We consider the recently proposed reinforcement learning (RL) framework of Contextual Markov Decision Processes (CMDP), where the agent has a sequence of episodic interactions with tabular environments chosen from a possibly infinite set. The parameters of these environments depend on a context vector that is available to the agent at the start of each episode. 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 models (GLMs). The proposed algorithm GL-ORL relies on efficient online updates and is also memory efficient. Our analysis of the algorithm gives new results in the logit link case and improves previous bounds in the linear case. Our algorithm uses efficient Online Newton Step updates to build confidence sets. Moreover, for any strongly convex link function, we also show a generic conversion from any online no-regret algorithm to confidence sets.
@cite_8 first proposed a no-regret online learning algorithm for average reward infinite horizon MDPs, and the problem has been extensively studied afterwards. More recently, there has been an increased focus on fixed horizon problems where the gap between the upper and lower bounds has been effectively closed. @cite_3 and @cite_4 , both provide optimal regret guarantees ( @math ) for repeated interaction with a finite episodic MDP by using empirical Bernstein based bonus and a fine-grained analysis. Both these methods do not directly carry over to our setting and therefore, improving the dependence on @math is an interesting problem for future work which would require a novel approach. Contextualized version of MDPs are relatively unexplored with @cite_5 and @cite_4 being the only works to the best of our knowledge.
@cite_5 @cite_4 @cite_3 @cite_8
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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. We consider the problem of provably optimal exploration in reinforcement learning for finite horizon MDPs. We show that an optimistic modification to value iteration achieves a regret bound of @math where @math is the time horizon, @math the number of states, @math the number of actions and @math the number of time-steps. This result improves over the best previous known bound @math achieved by the UCRL2 algorithm of , 2010. The key significance of our new results is that when @math and @math , it leads to a regret of @math that matches the established lower bound of @math up to a logarithmic factor. Our analysis contains two key insights. We use careful application of concentration inequalities to the optimal value function as a whole, rather than to the transitions probabilities (to improve scaling in @math ), and we define Bernstein-based "exploration bonuses" that use the empirical variance of the estimated values at the next states (to improve scaling in @math ). 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 @math , where @math is the number of rounds. To best of our knowledge, this is the first regret bound for this setting. 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. 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. We study the problem of online learning in finite episodic Markov decision processes (MDPs) where the loss function is allowed to change between episodes. The natural performance measure in this learning problem is the regret defined as the difference between the total loss of the best stationary policy and the total loss suffered by the learner. We assume that the learner is given access to a finite action space A and the state space X has a layered structure with L layers, so that state transitions are only possible between consecutive layers. We describe a variant of the recently proposed Relative Entropy Policy Search algorithm and show that its regret after T episodes is 2√L|X||A|T log (|X||A| L) in the bandit setting and 2L √T log(|X||A| L) in the full information setting, given that the learner has perfect knowledge of the transition probabilities of the underlying MDP. These guarantees largely improve previously known results under much milder assumptions and cannot be significantly improved under general assumptions. Abstract Documents in computer-readable form can be used to provide information about other documents, i.e. those they cite. To do this efficiently requires procedures for computer recognition of citing statements. This is not easy, especially for multi-sentence citing statements. Computer recognition procedures have been developed which are accurate to the following extent: 73 of the words in statements selected by computer procedures as being citing statements are words which are correctly attributable to the corresponding documents. The retrieval effectiveness of computer-recognized citing statements was tested in the following way. First, for eight retrieval requests in inorganiic chemistry, average recall by search of Chemical Abstracts Service indexing and Chemical Abstracts abstract text words was found to be 50 . Words from citing statements referring to the papers to be retrieved were then added to the index terms and abstract words as additional access points, and searching was repeated. Average recall increased to 70 . Only words from citing statements published within a year of the cited papers were used. The retrieval effect of citing statement words alone (published within a year) without index or abstract terms was the following: average recall was 40 . When just the words of the titles of the cited papers were added to those citing statement words, average recall increased to 50 . For undiscounted reinforcement learning in Markov decision processes (MDPs) we consider the total regret of a learning algorithm with respect to an optimal policy. In order to describe the transition structure of an MDP we propose a new parameter: An MDP has diameter D if for any pair of states s,s' there is a policy which moves from s to s' in at most D steps (on average). We present a reinforcement learning algorithm with total regret O(DS√AT) after T steps for any unknown MDP with S states, A actions per state, and diameter D. A corresponding lower bound of Ω(√DSAT) on the total regret of any learning algorithm is given as well. These results are complemented by a sample complexity bound on the number of suboptimal steps taken by our algorithm. This bound can be used to achieve a (gap-dependent) regret bound that is logarithmic in T. Finally, we also consider a setting where the MDP is allowed to change a fixed number of l times. We present a modification of our algorithm that is able to deal with this setting and show a regret bound of O(l1 3T2 3DS√A). We present a general framework for applying machine-learning algorithms to the verification of Markov decision processes (MDPs). The primary goal of these techniques is to improve performance by avoiding an exhaustive exploration of the state space. Our framework focuses on probabilistic reachability, which is a core property for verification, and is illustrated through two distinct instantiations. The first assumes that full knowledge of the MDP is available, and performs a heuristic-driven partial exploration of the model, yielding precise lower and upper bounds on the required probability. The second tackles the case where we may only sample the MDP, and yields probabilistic guarantees, again in terms of both the lower and upper bounds, which provides efficient stopping criteria for the approximation. The latter is the first extension of statistical model checking for unbounded properties in MDPs. In contrast with other related techniques, our approach is not restricted to time-bounded (finite-horizon) or discounted properties, nor does it assume any particular properties of the MDP. We also show how our methods extend to LTL objectives. We present experimental results showing the performance of our framework on several examples. We consider a Markov decision process (MDP) setting in which the reward function is allowed to change after each time step (possibly in an adversarial manner), yet the dynamics remain fixed. Similar to the experts setting, we address the question of how well an agent can do when compared to the reward achieved under the best stationary policy over time. We provide efficient algorithms, which have regret bounds with no dependence on the size of state space. Instead, these bounds depend only on a certain horizon time of the process and logarithmically on the number of actions. We show for the first time, to our knowledge, that it is possible to reconcile in online learning in zero-sum games two seemingly contradictory objectives: vanishing time-average regret and non-vanishing step sizes. This phenomenon, that we coin fast and furious" learning in games, sets a new benchmark about what is possible both in max-min optimization as well as in multi-agent systems. Our analysis does not depend on introducing a carefully tailored dynamic. Instead we focus on the most well studied online dynamic, gradient descent. Similarly, we focus on the simplest textbook class of games, two-agent two-strategy zero-sum games, such as Matching Pennies. Even for this simplest of benchmarks the best known bound for total regret, prior to our work, was the trivial one of @math , which is immediately applicable even to a non-learning agent. Based on a tight understanding of the geometry of the non-equilibrating trajectories in the dual space we prove a regret bound of @math matching the well known optimal bound for adaptive step sizes in the online setting. This guarantee holds for all fixed step-sizes without having to know the time horizon in advance and adapt the fixed step-size accordingly. As a corollary, we establish that even with fixed learning rates the time-average of mixed strategies, utilities converge to their exact Nash equilibrium values. Contextual bandits are widely used in Internet services from news recommendation to advertising, and to Web search. Generalized linear models (logistical regression in particular) have demonstrated stronger performance than linear models in many applications where rewards are binary. However, most theoretical analyses on contextual bandits so far are on linear bandits. In this work, we propose an upper confidence bound based algorithm for generalized linear contextual bandits, which achieves an @math regret over @math rounds with @math dimensional feature vectors. This regret matches the minimax lower bound, up to logarithmic terms, and improves on the best previous result by a @math factor, assuming the number of arms is fixed. A key component in our analysis is to establish a new, sharp finite-sample confidence bound for maximum-likelihood estimates in generalized linear models, which may be of independent interest. We also analyze a simpler upper confidence bound algorithm, which is useful in practice, and prove it to have optimal regret for certain cases. 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 study the stochastic combinatorial multi-armed bandit (CMAB) framework that allows a general nonlinear reward function, whose expected value may not depend only on the means of the input random variables but possibly on the entire distributions of these variables. Our framework enables a much larger class of reward functions such as the @math function and nonlinear utility functions. Existing techniques relying on accurate estimations of the means of random variables, such as the upper confidence bound (UCB) technique, do not work directly on these functions. We propose a new algorithm called stochastically dominant confidence bound (SDCB), which estimates the distributions of underlying random variables and their stochastically dominant confidence bounds. We prove that SDCB can achieve @math distribution-dependent regret and @math distribution-independent regret, where @math is the time horizon. We apply our results to the @math -MAX problem and expected utility maximization problems. In particular, for @math -MAX, we provide the first polynomial-time approximation scheme (PTAS) for its offline problem, and give the first @math bound on the @math -approximation regret of its online problem, for any @math . In a document network such as a citation network of scientific documents, web-logs, etc., the content produced by authors exhibits their interest in certain topics. In addition some authors influence other authors' interests. In this work, we propose to model the influence of cited authors along with the interests of citing authors. Moreover, we hypothesize that apart from the citations present in documents, the context surrounding the citation mention provides extra topical information about the cited authors. However, associating terms in the context to the cited authors remains an open problem. We propose novel document generation schemes that incorporate the context while simultaneously modeling the interests of citing authors and influence of the cited authors. Our experiments show significant improvements over baseline models for various evaluation criteria such as link prediction between document and cited author, and quantitatively explaining unseen text. A citing sentence is one that appears in a scientific article and cites previous work. Citing sentences have been studied and used in many applications. For example, they have been used in scientific paper summarization, automatic survey generation, paraphrase identification, and citation function classification. Citing sentences that cite multiple papers are common in scientific writing. This observation should be taken into consideration when using citing sentences in applications. For instance, when a citing sentence is used in a summary of a scientific paper, only the fragments of the sentence that are relevant to the summarized paper should be included in the summary. In this paper, we present and compare three different approaches for identifying the fragments of a citing sentence that are related to a given target reference. Our methods are: word classification, sequence labeling, and segment classification. Our experiments show that segment classification achieves the best results. We consider the problems of learning the optimal action-value function and the optimal policy in discounted-reward Markov decision processes (MDPs). We prove new PAC bounds on the sample-complexity of two well-known model-based reinforcement learning (RL) algorithms in the presence of a generative model of the MDP: value iteration and policy iteration. The first result indicates that for an MDP with N state-action pairs and the discount factor ??[0,1) only O(Nlog(N ?) ((1??)3 ? 2)) state-transition samples are required to find an ?-optimal estimation of the action-value function with the probability (w.p.) 1??. Further, we prove that, for small values of ?, an order of O(Nlog(N ?) ((1??)3 ? 2)) samples is required to find an ?-optimal policy w.p. 1??. We also prove a matching lower bound of ?(Nlog(N ?) ((1??)3 ? 2)) on the sample complexity of estimating the optimal action-value function with ? accuracy. To the best of our knowledge, this is the first minimax result on the sample complexity of RL: the upper bounds match the lower bound in terms of N, ?, ? and 1 (1??) up to a constant factor. Also, both our lower bound and upper bound improve on the state-of-the-art in terms of their dependence on 1 (1??). Thompson Sampling is one of the oldest heuristics for multi-armed bandit problems. It is a randomized algorithm based on Bayesian ideas, and has recently generated significant interest after several studies demonstrated it to have better empirical performance compared to the state-of-the-art methods. However, many questions regarding its theoretical performance remained open. In this paper, we design and analyze a generalization of Thompson Sampling algorithm for the stochastic contextual multi-armed bandit problem with linear payoff functions, when the contexts are provided by an adaptive adversary. This is among the most important and widely studied versions of the contextual bandits problem. We provide the first theoretical guarantees for the contextual version of Thompson Sampling. We prove a high probability regret bound of @math (or @math ), which is the best regret bound achieved by any computationally efficient algorithm available for this problem in the current literature, and is within a factor of @math (or @math ) of the information-theoretic lower bound for this problem. We consider the problem of learning in single-player and multiplayer multiarmed bandit models. Bandit problems are classes of online learning problems that capture exploration versus exploitation tradeoffs. In a multiarmed bandit model, players can pick among many arms, and each play of an arm generates an i.i.d. reward from an unknown distribution. The objective is to design a policy that maximizes the expected reward over a time horizon for a single player setting and the sum of expected rewards for the multiplayer setting. In the multiplayer setting, arms may give different rewards to different players. There is no separate channel for coordination among the players. Any attempt at communication is costly and adds to regret. We propose two decentralizable policies, @math ( @math - @math ) and @math - @math , that can be used in both single player and multiplayer settings. These policies are shown to yield expected regret that grows at most as O( @math ). It is well known that @math is the lower bound on the rate of growth of regret even in a centralized case. The proposed algorithms improve on prior work where regret grew at O( @math ). More fundamentally, these policies address the question of additional cost incurred in decentralized online learning, suggesting that there is at most an @math -factor cost in terms of order of regret. This solves a problem of relevance in many domains and had been open for a while. We consider the problem of learning in single-player and multiplayer multiarmed bandit models. Bandit problems are classes of online learning problems that capture exploration versus exploitation tradeoffs. In a multiarmed bandit model, players can pick among many arms, and each play of an arm generates an i.i.d. reward from an unknown distribution. The objective is to design a policy that maximizes the expected reward over a time horizon for a single-player setting and the sum of expected rewards for the multiplayer setting. In the multiplayer setting, arms may give different rewards to different players. There is no separate channel for coordination among the players. Any attempt at communication is costly and adds to regret. We propose two decentralizable policies, @math ( @math - @math ) and @math - @math , that can be used in both single-player and multiplayer settings. These policies are shown to yield expected regret that grows, at most, as @math (and @math under some assumption). It is well known that @math is the lower bound on the rate of growth of regret even in a centralized case. The proposed algorithms improve on prior work where regret grew at @math . More fundamentally, these policies address the question of additional cost incurred in decentralized online learning, suggesting that there is, at most, an @math -factor cost in terms of the order of regret. This solves a problem of relevance in many domains and had been open for a while.