We consider off-policy evaluation in the contextual bandit setting for the purpose of obtaining a robust off-policy selection strategy, where the selection strategy is evaluated based on the value of the chosen policy in a set of proposal (target) policies. We propose a new method to compute a lower bound on the value of an arbitrary target policy given some logged data in contextual bandits for a desired coverage. The lower bound is built around the so-called Self-normalized Importance Weighting (SN) estimator. It combines the use of a semi-empirical Efron-Stein tail inequality to control the concentration and Harris' inequality to control the bias. The new approach is evaluated on a number of synthetic and real datasets and is found to be superior to its main competitors, both in terms of tightness of the confidence intervals and the quality of the policies chosen.

This note proposes a new proof and new perspectives on the so-called Elliptical Potential Lemma. This result is important in online learning, especially for linear stochastic bandits. The original proof of the result, however short and elegant, does not give much flexibility on the type of potentials considered and we believe that this new interpretation can be of interest for future research in this field.

We present a novel view on principal component analysis (PCA) as a competitive game in which each approximate eigenvector is controlled by a player whose goal is to maximize their own utility function. We analyze the properties of this PCA game and the behavior of its gradient based updates. The resulting algorithm--which combines elements from Oja's rule with a generalized Gram-Schmidt orthogonalization--is naturally decentralized and hence parallelizable through message passing. We demonstrate the scalability of the algorithm with experiments on large image datasets and neural network activations. We discuss how this new view of PCA as a differentiable game can lead to further algorithmic developments and insights.

Delayed feedback in online learning have been addressed Online recommender systems often face long delays both in the full information setting (see, e.g., Joulani et al., in receiving feedback, especially when optimizing 2013, and the references therein), and in the bandit setting for some long-term metrics. While mitigating (see, e.g., Mandel et al., 2015; Vernade et al., 2017; Cesa-the effects of delays in learning is wellunderstood Bianchi et al., 2019, and the references therein), assuming in stationary environments, the problem both stochastic and adversarial environments. The main becomes much more challenging when the takeaway message from these studies is that, for bandits, environment changes. In fact, if the timescale the impact of a constant delay D results in an extra additive of the change is comparable to the delay, it is O( DT) term in the regret in adversarial settings, or an additive impossible to learn about the environment, since O(D) term in stochastic settings. The aforementioned the available observations are already obsolete.

Significant work has been recently dedicated to the stochastic delayed bandit setting because of its relevance in applications. The applicability of existing algorithms is however restricted by the fact that strong assumptions are often made on the delay distributions, such as full observability, restrictive shape constraints, or uniformity over arms. In this work, we weaken them significantly and only assume that there is a bound on the tail of the delay. In particular, we cover the important case where the delay distributions vary across arms, and the case where the delays are heavy-tailed. Addressing these difficulties, we propose a simple but efficient UCB-based algorithm called the PatientBandits. We provide both problems-dependent and problems-independent bounds on the regret as well as performance lower bounds.

We consider a stochastic linear bandit model in which the available actions correspond to arbitrary context vectors whose associated rewards follow a non-stationary linear regression model. In this setting, the unknown regression parameter is allowed to vary in time. To address this problem, we propose D-LinUCB, a novel optimistic algorithm based on discounted linear regression, where exponential weights are used to smoothly forget the past. This involves studying the deviations of the sequential weighted least-squares estimator under generic assumptions. As a by-product, we obtain novel deviation results that can be used beyond non-stationary environments. We provide theoretical guarantees on the behavior of D-LinUCB in both slowly-varying and abruptly-changing environments. We obtain an upper bound on the dynamic regret that is of order $d^{2/3} B_T^{1/3}T^{2/3}$, where $B_T$ is a measure of non-stationarity (d and T being, respectively, dimension and horizon). This rate is known to be optimal. We also illustrate the empirical performance of D-LinUCB and compare it with recently proposed alternatives in simulated environments.

Delayed feedback is an ubiquitous problem in many industrial systems employing bandit algorithms. Most of those systems seek to optimize binary indicators as clicks. In that case, when the reward is not sent immediately, the learner cannot distinguish a negative signal from a not-yet-sent positive one: she might be waiting for a feedback that will never come. In this paper, we define and address the contextual bandit problem with delayed and censored feedback by providing a new UCB-based algorithm. In order to demonstrate its effectiveness, we provide a finite time regret analysis and an empirical evaluation that compares it against a baseline commonly used in practice.

This paper is devoted to the study of the max K-armed bandit problem, which consists in sequentially allocating resources in order to detect extreme values. Our contribution is twofold. We first significantly refine the analysis of the ExtremeHunter algorithm carried out in Carpentier and Valko (2014), and next propose an alternative approach, showing that, remarkably, Extreme Bandits can be reduced to a classical version of the bandit problem to a certain extent. Beyond the formal analysis, these two approaches are compared through numerical experiments.

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-$1$ matrix. We propose the learning problem of a Bernoulli rank-$1$ 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-$1$ 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 $\mu$ is bounded away from zero. The issue with Rank1Elim is that it fails to be competitive with straightforward bandit strategies as $\mu \rightarrow 0$. 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 $\mu$. 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 propose stochastic rank-$1$ bandits, a class of online learning problems where at each step a learning agent chooses a pair of row and column arms, and receives the product of their values as a reward. The main challenge of the problem is that the individual values of the row and column are unobserved. We assume that these values are stochastic and drawn independently. We propose a computationally-efficient algorithm for solving our problem, which we call Rank1Elim. We derive a $O((K + L) (1 / \Delta) \log n)$ upper bound on its $n$-step regret, where $K$ is the number of rows, $L$ is the number of columns, and $\Delta$ is the minimum of the row and column gaps; under the assumption that the mean row and column rewards are bounded away from zero. To the best of our knowledge, we present the first bandit algorithm that finds the maximum entry of a rank-$1$ matrix whose regret is linear in $K + L$, $1 / \Delta$, and $\log n$. We also derive a nearly matching lower bound. Finally, we evaluate Rank1Elim empirically on multiple problems. We observe that it leverages the structure of our problems and can learn near-optimal solutions even if our modeling assumptions are mildly violated.