of the reward function into a model-free reinforcement learning (RL) algorithm, resulting in a model-free PBRL algorithm. Our new algorithm is based on Relative Entropy Policy Search (REPS), enabling us to utilize stochastic policies and to directly control the greediness of the policy update. REPS decreases exploration of the policy slowly by limiting the relative entropy of the policy update, which ensures that the algorithm is provided with a versatile set of trajectories, and consequently with informative preferences. The preference-based estimation is computed using a sample-based Bayesian method, which can also estimate the uncertainty of the utility. Additionally, we also compare to a linear solvable approximation, based on inverse RL. We show that both approaches perform favourably to the current state-of-the-art. The overall result is an algorithm that can learn non-parametric continuous action policies from a small number of preferences. We consider the problem of learning an unknown Markov Decision Process (MDP) that is weakly communicating in the infinite horizon setting. We propose a Thompson Sampling-based reinforcement learning algorithm with dynamic episodes (TSDE). At the beginning of each episode, the algorithm generates a sample from the posterior distribution over the unknown model parameters. It then follows the optimal stationary policy for the sampled model for the rest of the episode. The duration of each episode is dynamically determined by two stopping criteria. The first stopping criterion controls the growth rate of episode length. The second stopping criterion happens when the number of visits to any state-action pair is doubled. We establish @math bounds on expected regret under a Bayesian setting, where @math and @math are the sizes of the state and action spaces, @math is time, and @math is the bound of the span. This regret bound matches the best available bound for weakly communicating MDPs. Numerical results show it to perform better than existing algorithms for infinite horizon MDPs. This paper is devoted to fair optimization in Multiobjective Markov Decision Processes (MOMDPs). A MOMDP is an extension of the MDP model for planning under uncertainty while trying to optimize several reward functions simultaneously. This applies to multiagent problems when rewards define individual utility functions, or in multicriteria problems when rewards refer to different features. In this setting, we study the determination of policies leading to Lorenz-non-dominated tradeoffs. Lorenz dominance is a refinement of Pareto dominance that was introduced in Social Choice for the measurement of inequalities. In this paper, we introduce methods to efficiently approximate the sets of Lorenz-non-dominated solutions of infinite-horizon, discounted MOMDPs. The approximations are polynomial-sized subsets of those solutions. Learning, planning, and representing knowledge at multiple levels of temporal ab- straction are key, longstanding challenges for AI. In this paper we consider how these challenges can be addressed within the mathematical framework of reinforce- ment learning and Markov decision processes (MDPs). We extend the usual notion of action in this framework to include options—closed-loop policies for taking ac- tion over a period of time. Examples of options include picking up an object, going to lunch, and traveling to a distant city, as well as primitive actions such as mus- cle twitches and joint torques. Overall, we show that options enable temporally abstract knowledge and action to be included in the reinforcement learning frame- work in a natural and general way. In particular, we show that options may be used interchangeably with primitive actions in planning methods such as dynamic pro- gramming and in learning methods such as Q-learning. Formally, a set of options defined over an MDP constitutes a semi-Markov decision process (SMDP), and the theory of SMDPs provides the foundation for the theory of options. However, the most interesting issues concern the interplay between the underlying MDP and the SMDP and are thus beyond SMDP theory. We present results for three such cases: 1) we show that the results of planning with options can be used during execution to interrupt options and thereby perform even better than planned, 2) we introduce new intra-option methods that are able to learn about an option from fragments of its execution, and 3) we propose a notion of subgoal that can be used to improve the options themselves. All of these results have precursors in the existing literature; the contribution of this paper is to establish them in a simpler and more general setting with fewer changes to the existing reinforcement learning framework. In particular, we show that these results can be obtained without committing to (or ruling out) any particular approach to state abstraction, hierarchy, function approximation, or the macro-utility problem. In this paper, we address the problem of suboptimal behavior during online partially observable Markov decision process (POMDP) planning caused by time constraints on planning. Taking inspiration from the related field of reinforcement learning (RL), our solution is to shape the agent's reward function in order to lead the agent to large future rewards without having to spend as much time explicitly estimating cumulative future rewards, enabling the agent to save time to improve the breadth planning and build higher quality plans. Specifically, we extend potential-based reward shaping (PBRS) from RL to online POMDP planning. In our extension, information about belief states is added to the function optimized by the agent during planning. This information provides hints of where the agent might find high future rewards beyond its planning horizon, and thus achieve greater cumulative rewards. We develop novel potential functions measuring information useful to agent metareasoning in POMDPs (reflecting on agent knowledge and or histories of experience with the environment), theoretically prove several important properties and benefits of using PBRS for online POMDP planning, and empirically demonstrate these results in a range of classic benchmark POMDP planning problems. Consider a Markov decision process (MDP) that admits a set of state-action features, which can linearly express the process's probabilistic transition model. We propose a parametric Q-learning algorithm that finds an approximate-optimal policy using a sample size proportional to the feature dimension @math and invariant with respect to the size of the state space. To further improve its sample efficiency, we exploit the monotonicity property and intrinsic noise structure of the Bellman operator, provided the existence of anchor state-actions that imply implicit non-negativity in the feature space. We augment the algorithm using techniques of variance reduction, monotonicity preservation, and confidence bounds. It is proved to find a policy which is @math -optimal from any initial state with high probability using @math sample transitions for arbitrarily large-scale MDP with a discount factor @math . A matching information-theoretical lower bound is proved, confirming the sample optimality of the proposed method with respect to all parameters (up to polylog factors). Reinforcement learning (RL) agents have traditionally been tasked with maximizing the value function of a Markov decision process (MDP), either in continuous settings, with fixed discount factor @math , or in episodic settings, with @math . While this has proven effective for specific tasks with well-defined objectives (e.g., games), it has never been established that fixed discounting is suitable for general purpose use (e.g., as a model of human preferences). This paper characterizes rationality in sequential decision making using a set of seven axioms and arrives at a form of discounting that generalizes traditional fixed discounting. In particular, our framework admits a state-action dependent "discount" factor that is not constrained to be less than 1, so long as there is eventual long run discounting. Although this broadens the range of possible preference structures in continuous settings, we show that there exists a unique "optimizing MDP" with fixed @math whose optimal value function matches the true utility of the optimal policy, and we quantify the difference between value and utility for suboptimal policies. Our work can be seen as providing a normative justification for (a slight generalization of) Martha White's RL task formalism (2017) and other recent departures from the traditional RL, and is relevant to task specification in RL, inverse RL and preference-based RL. We consider the problem of power allocation over one or more time-varying channels with unknown distributions in energy harvesting communications. In the single-channel case, the transmitter chooses the transmit power based on the amount of stored energy in its battery with the goal of maximizing the average rate over time. We model this problem as a Markov decision process (MDP) with transmitter as the agent, battery status as the state, transmits power as the action and rate as the reward. The average reward maximization problem can be modeled by a linear program (LP) that uses the transition probabilities for the state-action pairs and their reward values to select a power allocation policy. This problem is challenging because the uncertainty in channels implies that the mean rewards associated with the state-action pairs are unknown. We therefore propose two online learning algorithms: linear program of sample means (LPSM) and Epoch-LPSM that learn these rewards and adapt their policies over time. For both algorithms, we prove that their regret is upper-bounded by a constant. To our knowledge this is the first result showing constant regret learning algorithms for MDPs with unknown mean rewards. We also prove an even stronger result about LPSM: that its policy matches the optimal policy exactly in finite expected time. Epoch-LPSM incurs a higher regret compared with the LPSM, while reducing the computational requirements substantially. We further consider a multi-channel scenario, where the agent also chooses a channel in each slot, and present our multi-channel LPSM (MC-LPSM) algorithm that explores different channels and uses that information to solve the LP during exploitation. MC-LPSM incurs a regret that scales logarithmically in time and linearly in the number of channels. Through a matching lower bound on the regret of any algorithm, we also prove the asymptotic order optimality of MC-LPSM. In this paper we present a general treatment of the preference aggregation problem, in which multiple preferences over objects must be combined into a single consensus ranking. We consider two instances of this problem: unsupervised aggregation where no information about a target ranking is available, and supervised aggregation where ground truth preferences are provided. For each problem class we develop novel learning methods that are applicable to a wide range of preference types. Specifically, for unsupervised aggregation we introduce the Multinomial Preference model (MPM) which uses a multinomial generative process to model the observed preferences. For the supervised problem we develop a supervised extension for MPM and then propose two fully supervised models. The first model employs SVD factorization to derive effective item features, transforming the aggregation problems into a learning-to-rank one. The second model aims to eliminate the costly SVD factorization and instantiates a probabilistic CRF framework, deriving unary and pairwise potentials directly from the observed preferences. Using a probabilistic framework allows us to directly optimize the expectation of any target metric, such as NDCG or ERR. All the proposed models operate on pairwise preferences and can thus be applied to a wide range of preference types. We empirically validate the models on rank aggregation and collaborative filtering data sets and demonstrate superior empirical accuracy. We consider Markov decision processes (MDPs) with multiple discounted reward objectives. Such MDPs occur in design problems where one wishes to simultaneously optimize several criteria, for example, latency and power. The possible trade-offs between the different objectives are characterized by the Pareto curve. We show that every Pareto-optimal point can be achieved by a memoryless strategy; however, unlike in the single-objective case, the memoryless strategy may require randomization. Moreover, we show that the Pareto curve can be approximated in polynomial time in the size of the MDP. Additionally, we study the problem if a given value vector is realizable by any strategy, and show that it can be decided in polynomial time; but the question whether it is realizable by a deterministic memoryless strategy is NP-complete. These results provide efficient algorithms for design exploration in MDP models with multiple objectives. We propose RUDDER, a novel reinforcement learning approach for delayed rewards in finite Markov decision processes (MDPs). RUDDER is based on two main ideas: (i) a backward view approach and (ii) the concept of return-equivalent MDPs. Forward view approaches, like deep Q-networks (DQNs) or Monte Carlo tree search (MCTS), have to average over a large number of probabilistic future state-action paths that increases exponentially with the delay of the reward. Backward view approaches, in contrast, identify actions and states that cause a delayed reward by analyzing already chosen paths. RUDDER's backward view transforms tasks of estimating future returns into regression tasks at which deep learning excels. RUDDER decomposes the return into new, non-delayed rewards by redistributing the original reward across the episode, thereby creating a new MDP that is return-equivalent to the original MDP. "Return-equivalent MDPs" is a new concept ensuring that both MDPs have the same optimal policies. If the return decomposition is optimal, then the new MDP will be stripped of any delayed rewards. In this case, action-value estimates are unbiased and the future expected return is always zero. On several artificial tasks with delayed rewards RUDDER significantly outperforms Monte Carlo, MCTS, temporal difference, TD( ), and reward shaping approaches. RUDDER is even exponentially faster than the last three. RUDDER on top of a Proximal Policy Optimization (PPO) baseline improves the scores on Atari games and excels for delayed rewards. For long delayed rewards as in Bowling, Frostbite, PrivateEye and Venture RUDDER yields exceptional results. We describe an algorithm for learning in the presence of multiple criteria. Our technique generalizes previous approaches in that it can learn optimal policies for all linear preference assignments over the multiple reward criteria at once. The algorithm can be viewed as an extension to standard reinforcement learning for MDPs where instead of repeatedly backing up maximal expected rewards, we back up the set of expected rewards that are maximal for some set of linear preferences (given by a weight vector, w). We present the algorithm along with a proof of correctness showing that our solution gives the optimal policy for any linear preference function. The solution reduces to the standard value iteration algorithm for a specific weight vector, w. This article addresses reinforcement learning problems based on factored Markov decision processes MDPs in which the agent must choose among a set of candidate abstractions, each build up from a different combination of state components. We present and evaluate a new approach that can perform effective abstraction selection that is more resource-efficient and or more general than existing approaches. The core of the approach is to make selection of an abstraction part of the learning agent's decision-making process by augmenting the agent's action space with internal actions that select the abstraction it uses. We prove that under certain conditions this approach results in a derived MDP whose solution yields both the optimal abstraction for the original MDP and the optimal policy under that abstraction. We examine our approach in three domains of increasing complexity: contextual bandit problems, episodic MDPs, and general MDPs with context-specific structure. © 2013 Wiley Periodicals, Inc. In this paper we consider the problem of computing an ϵ-optimal policy of a discounted Markov Decision Process (DMDP) provided we can only access its transition function through a generative sampling model that given any state-action pair samples from the transition function in O(1) time. Given such a DMDP with states , actions , discount factor γ∈(0,1), and rewards in range [0,1] we provide an algorithm which computes an ϵ-optimal policy with probability 1−δ where both the run time spent and number of sample taken is upper bounded by O[ | || | (1−γ)3ϵ2 log( | || | (1−γ)δϵ )log( 1 (1−γ)ϵ )] . For fixed values of ϵ, this improves upon the previous best known bounds by a factor of (1−γ)−1 and matches the sample complexity lower bounds proved in azar2013minimax up to logarithmic factors. We also extend our method to computing ϵ-optimal policies for finite-horizon MDP with a generative model and provide a nearly matching sample complexity lower bound. We propose a new approach to the problem of searching a space of policies for a Markov decision process (MDP) or a partially observable Markov decision process (POMDP), given a model. Our approach is based on the following observation: Any (PO)MDP can be transformed into an "equivalent" POMDP in which all state transitions (given the current state and action) are deterministic. This reduces the general problem of policy search to one in which we need only consider POMDPs with deterministic transitions. We give a natural way of estimating the value of all policies in these transformed POMDPs. Policy search is then simply performed by searching for a policy with high estimated value. We also establish conditions under which our value estimates will be good, recovering theoretical results similar to those of Kearns, Mansour and Ng (1999), but with "sample complexity" bounds that have only a polynomial rather than exponential dependence on the horizon time. Our method applies to arbitrary POMDPs, including ones with infinite state and action spaces. We also present empirical results for our approach on a small discrete problem, and on a complex continuous state continuous action problem involving learning to ride a bicycle. We present an algorithm called Optimistic Linear Programming (OLP) for learning to optimize average reward in an irreducible but otherwise unknown Markov decision process (MDP). OLP uses its experience so far to estimate the MDP. It chooses actions by optimistically maximizing estimated future rewards over a set of next-state transition probabilities that are close to the estimates, a computation that corresponds to solving linear programs. We show that the total expected reward obtained by OLP up to time T is within C(P) log T of the reward obtained by the optimal policy, where C(P) is an explicit, MDP-dependent constant. OLP is closely related to an algorithm proposed by Burnetas and Katehakis with four key differences: OLP is simpler, it does not require knowledge of the supports of transition probabilities, the proof of the regret bound is simpler, but our regret bound is a constant factor larger than the regret of their algorithm. OLP is also similar in flavor to an algorithm recently proposed by Auer and Ortner. But OLP is simpler and its regret bound has a better dependence on the size of the MDP. We consider the problem of imitation learning where the examples, demonstrated by an expert, cover only a small part of a large state space. Inverse Reinforcement Learning (IRL) provides an ecient tool for generalizing the demonstration, based on the assumption that the expert is optimally acting in a Markov Decision Process (MDP). Most of the past work on IRL requires that a (near)optimal policy can be computed for dierent reward functions. However, this requirement can hardly be satised in systems with a large, or continuous, state space. In this paper, we propose a model-free IRL algorithm, where the relative entropy between the empirical distribution of the state-action trajectories under a baseline policy and their distribution under the learned policy is minimized by stochastic gradient descent. We compare this new approach to well-known IRL algorithms using learned MDP models. Empirical results on simulated car racing, gridworld and ball-in-a-cup problems show that our approach is able to learn good policies from a small number of demonstrations. In this paper, we propose an exact solution method to generate fair policies in Multiobjective Markov Decision Processes (MMDPs). MMDPs consider n immediate reward functions, representing either individual payoffs in a multiagent problem or rewards with respect to different objectives. In this context, we focus on the determination of a policy that fairly shares regrets among agents or objectives, the regret being defined on each dimension as the opportunity loss with respect to optimal expected rewards. To this end, we propose to minimize the ordered weighted average of regrets (OWR). The OWR criterion indeed extends the minimax regret, relaxing egalitarianism for a milder notion of fairness. After showing that OWR-optimality is state-dependent and that the Bellman principle does not hold for OWR-optimal policies, we propose a linear programming reformulation of the problem. We also provide experimental results showing the efficiency of our approach. The partially observable Markov decision process (POMDP) provides a principled general framework for planning under uncertainty, but solving POMDPs optimally is computationally intractable, due to the "curse of dimensionality" and the "curse of history". To overcome these challenges, we introduce the Determinized Sparse Partially Observable Tree (DESPOT), a sparse approximation of the standard belief tree, for online planning under uncertainty. A DESPOT focuses online planning on a set of randomly sampled scenarios and compactly captures the "execution" of all policies under these scenarios. We show that the best policy obtained from a DESPOT is near-optimal, with a regret bound that depends on the representation size of the optimal policy. Leveraging this result, we give an anytime online planning algorithm, which searches a DESPOT for a policy that optimizes a regularized objective function. Regularization balances the estimated value of a policy under the sampled scenarios and the policy size, thus avoiding overfitting. The algorithm demonstrates strong experimental results, compared with some of the best online POMDP algorithms available. It has also been incorporated into an autonomous driving system for real-time vehicle control. The source code for the algorithm is available online.