Mobile Context-Aware Recommender Systems can be naturally modelled as an exploration exploitation trade-off (exr exp) problem, where the system has to choose between maximizing its expected rewards dealing with its current knowledge (exploitation) and learning more about the unknown user's preferences to improve its knowledge (exploration). This problem has been addressed by the reinforcement learning community but they do not consider the risk level of the current user's situation, where it may be dangerous to recommend items the user may not desire in her current situation if the risk level is high. We introduce in this paper an algorithm named R-UCB that considers the risk level of the user's situation to adaptively balance between exr and exp. The detailed analysis of the experimental results reveals several important discoveries in the exr exp behaviour.
In @cite_12 , the authors propose to solve bandit problem in dynamic environment by combining the UCB with the @math -greedy strategy and they dynamically update the @math exploration value. At each iteration, they run a sampling procedure to select a new @math from a finite set of candidates. The probabilities associated to the candidates are uniformly initialized and updated with the Exponentiated Gradient (EG) @cite_16 . This updating rule increases the probability of a candidate @math if it leads to a user's click. Compared to both @math -beginning and @math -decreasing, this technique gives better results. @cite_9 @cite_22 describe a smart way to balance exr exp, but do not consider the user's situation and its associated risk during the recommendation.
@cite_9 @cite_16 @cite_22 @cite_12
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In sequential decision problems in an unknown environment, the decision maker often faces a dilemma over whether to explore to discover more about the environment, or to exploit current knowledge. We address the exploration-exploitation dilemma in a general setting encompassing both standard and contextualised bandit problems. The contextual bandit problem has recently resurfaced in attempts to maximise click-through rates in web based applications, a task with significant commercial interest. In this article we consider an approach of Thompson (1933) which makes use of samples from the posterior distributions for the instantaneous value of each action. We extend the approach by introducing a new algorithm, Optimistic Bayesian Sampling (OBS), in which the probability of playing an action increases with the uncertainty in the estimate of the action value. This results in better directed exploratory behaviour. We prove that, under unrestrictive assumptions, both approaches result in optimal behaviour with respect to the average reward criterion of Yang and Zhu (2002). We implement OBS and measure its performance in simulated Bernoulli bandit and linear regression domains, and also when tested with the task of personalised news article recommendation on a Yahoo! Front Page Today Module data set. We find that OBS performs competitively when compared to recently proposed benchmark algorithms and outperforms Thompson's method throughout. 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. We improve the theoretical analysis and empirical performance of algorithms for the stochastic multi-armed bandit problem and the linear stochastic multi-armed bandit problem. In particular, we show that a simple modification of Auer's UCB algorithm (Auer, 2002) achieves with high probability constant regret. More importantly, we modify and, consequently, improve the analysis of the algorithm for the for linear stochastic bandit problem studied by Auer (2002), (2008), Rusmevichientong and Tsitsiklis (2010), (2010). Our modification improves the regret bound by a logarithmic factor, though experiments show a vast improvement. In both cases, the improvement stems from the construction of smaller confidence sets. For their construction we use a novel tail inequality for vector-valued martingales. We study exploration in Multi-Armed Bandits in a setting where @math players collaborate in order to identify an @math -optimal arm. Our motivation comes from recent employment of bandit algorithms in computationally intensive, large-scale applications. Our results demonstrate a non-trivial tradeoff between the number of arm pulls required by each of the players, and the amount of communication between them. In particular, our main result shows that by allowing the @math players to communicate only once, they are able to learn @math times faster than a single player. That is, distributing learning to @math players gives rise to a factor @math parallel speed-up. We complement this result with a lower bound showing this is in general the best possible. On the other extreme, we present an algorithm that achieves the ideal factor @math speed-up in learning performance, with communication only logarithmic in @math . 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 . Multi-armed bandit problems are considered as a paradigm of the trade-off between exploring the environment to find profitable actions and exploiting what is already known. In the stationary case, the distributions of the rewards do not change in time, Upper-Confidence Bound (UCB) policies have been shown to be rate optimal. A challenging variant of the MABP is the non-stationary bandit problem where the gambler must decide which arm to play while facing the possibility of a changing environment. In this paper, we consider the situation where the distributions of rewards remain constant over epochs and change at unknown time instants. We analyze two algorithms: the discounted UCB and the sliding-window UCB. We establish for these two algorithms an upper-bound for the expected regret by upper-bounding the expectation of the number of times a suboptimal arm is played. For that purpose, we derive a Hoeffding type inequality for self normalized deviations with a random number of summands. We establish a lower-bound for the regret in presence of abrupt changes in the arms reward distributions. We show that the discounted UCB and the sliding-window UCB both match the lower-bound up to a logarithmic factor. 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. 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. Many problems, such as cognitive radio, parameter control of a scanning tunnelling microscope or internet advertisement, can be modelled as non-stationary bandit problems where the distributions of rewards changes abruptly at unknown time instants. In this paper, we analyze two algorithms designed for solving this issue: discounted UCB (D-UCB) and sliding-window UCB (SW-UCB). We establish an upperbound for the expected regret by upper-bounding the expectation of the number of times suboptimal arms are played. The proof relies on an interesting Hoeffding type inequality for self normalized deviations with a random number of summands. We establish a lower-bound for the regret in presence of abrupt changes in the arms reward distributions. We show that the discounted UCB and the sliding-window UCB both match the lower-bound up to a logarithmic factor. Numerical simulations show that D-UCB and SW-UCB perform significantly better than existing soft-max methods like EXP3.S. We show how an ensemble of @math -functions can be leveraged for more effective exploration in deep reinforcement learning. We build on well established algorithms from the bandit setting, and adapt them to the @math -learning setting. We propose an exploration strategy based on upper-confidence bounds (UCB). Our experiments show significant gains on the Atari benchmark. 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. The multi-armed bandit (MAB) problem features the classical tradeoff between exploration and exploitation. The input specifies several stochastic arms which evolve with each pull, and the goal is to maximize the expected reward after a fixed budget of pulls. The celebrated work of , surveyed in [8], presumes a condition on the arms called the martingale assumption. In [9], A. obtained an LP-based 1 48-approximation for the problem with the martingale assumption removed. We improve the algorithm to a 4 27-approximation, with simpler analysis. Our algorithm also generalizes to the case of MAB superprocesses with (stochastic) multi-period actions. This generalization captures the framework introduced by Guha and Munagala in [11], and yields new results for their budgeted learning problems. Also, we obtain a (1 2 -- e)-approximation for the variant of MAB where preemption (playing an arm, switching to another arm, then coming back to the first arm) is not allowed. This contains the stochastic knapsack problem of Dean, Goemans, and Vondrak in [6] with correlated rewards, where we are given a knapsack of fixed size, and a set of jobs each with a joint distribution for its size and reward. The actual size and reward of a job can only be discovered in real-time as it is being scheduled, and the objective is to maximize expected reward before the knapsack size is exhausted. Our (1 2 -- e)-approximation improves the 1 16 and 1 8 approximations in [9] for correlated stochastic knapsack with cancellation and no cancellation, respectively, providing to our knowledge the first tight algorithm for these problems that matches the integrality gap of 2. We sample probabilities from an exponential-sized dynamic programming solution, whose existence is guaranteed by an LP projection argument. We hope this technique can also be applied to other dynamic programming problems which can be projected down onto a small LP. 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 adversarial convex bandit problem and we build the first @math -time algorithm with @math -regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves @math -regret, and we show that a simple variant of this algorithm can be run in @math -time per step at the cost of an additional @math factor in the regret. These results improve upon the @math -regret and @math -time result of the first two authors, and the @math -regret and @math -time result of Hazan and Li. Furthermore we conjecture that another variant of the algorithm could achieve @math -regret, and moreover that this regret is unimprovable (the current best lower bound being @math and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order @math . 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. We develop a learning principle and an efficient algorithm for batch learning from logged bandit feedback. This learning setting is ubiquitous in online systems (e.g., ad placement, web search, recommendation), where an algorithm makes a prediction (e.g., ad ranking) for a given input (e.g., query) and observes bandit feedback (e.g., user clicks on presented ads). We first address the counterfactual nature of the learning problem (, 2013) through propensity scoring. Next, we prove generalization error bounds that account for the variance of the propensity-weighted empirical risk estimator. In analogy to the Structural Risk Minimization principle of Wapnik and Tscherwonenkis (1979), these constructive bounds give rise to the Counterfactual Risk Minimization (CRM) principle. We show how CRM can be used to derive a new learning method--called Policy Optimizer for Exponential Models (POEM)--for learning stochastic linear rules for structured output prediction. We present a decomposition of the POEM objective that enables efficient stochastic gradient optimization. The effectiveness and efficiency of POEM is evaluated on several simulated multi-label classification problems, as well as on a real-world information retrieval problem. The empirical results show that the CRM objective implemented in POEM provides improved robustness and generalization performance compared to the state-of-the-art. We develop a learning principle and an efficient algorithm for batch learning from logged bandit feedback. This learning setting is ubiquitous in online systems (e.g., ad placement, web search, recommendation), where an algorithm makes a prediction (e.g., ad ranking) for a given input (e.g., query) and observes bandit feedback (e.g., user clicks on presented ads). We first address the counterfactual nature of the learning problem through propensity scoring. Next, we prove generalization error bounds that account for the variance of the propensity-weighted empirical risk estimator. These constructive bounds give rise to the Counterfactual Risk Minimization (CRM) principle. We show how CRM can be used to derive a new learning method -- called Policy Optimizer for Exponential Models (POEM) -- for learning stochastic linear rules for structured output prediction. We present a decomposition of the POEM objective that enables efficient stochastic gradient optimization. POEM is evaluated on several multi-label classification problems showing substantially improved robustness and generalization performance compared to the state-of-the-art. 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 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 max() 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 O(log T) distribution-dependent regret and O(√T) distribution-independent regret, where T is the time horizon. We apply our results to the K-MAX problem and expected utility maximization problems. In particular, for K-MAX, we provide the first polynomial-time approximation scheme (PTAS) for its offline problem, and give the first O(√T) bound on the (1 — e)-approximation regret of its online problem, for any e > 0. We propose the contextual combinatorial cascading bandits, a combinatorial online learning game, where at each time step a learning agent is given a set of contextual information, then selects a list of items, and observes stochastic outcomes of a prefix in the selected items by some stopping criterion. In online recommendation, the stopping criterion might be the first item a user selects; in network routing, the stopping criterion might be the first edge blocked in a path. We consider position discounts in the list order, so that the agent's reward is discounted depending on the position where the stopping criterion is met. We design a UCB-type algorithm, C3-UCB, for this problem, prove an n-step regret bound O(√n) in the general setting, and give finer analysis for two special cases. Our work generalizes existing studies in several directions, including contextual information, position discounts, and a more general cascading bandit model. Experiments on synthetic and real datasets demonstrate the advantage of involving contextual information and position discounts.