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.
Our reward model is based on (stochastic) linear bandits which has seen a vast number of studies since @cite_9 . Our work leverages ideas from @cite_12 for both the reward model and intermediate results for the GLM case. Extending the linear bandit problem, @cite_6 first proposed the GLM bandit problem and provided a no-regret algorithm in the OFU paradigm. This paper builds up on the ideas from @cite_10 and @cite_15 who also studied the logistic bandit and GLM Bernoulli bandit case. We provide the extension of both methodologies to a generic multinomial GLM setting. Consequently, our bounds also show a dependence on the strong convexity parameter @math which was recently shown to be unavoidable by @cite_0 for proper learning in online logistic regression. Further, @cite_11 have recently proposed an optimal regret bound for the GLM Bernoulli bandit problem by using phases for removing the statistical dependence between parameter estimates and covariates. However, the proposed algorithm is inefficient and obtaining such an optimal algorithm is still an open problem.
@cite_9 @cite_6 @cite_0 @cite_15 @cite_10 @cite_12 @cite_11
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We consider structured multi-armed bandit problems based on the Generalized Linear Model (GLM) framework of statistics. For these bandits, we propose a new algorithm, called GLM-UCB. We derive finite time, high probability bounds on the regret of the algorithm, extending previous analyses developed for the linear bandits to the non-linear case. The analysis highlights a key difficulty in generalizing linear bandit algorithms to the non-linear case, which is solved in GLM-UCB by focusing on the reward space rather than on the parameter space. Moreover, as the actual effectiveness of current parameterized bandit algorithms is often poor in practice, we provide a tuning method based on asymptotic arguments, which leads to significantly better practical performance. We present two numerical experiments on real-world data that illustrate the potential of the GLM-UCB approach. 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. In this paper, we study a special bandit setting of online stochastic linear optimization, where only one-bit of information is revealed to the learner at each round. This problem has found many applications including online advertisement and online recommendation. We assume the binary feedback is a random variable generated from the logit model, and aim to minimize the regret defined by the unknown linear function. Although the existing method for generalized linear bandit can be applied to our problem, the high computational cost makes it impractical for real-world problems. To address this challenge, we develop an efficient online learning algorithm by exploiting particular structures of the observation model. Specifically, we adopt online Newton step to estimate the unknown parameter and derive a tight confidence region based on the exponential concavity of the logistic loss. Our analysis shows that the proposed algorithm achieves a regret bound of @math , which matches the optimal result of stochastic linear bandits. We consider a linear stochastic bandit problem where the dimension @math of the unknown parameter @math is larger than the sampling budget @math . In such cases, it is in general impossible to derive sub-linear regret bounds since usual linear bandit algorithms have a regret in @math . In this paper we assume that @math is @math sparse, i.e. has at most @math non-zero components, and that the space of arms is the unit ball for the @math norm. We combine ideas from Compressed Sensing and Bandit Theory and derive algorithms with regret bounds in @math . Adversarial bandit problems studied by [4] are multi-armed bandit problems in which no stochastic assumption is made on the nature of the process generating the rewards for actions. In this paper, we extend their theories to the case where k(≥1) distinct actions are selected at each time step. As algorithms to solve our problem, we analyze an extension of Exp3 [4] and an application of a bandit online linear optimization algorithm [1] in addition to two existing algorithms (Exp3, ComBand [5]) in terms of time and space efficiency and the regret for the best fixed action set. The extension of Exp3, called Exp3.M, performs best with respect to all the measures: it runs in O(K(log k+1)) time and O(K) space, and suffers at most O(√kTK log(K k)) regret, where K is the number of possible actions and T is the number of iterations. The upper bound of the regret we proved for Exp3.M is an extension of that proved by for Exp3. Learning linear predictors with the logistic loss---both in stochastic and online settings---is a fundamental task in machine learning and statistics, with direct connections to classification and boosting. Existing "fast rates" for this setting exhibit exponential dependence on the predictor norm, and (2014) showed that this is unfortunately unimprovable. Starting with the simple observation that the logistic loss is @math -mixable, we design a new efficient improper learning algorithm for online logistic regression that circumvents the aforementioned lower bound with a regret bound exhibiting a doubly-exponential improvement in dependence on the predictor norm. This provides a positive resolution to a variant of the COLT 2012 open problem of McMahan and Streeter (2012) when improper learning is allowed. This improvement is obtained both in the online setting and, with some extra work, in the batch statistical setting with high probability. We also show that the improved dependence on predictor norm is near-optimal. Leveraging this improved dependency on the predictor norm yields the following applications: (a) we give algorithms for online bandit multiclass learning with the logistic loss with an @math relative mistake bound across essentially all parameter ranges, thus providing a solution to the COLT 2009 open problem of Abernethy and Rakhlin (2009), and (b) we give an adaptive algorithm for online multiclass boosting with optimal sample complexity, thus partially resolving an open problem of (2015) and (2017). Finally, we give information-theoretic bounds on the optimal rates for improper logistic regression with general function classes, thereby characterizing the extent to which our improvement for linear classes extends to other parametric and even nonparametric settings. We consider a multi-armed bandit problem in a setting where each arm produces a noisy reward realization which depends on an observable random covariate. As opposed to the traditional static multi-armed bandit problem, this setting allows for dynamically changing rewards that better describe applications where side information is available. We adopt a nonparametric model where the expected rewards are smooth functions of the covariate and where the hardness of the problem is captured by a margin parameter. To maximize the expected cumulative reward, we introduce a policy called Adaptively Binned Successive Elimination (abse) that adaptively decomposes the global problem into suitably “localized” static bandit problems. This policy constructs an adaptive partition using a variant of the Successive Elimination (se) policy. Our results include sharper regret bounds for the se policy in a static bandit problem and minimax optimal regret bounds for the abse policy in the dynamic problem. We study contextual bandits with budget and time constraints, referred to as constrained contextual bandits.The time and budget constraints significantly complicate the exploration and exploitation tradeoff because they introduce complex coupling among contexts over time.Such coupling effects make it difficult to obtain oracle solutions that assume known statistics of bandits. To gain insight, we first study unit-cost systems with known context distribution. When the expected rewards are known, we develop an approximation of the oracle, referred to Adaptive-Linear-Programming (ALP), which achieves near-optimality and only requires the ordering of expected rewards. With these highly desirable features, we then combine ALP with the upper-confidence-bound (UCB) method in the general case where the expected rewards are unknown a priori . We show that the proposed UCB-ALP algorithm achieves logarithmic regret except for certain boundary cases. Further, we design algorithms and obtain similar regret analysis results for more general systems with unknown context distribution and heterogeneous costs. To the best of our knowledge, this is the first work that shows how to achieve logarithmic regret in constrained contextual bandits. Moreover, this work also sheds light on the study of computationally efficient algorithms for general constrained contextual bandits. We tackle the problem of online reward maximisation over a large finite set of actions described by their contexts. We focus on the case when the number of actions is too big to sample all of them even once. However we assume that we have access to the similarities between actions' contexts and that the expected reward is an arbitrary linear function of the contexts' images in the related reproducing kernel Hilbert space (RKHS). We propose KernelUCB, a kernelised UCB algorithm, and give a cumulative regret bound through a frequentist analysis. For contextual bandits, the related algorithm GP-UCB turns out to be a special case of our algorithm, and our finite-time analysis improves the regret bound of GP-UCB for the agnostic case, both in the terms of the kernel-dependent quantity and the RKHS norm of the reward function. Moreover, for the linear kernel, our regret bound matches the lower bound for contextual linear bandits. The question of the optimality of Thompson Sampling for solving the stochastic multi-armed bandit problem had been open since 1933. In this paper we answer it positively for the case of Bernoulli rewards by providing the first finite-time analysis that matches the asymptotic rate given in the Lai and Robbins lower bound for the cumulative regret. The proof is accompanied by a numerical comparison with other optimal policies, experiments that have been lacking in the literature until now for the Bernoulli case. This work deals with four classical prediction settings, namely full information, bandit, label efficient and bandit label efficient as well as four different notions of regret: pseudo-regret, expected regret, high probability regret and tracking the best expert regret. We introduce a new forecaster, INF (Implicitly Normalized Forecaster) based on an arbitrary function ψ for which we propose a unified analysis of its pseudo-regret in the four games we consider. In particular, for ψ(x)=exp(η x) + γ K, INF reduces to the classical exponentially weighted average forecaster and our analysis of the pseudo-regret recovers known results while for the expected regret we slightly tighten the bounds. On the other hand with ψ(x)=(η -x)q + γ K, which defines a new forecaster, we are able to remove the extraneous logarithmic factor in the pseudo-regret bounds for bandits games, and thus fill in a long open gap in the characterization of the minimax rate for the pseudo-regret in the bandit game. We also provide high probability bounds depending on the cumulative reward of the optimal action. Finally, we consider the stochastic bandit game, and prove that an appropriate modification of the upper confidence bound policy UCB1 (, 2002a) achieves the distribution-free optimal rate while still having a distribution-dependent rate logarithmic in the number of plays. We introduce the study of fairness in multi-armed bandit problems. Our fairness definition can be interpreted as demanding that given a pool of applicants (say, for college admission or mortgages), a worse applicant is never favored over a better one, despite a learning algorithm's uncertainty over the true payoffs. We prove results of two types. First, in the important special case of the classic stochastic bandits problem (i.e., in which there are no contexts), we provide a provably fair algorithm based on "chained" confidence intervals, and provide a cumulative regret bound with a cubic dependence on the number of arms. We further show that any fair algorithm must have such a dependence. When combined with regret bounds for standard non-fair algorithms such as UCB, this proves a strong separation between fair and unfair learning, which extends to the general contextual case. In the general contextual case, we prove a tight connection between fairness and the KWIK (Knows What It Knows) learning model: a KWIK algorithm for a class of functions can be transformed into a provably fair contextual bandit algorithm, and conversely any fair contextual bandit algorithm can be transformed into a KWIK learning algorithm. This tight connection allows us to provide a provably fair algorithm for the linear contextual bandit problem with a polynomial dependence on the dimension, and to show (for a different class of functions) a worst-case exponential gap in regret between fair and non-fair learning algorithms 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. 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. The multi-armed bandit problem is a popular model for studying exploration exploitation trade-off in sequential decision problems. Many algorithms are now available for this well-studied problem. One of the earliest algorithms, given by W. R. Thompson, dates back to 1933. This algorithm, referred to as Thompson Sampling, is a natural Bayesian algorithm. The basic idea is to choose an arm to play according to its probability of being the best arm. Thompson Sampling algorithm has experimentally been shown to be close to optimal. In addition, it is efficient to implement and exhibits several desirable properties such as small regret for delayed feedback. However, theoretical understanding of this algorithm was quite limited. In this paper, for the first time, we show that Thompson Sampling algorithm achieves logarithmic expected regret for the multi-armed bandit problem. More precisely, for the two-armed bandit problem, the expected regret in time @math is @math . And, for the @math -armed bandit problem, the expected regret in time @math is @math . Our bounds are optimal but for the dependence on @math and the constant factors in big-Oh. We consider stochastic multi-armed bandits where the expected reward is a unimodal function over partially ordered arms. This important class of problems has been recently investigated in (Cope 2009, Yu 2011). The set of arms is either discrete, in which case arms correspond to the vertices of a finite graph whose structure represents similarity in rewards, or continuous, in which case arms belong to a bounded interval. For discrete unimodal bandits, we derive asymptotic lower bounds for the regret achieved under any algorithm, and propose OSUB, an algorithm whose regret matches this lower bound. Our algorithm optimally exploits the unimodal structure of the problem, and surprisingly, its asymptotic regret does not depend on the number of arms. We also provide a regret upper bound for OSUB in non-stationary environments where the expected rewards smoothly evolve over time. The analytical results are supported by numerical experiments showing that OSUB performs significantly better than the state-of-the-art algorithms. For continuous sets of arms, we provide a brief discussion. We show that combining an appropriate discretization of the set of arms with the UCB algorithm yields an order-optimal regret, and in practice, outperforms recently proposed algorithms designed to exploit the unimodal structure. We study a variant of the stochastic @math -armed bandit problem, which we call "bandits with delayed, aggregated anonymous feedback". In this problem, when the player pulls an arm, a reward is generated, however it is not immediately observed. Instead, at the end of each round the player observes only the sum of a number of previously generated rewards which happen to arrive in the given round. The rewards are stochastically delayed and due to the aggregated nature of the observations, the information of which arm led to a particular reward is lost. The question is what is the cost of the information loss due to this delayed, aggregated anonymous feedback? Previous works have studied bandits with stochastic, non-anonymous delays and found that the regret increases only by an additive factor relating to the expected delay. In this paper, we show that this additive regret increase can be maintained in the harder delayed, aggregated anonymous feedback setting when the expected delay (or a bound on it) is known. We provide an algorithm that matches the worst case regret of the non-anonymous problem exactly when the delays are bounded, and up to logarithmic factors or an additive variance term for unbounded delays. The probability that a user will click a search result depends both on its relevance and its position on the results page. The position based model explains this behavior by ascribing to every item an attraction probability, and to every position an examination probability. To be clicked, a result must be both attractive and examined. The probabilities of an item-position pair being clicked thus form the entries of a rank- @math matrix. We propose the learning problem of a Bernoulli rank- @math bandit where at each step, the learning agent chooses a pair of row and column arms, and receives the product of their Bernoulli-distributed values as a reward. This is a special case of the stochastic rank- @math bandit problem considered in recent work that proposed an elimination based algorithm Rank1Elim, and showed that Rank1Elim's regret scales linearly with the number of rows and columns on "benign" instances. These are the instances where the minimum of the average row and column rewards @math is bounded away from zero. The issue with Rank1Elim is that it fails to be competitive with straightforward bandit strategies as @math . In this paper we propose Rank1ElimKL which simply replaces the (crude) confidence intervals of Rank1Elim with confidence intervals based on Kullback-Leibler (KL) divergences, and with the help of a novel result concerning the scaling of KL divergences we prove that with this change, our algorithm will be competitive no matter the value of @math . Experiments with synthetic data confirm that on benign instances the performance of Rank1ElimKL is significantly better than that of even Rank1Elim, while experiments with models derived from real data confirm that the improvements are significant across the board, regardless of whether the data is benign or not. 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 version of the contextual bandits problem. We prove a high probability regret bound of O(d2 e√T1+e) in time T for any 0 < e < 1, where d is the dimension of each context vector and e is a parameter used by the algorithm. Our results provide the first theoretical guarantees for the contextual version of Thompson Sampling, and are close to the lower bound of Ω(d√T) for this problem. This essentially solves a COLT open problem of Chapelle and Li [COLT 2012].