We consider a variant of the multi-armed bandit model, which we call multi-armed bandit problem with known trend, where the gambler knows the shape of the reward function of each arm but not its distribution. This new problem is motivated by different online problems like active learning, music and interface recommendation applications, where when an arm is sampled by the model the received reward change according to a known trend. By adapting the standard multi-armed bandit algorithm UCB1 to take advantage of this setting, we propose the new algorithm named A-UCB that assumes a stochastic model. We provide upper bounds of the regret which compare favourably with the ones of UCB1. We also confirm that experimentally with different simulations
In @cite_11 author study specific classes of drifting restless bandits selected for their relevance to modelling an online website optimization process. The contribution was a feasible weighted least squares technique capable of utilizing contextual arm parameters while considering the parameter space drifting non-stationary within reasonable bounds.
@cite_11
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Restless bandits model the exploration vs. exploitation trade-off in a changing (non-stationary) world. Restless bandits have been studied in both the context of continuously-changing (drifting) and change-point (sudden) restlessness. In this work, we study specific classes of drifting restless bandits selected for their relevance to modelling an online website optimization process. The contribution in this work is a simple, feasible weighted least squares technique capable of utilizing contextual arm parameters while considering the parameter space drifting non-stationary within reasonable bounds. We produce a reference implementation, then evaluate and compare its performance in several different true world states, finding experimentally that performance is robust to time drifting factors similar to those seen in many real world cases. We consider the multi armed bandit problem in non-stationary environments. Based on the Bayesian method, we propose a variant of Thompson Sampling which can be used in both rested and restless bandit scenarios. Applying discounting to the parameters of prior distribution, we describe a way to systematically reduce the effect of past observations. Further, we derive the exact expression for the probability of picking sub-optimal arms. By increasing the exploitative value of Bayes' samples, we also provide an optimistic version of the algorithm. Extensive empirical analysis is conducted under various scenarios to validate the utility of proposed algorithms. A comparison study with various state-of-the-arm algorithms is also included. In this paper, we study the online learning problem involving rested and restless bandits, in both a centralized and a decentralized setting. In a centralized setting, the system consists of a single player user and a set of K finite-state discrete-time Markov chains (arms) with unknown state spaces (rewards) and statistics. The objective of the player is to decide in each step which M of the K arms to play over a sequence of trials so as to maximize its long-term reward. In a decentralized setting, multiple uncoordinated players each makes its own decision on which arm to play in a step, and if two or more players select the same arm simultaneously, a collision results and none of the players selecting that arm gets a reward. The objective of each player again is to maximize its long-term reward. We first show that logarithmic regret algorithms exist both for the centralized rested and restless bandit problems. For the decentralized setting, we propose an algorithm with logarithmic regret with respect to the optimal centralized arm allocation. Numerical results and extensive discussion are also provided to highlight insights obtained from this study. In the classic Bayesian restless multi-armed bandit (RMAB) problem, there are N arms, with rewards on all arms evolving at each time as Markov chains with known parameters. A player seeks to activate K ≥ 1 arms at each time in order to maximize the expected total reward obtained over multiple plays. RMAB is a challenging problem that is known to be PSPACE-hard in general. We consider in this work the even harder non-Bayesian RMAB, in which the parameters of the Markov chain are assumed to be unknown a priori. We develop an original approach to this problem that is applicable when the corresponding Bayesian problem has the structure that, depending on the known parameter values, the optimal solution is one of a prescribed finite set of policies. In such settings, we propose to learn the optimal policy for the non-Bayesian RMAB by employing a suitable meta-policy which treats each policy from this finite set as an arm in a different non-Bayesian multi-armed bandit problem for which a single-arm selection policy is optimal. We demonstrate this approach by developing a novel sensing policy for opportunistic spectrum access over unknown dynamic channels. We prove that our policy achieves near-logarithmic regret (the difference in expected reward compared to a model-aware genie), which leads to the same average reward that can be achieved by the optimal policy under a known model. This is the first such result in the literature for a non-Bayesian RMAB. Motivated by online recommendation and advertising systems, we consider a causal model for stochastic contextual bandits with a latent low-dimensional confounder. In our model, there are @math observed contexts and @math arms of the bandit. The observed context influences the reward obtained through a latent confounder variable with cardinality @math ( @math ). The arm choice and the latent confounder causally determines the reward while the observed context is correlated with the confounder. Under this model, the @math mean reward matrix @math (for each context in @math and each arm in @math ) factorizes into non-negative factors @math ( @math ) and @math ( @math ). This insight enables us to propose an @math -greedy NMF-Bandit algorithm that designs a sequence of interventions (selecting specific arms), that achieves a balance between learning this low-dimensional structure and selecting the best arm to minimize regret. Our algorithm achieves a regret of @math at time @math , as compared to @math for conventional contextual bandits, assuming a constant gap between the best arm and the rest for each context. These guarantees are obtained under mild sufficiency conditions on the factors that are weaker versions of the well-known Statistical RIP condition. We further propose a class of generative models that satisfy our sufficient conditions, and derive a lower bound of @math . These are the first regret guarantees for online matrix completion with bandit feedback, when the rank is greater than one. We further compare the performance of our algorithm with the state of the art, on synthetic and real world data-sets. Unlabeled samples can be intelligently selected for labeling to minimize classification error. In many real-world applications, a large number of unlabeled samples arrive in a streaming manner, making it impossible to maintain all the data in a candidate pool. In this work, we focus on binary classification problems and study selective labeling in data streams where a decision is required on each sample sequentially. We consider the unbiasedness property in the sampling process, and design optimal instrumental distributions to minimize the variance in the stochastic process. Meanwhile, Bayesian linear classifiers with weighted maximum likelihood are optimized online to estimate parameters. In empirical evaluation, we collect a data stream of user-generated comments on a commercial news portal in 30 consecutive days, and carry out offline evaluation to compare various sampling strategies, including unbiased active learning, biased variants, and random sampling. Experimental results verify the usefulness of online active learning, especially in the non-stationary situation with concept drift. We consider a two–armed bandit problem which involves sequential sampling from two non-homogeneous populations. The response in each is determined by a random covariate vector and a vector of parameters whose values are not known a priori. The goal is to maximize cumulative expected reward. We study this problem in a minimax setting, and develop rate-optimal polices that combine myopic action based on least squares estimates with a suitable “forced sampling” strategy. It is shown that the regret grows logarithmically in the time horizon n and no policy can achieve a slower growth rate over all feasible problem instances. In this setting of linear response bandits, the identity of the sub-optimal action changes with the values of the covariate vector, and the optimal policy is subject to sampling from the inferior population at a rate that grows like n. The restless bandit problem is one of the most well-studied generalizations of the celebrated stochastic multi-armed bandit (MAB) problem in decision theory. In its ultimate generality, the restless bandit problem is known to be PSPACE-Hard to approximate to any nontrivial factor, and little progress has been made on this problem despite its significance in modeling activity allocation under uncertainty. In this article, we consider the Feedback MAB problem, where the reward obtained by playing each of n independent arms varies according to an underlying on off Markov process whose exact state is only revealed when the arm is played. The goal is to design a policy for playing the arms in order to maximize the infinite horizon time average expected reward. This problem is also an instance of a Partially Observable Markov Decision Process (POMDP), and is widely studied in wireless scheduling and unmanned aerial vehicle (UAV) routing. Unlike the stochastic MAB problem, the Feedback MAB problem does not admit to greedy index-based optimal policies. We develop a novel duality-based algorithmic technique that yields a surprisingly simple and intuitive (2+e)-approximate greedy policy to this problem. We show that both in terms of approximation factor and computational efficiency, our policy is closely related to the Whittle index, which is widely used for its simplicity and efficiency of computation. Subsequently we define a multi-state generalization, that we term Monotone bandits, which remains subclass of the restless bandit problem. We show that our policy remains a 2-approximation in this setting, and further, our technique is robust enough to incorporate various side-constraints such as blocking plays, switching costs, and even models where determining the state of an arm is a separate operation from playing it. Our technique is also of independent interest for other restless bandit problems, and we provide an example in nonpreemptive machine replenishment. Interestingly, in this case, our policy provides a constant factor guarantee, whereas the Whittle index is provably polynomially worse. By presenting the first O(1) approximations for nontrivial instances of restless bandits as well as of POMDPs, our work initiates the study of approximation algorithms in both these contexts. We consider the restless multi-armed bandit (RMAB) problem with unknown dynamics. At each time, a player chooses K out of N (N > K) arms to play. The state of each arm determines the reward when the arm is played and transits according to Markovian rules no matter the arm is engaged or passive. The Markovian dynamics of the arms are unknown to the player. The objective is to maximize the long-term expected reward by designing an optimal arm selection policy. The performance of a policy is measured by regret, defined as the reward loss with respect to the case where the player knows which K arms are the most rewarding and always plays these K best arms. We construct a policy, referred to as Restless Upper Confidence Bound (RUCB), that achieves a regret with logarithmic order of time when an arbitrary nontrivial bound on certain system parameters is known. When no knowledge about the system is available, we extend the RUCB policy to achieve a regret arbitrarily close to the logarithmic order. In both cases, the system achieves the maximum average reward offered by the K best arms. Potential applications of these results include cognitive radio networks, opportunistic communications in unknown fading environments, and financial investment. We consider a class of restless multi-armed bandit (RMAB) problems with unknown arm dynamics. At each time, a player chooses an arm out of @math arms to play, referred to as an active arm, and receives a random reward from a finite set of reward states. The reward state of the active arm transits according to an unknown Markovian dynamic. The reward state of passive arms (which are not chosen to play at time t) evolves according to an arbitrary unknown random process. The objective is an arm-selection policy that minimizes the regret, defined as the reward loss with respect to a player that always plays the most rewarding arm. This class of RMAB problems has been studied recently in the context of communication networks and financial investment applications. We develop a strategy that selects arms to be played in a consecutive manner in which the selection sequencing rules are adaptively updated controlled by the current sample reward means, referred to as Adaptive Sequencing Rules (ASR) algorithm. By designing judiciously the adaptive sequencing rules of the chosen arms, we show that ASR algorithm achieves a logarithmic regret order with time and a finite-sample bound on the regret is established. Although existing methods have shown a logarithmic regret order with time in this RMAB setting, the theoretical analysis presents significant improvement in the regret scaling with respect to the system parameters under ASR. Extensive simulation results support the theoretical study and demonstrate strong performance of the algorithm as compared to existing methods. This paper studies a generalized class of restless multi-armed bandits with hidden states and allow cumulative feedback, as opposed to the conventional instantaneous feedback. We call them lazy restless bandits (LRBs) as the events of decision making are sparser than the events of state transition. Hence, feedback after each decision event is the cumulative effect of the following state transition events. The states of arms are hidden from the decision maker and rewards for actions are state dependent. The decision maker needs to choose one arm in each decision interval, such that the long-term cumulative reward is maximized. As the states are hidden, the decision maker maintains and updates its belief about them. It is shown that LRBs admit an optimal policy which has threshold structure in belief space. The Whittle-index policy for solving the LRB problem is analyzed; indexability of LRBs is shown. Further, the closed-form index expressions are provided for two sets of special cases; for more general cases, an algorithm for index computation is provided. An extensive simulation study is presented; Whittle-index, modified Whittle-index, and myopic policies are compared. The Lagrangian relaxation of the problem provides an upper bound on the optimal value function; it is used to assess the degree of sub-optimality various policies. The capabilities of electric loads participating in load curtailment programs are often unknown until the loads have been told to curtail (i.e., deployed) and observed. In programs in which payments are made each time a load is deployed, we aim to pick the "best" loads to deploy in each time step. Our choice is a tradeoff between exploration and exploitation, i.e., curtailing poorly characterized loads in order to better characterize them in the hope of benefiting in the future versus curtailing well-characterized loads so that we benefit now. We formulate this problem as a multi-armed restless bandit problem with controlled bandits. In contrast to past work that has assumed all load parameters are known allowing the use of optimization approaches, we assume the parameters of the controlled system are unknown and develop an online learning approach. Our problem has two features not commonly addressed in the bandit literature: the arms processes evolve according to different probabilistic laws depending on the control, and the reward feedback observed by the decision-maker is the total realized curtailment, not the curtailment of each load. We develop an adaptive demand response learning algorithm and an extended version that works with aggregate feedback, both aimed at approximating the Whittle index policy. We show numerically that the regret of our algorithms with respect to the Whittle index policy is of logarithmic order in time, and significantly outperforms standard learning algorithms like UCB1. I. INTRODUCTION The class of restless bandits as proposed by Whittle [21] have long been known to be intractable. This paper presents an optimality result which extends that of Weber and Weiss [19] for restless bandits to a more general setting in which individual bandits have multiple levels of activation but are subject to an overall resource constraint. The contribution is motivated by recent works of [10, 11] who discuss the performance of index heuristics for resource allocation in such systems. Hitherto, index heuristics have been shown, under a condition of full indexability, to be optimal for a natural Lagrangian relaxation of such problems in which resource is purchased rather than constrained. We find that under key assumptions about the nature of solutions to a deterministic differential equation that the index heuristics above are asymptotically optimal in a sense described by Whittle. We then demonstrate that these assumptions always hold for three-state bandits. We show that several well-known optimization problems related to the optimal control of queues are provably intractable-independently of any unproven conjecture such as P â NP. In particular, we show that several versions of the problem of optimally controlling a simple network of queues with simple arrival and service distributions and multiple customer classes is complete for exponential time. This is perhaps the first such intractability result for a well-known optimization problem. We also show that the restless bandit problem the generalization of the multi-armed bandit problem to the case in which the unselected processes are not quiescent is complete for polynomial space. We consider stochastic multi-armed bandit problems with complex actions over a set of basic arms, where the decision maker plays a complex action rather than a basic arm in each round. The reward of the complex action is some function of the basic arms' rewards, and the feedback observed may not necessarily be the reward perarm. For instance, when the complex actions are subsets of the arms, we may only observe the maximum reward over the chosen subset. Thus, feedback across complex actions may be coupled due to the nature of the reward function. We prove a frequentist regret bound for Thompson sampling in a very general setting involving parameter, action and observation spaces and a likelihood function over them. The bound holds for discretely-supported priors over the parameter space without additional structural properties such as closed-form posteriors, conjugate prior structure or independence across arms. The regret bound scales logarithmically with time but, more importantly, with an improved constant that non-trivially captures the coupling across complex actions due to the structure of the rewards. As applications, we derive improved regret bounds for classes of complex bandit problems involving selecting subsets of arms, including the first nontrivial regret bounds for nonlinear MAX reward feedback from subsets. Using particle filters for computing posterior distributions which lack an explicit closed-form, we present numerical results for the performance of Thompson sampling for subset-selection and job scheduling problems. Multi-armed bandit (MAB) is a class of online learning problems where a learning agent aims to maximize its expected cumulative reward while repeatedly selecting to pull arms with unknown reward distributions. In this paper, we consider a scenario in which the arms' reward distributions may change in a piecewise-stationary fashion at unknown time steps. By connecting change-detection techniques with classic UCB algorithms, we motivate and propose a learning algorithm called M-UCB, which can detect and adapt to changes, for the considered scenario. We also establish an @math regret bound for M-UCB, where @math is the number of time steps, @math is the number of arms, and @math is the number of stationary segments. and @math is the gap between the expected rewards of the optimal and best suboptimal arms. Comparison with the best available lower bound shows that M-UCB is nearly optimal in @math up to a logarithmic factor. We also compare M-UCB with state-of-the-art algorithms in a numerical experiment based on a public Yahoo! dataset. In this experiment, M-UCB achieves about @math regret reduction with respect to the best performing state-of-the-art algorithm. 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 Abstract Nowadays huge volumes of data are produced in the form of fast streams, which are further affected by non-stationary phenomena. The resulting lack of stationarity in the distribution of the produced data calls for efficient and scalable algorithms for online analysis capable of adapting to such changes (concept drift). The online learning field has lately turned its focus on this challenging scenario, by designing incremental learning algorithms that avoid becoming obsolete after a concept drift occurs. Despite the noted activity in the literature, a need for new efficient and scalable algorithms that adapt to the drift still prevails as a research topic deserving further effort. Surprisingly, Spiking Neural Networks, one of the major exponents of the third generation of artificial neural networks, have not been thoroughly studied as an online learning approach, even though they are naturally suited to easily and quickly adapting to changing environments. This work covers this research gap by adapting Spiking Neural Networks to meet the processing requirements that online learning scenarios impose. In particular the work focuses on limiting the size of the neuron repository and making the most of this limited size by resorting to data reduction techniques. Experiments with synthetic and real data sets are discussed, leading to the empirically validated assertion that, by virtue of a tailored exploitation of the neuron repository, Spiking Neural Networks adapt better to drifts, obtaining higher accuracy scores than naive versions of Spiking Neural Networks for online learning environments. Multi-armed bandit problems are the most basic examples of sequential decision problems with an exploration-exploitation trade-off. This is the balance between staying with the option that gave highest payoffs in the past and exploring new options that might give higher payoffs in the future. Although the study of bandit problems dates back to the Thirties, exploration-exploitation trade-offs arise in several modern applications, such as ad placement, website optimization, and packet routing. Mathematically, a multi-armed bandit is defined by the payoff process associated with each option. In this survey, we focus on two extreme cases in which the analysis of regret is particularly simple and elegant: i.i.d. payoffs and adversarial payoffs. Besides the basic setting of finitely many actions, we also analyze some of the most important variants and extensions, such as the contextual bandit model. 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].