Olivier Cappé

Aurélien Garivier, Olivier Cappé

This paper presents a finite-time analysis of the KL-UCB algorithm, an online, horizon-free index policy for stochastic bandit problems. We prove two distinct results: first, for arbitrary bounded rewards, the KL-UCB algorithm satisfies a uniformly better regret bound than UCB or UCB2; second, in the special case of Bernoulli rewards, it reaches the lower bound of Lai and Robbins. Furthermore, we show that simple adaptations of the KL-UCB algorithm are also optimal for specific classes of (possibly unbounded) rewards, including those generated from exponential families of distributions. A large-scale numerical study comparing KL-UCB with its main competitors (UCB, UCB2, UCB-Tuned, UCB-V, DMED) shows that KL-UCB is remarkably efficient and stable, including for short time horizons. KL-UCB is also the only method that always performs better than the basic UCB policy. Our regret bounds rely on deviations results of independent interest which are stated and proved in the Appendix. As a by-product, we also obtain an improved regret bound for the standard UCB algorithm.

Juliette Achddou, Olivier Cappe, Aurélien Garivier

Motivated by programmatic advertising optimization, we consider the task of sequentially allocating budget across a set of resources. At every time step, a feasible allocation is chosen and only a corresponding random return is observed. The goal is to maximize the cumulative expected sum of returns. This is a realistic model for budget allocation across subdivisions of marketing campaigns, with the objective of maximizing the number of conversions. We study direct search (also known as pattern search) methods for linearly constrained and derivative-free optimization in the presence of noise, which apply in particular to sequential budget allocation. These algorithms, which do not rely on hierarchical partitioning of the resource space, are easy to implement; they respect the operational constraints of resource allocation by avoiding evaluation outside of the feasible domain; and they are also compatible with warm start by being (approximate) descent algorithms. However, they have not yet been analyzed from the perspective of cumulative regret. We show that direct search methods achieves finite regret in the deterministic and unconstrained case. In the presence of evaluation noise and linear constraints, we propose a simple extension of direct search that achieves a regret upper-bound of the order of $T^{2/3}$. We also propose an accelerated version of the algorithm, relying on repeated sequential testing, that significantly improves the practical behavior of the approach.

Yoan Russac, Christina Katsimerou, Dennis Bohle et al.

Motivated by A/B/n testing applications, we consider a finite set of distributions (called \emph{arms}), one of which is treated as a \emph{control}. We assume that the population is stratified into homogeneous subpopulations. At every time step, a subpopulation is sampled and an arm is chosen: the resulting observation is an independent draw from the arm conditioned on the subpopulation. The quality of each arm is assessed through a weighted combination of its subpopulation means. We propose a strategy for sequentially choosing one arm per time step so as to discover as fast as possible which arms, if any, have higher weighted expectation than the control. This strategy is shown to be asymptotically optimal in the following sense: if $τ_δ$ is the first time when the strategy ensures that it is able to output the correct answer with probability at least $1-δ$, then $\mathbb{E}[τ_δ]$ grows linearly with $\log(1/δ)$ at the exact optimal rate. This rate is identified in the paper in three different settings: (1) when the experimenter does not observe the subpopulation information, (2) when the subpopulation of each sample is observed but not chosen, and (3) when the experimenter can select the subpopulation from which each response is sampled. We illustrate the efficiency of the proposed strategy with numerical simulations on synthetic and real data collected from an A/B/n experiment.

Juliette Achddou, Olivier Cappé, Aurélien Garivier

First-price auctions have largely replaced traditional bidding approaches based on Vickrey auctions in programmatic advertising. As far as learning is concerned, first-price auctions are more challenging because the optimal bidding strategy does not only depend on the value of the item but also requires some knowledge of the other bids. They have already given rise to several works in sequential learning, many of which consider models for which the value of the buyer or the opponents' maximal bid is chosen in an adversarial manner. Even in the simplest settings, this gives rise to algorithms whose regret grows as $\sqrt{T}$ with respect to the time horizon $T$. Focusing on the case where the buyer plays against a stationary stochastic environment, we show how to achieve significantly lower regret: when the opponents' maximal bid distribution is known we provide an algorithm whose regret can be as low as $\log^2(T)$; in the case where the distribution must be learnt sequentially, a generalization of this algorithm can achieve $T^{1/3+ ε}$ regret, for any $ε>0$. To obtain these results, we introduce two novel ideas that can be of interest in their own right. First, by transposing results obtained in the posted price setting, we provide conditions under which the first-price biding utility is locally quadratic around its optimum. Second, we leverage the observation that, on small sub-intervals, the concentration of the variations of the empirical distribution function may be controlled more accurately than by using the classical Dvoretzky-Kiefer-Wolfowitz inequality. Numerical simulations confirm that our algorithms converge much faster than alternatives proposed in the literature for various bid distributions, including for bids collected on an actual programmatic advertising platform.

Dorian Baudry, Yoan Russac, Olivier Cappé

There has been a recent surge of interest in nonparametric bandit algorithms based on subsampling. One drawback however of these approaches is the additional complexity required by random subsampling and the storage of the full history of rewards. Our first contribution is to show that a simple deterministic subsampling rule, proposed in the recent work of Baudry et al. (2020) under the name of ''last-block subsampling'', is asymptotically optimal in one-parameter exponential families. In addition, we prove that these guarantees also hold when limiting the algorithm memory to a polylogarithmic function of the time horizon. These findings open up new perspectives, in particular for non-stationary scenarios in which the arm distributions evolve over time. We propose a variant of the algorithm in which only the most recent observations are used for subsampling, achieving optimal regret guarantees under the assumption of a known number of abrupt changes. Extensive numerical simulations highlight the merits of this approach, particularly when the changes are not only affecting the means of the rewards.

Yoan Russac, Louis Faury, Olivier Cappé et al.

Contextual sequential decision problems with categorical or numerical observations are ubiquitous and Generalized Linear Bandits (GLB) offer a solid theoretical framework to address them. In contrast to the case of linear bandits, existing algorithms for GLB have two drawbacks undermining their applicability. First, they rely on excessively pessimistic concentration bounds due to the non-linear nature of the model. Second, they require either non-convex projection steps or burn-in phases to enforce boundedness of the estimators. Both of these issues are worsened when considering non-stationary models, in which the GLB parameter may vary with time. In this work, we focus on self-concordant GLB (which include logistic and Poisson regression) with forgetting achieved either by the use of a sliding window or exponential weights. We propose a novel confidence-based algorithm for the maximum-likehood estimator with forgetting and analyze its perfomance in abruptly changing environments. These results as well as the accompanying numerical simulations highlight the potential of the proposed approach to address non-stationarity in GLB.

Louis Filstroff, Olivier Gouvert, Cédric Févotte et al.

Non-negative matrix factorization (NMF) has become a well-established class of methods for the analysis of non-negative data. In particular, a lot of effort has been devoted to probabilistic NMF, namely estimation or inference tasks in probabilistic models describing the data, based for example on Poisson or exponential likelihoods. When dealing with time series data, several works have proposed to model the evolution of the activation coefficients as a non-negative Markov chain, most of the time in relation with the Gamma distribution, giving rise to so-called temporal NMF models. In this paper, we review four Gamma Markov chains of the NMF literature, and show that they all share the same drawback: the absence of a well-defined stationary distribution. We then introduce a fifth process, an overlooked model of the time series literature named BGAR(1), which overcomes this limitation. These temporal NMF models are then compared in a MAP framework on a prediction task, in the context of the Poisson likelihood.

Yoan Russac, Olivier Cappé, Aurélien Garivier

The statistical framework of Generalized Linear Models (GLM) can be applied to sequential problems involving categorical or ordinal rewards associated, for instance, with clicks, likes or ratings. In the example of binary rewards, logistic regression is well-known to be preferable to the use of standard linear modeling. Previous works have shown how to deal with GLMs in contextual online learning with bandit feedback when the environment is assumed to be stationary. In this paper, we relax this latter assumption and propose two upper confidence bound based algorithms that make use of either a sliding window or a discounted maximum-likelihood estimator. We provide theoretical guarantees on the behavior of these algorithms for general context sequences and in the presence of abrupt changes. These results take the form of high probability upper bounds for the dynamic regret that are of order d^2/3 G^1/3 T^2/3 , where d, T and G are respectively the dimension of the unknown parameter, the number of rounds and the number of breakpoints up to time T. The empirical performance of the algorithms is illustrated in simulated environments.

Yoan Russac, Claire Vernade, Olivier Cappé

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.

Claire Vernade, Olivier Cappé, Vianney Perchet

Online advertising and product recommendation are important domains of applications for multi-armed bandit methods. In these fields, the reward that is immediately available is most often only a proxy for the actual outcome of interest, which we refer to as a conversion. For instance, in web advertising, clicks can be observed within a few seconds after an ad display but the corresponding sale --if any-- will take hours, if not days to happen. This paper proposes and investigates a new stochas-tic multi-armed bandit model in the framework proposed by Chapelle (2014) --based on empirical studies in the field of web advertising-- in which each action may trigger a future reward that will then happen with a stochas-tic delay. We assume that the probability of conversion associated with each action is unknown while the distribution of the conversion delay is known, distinguishing between the (idealized) case where the conversion events may be observed whatever their delay and the more realistic setting in which late conversions are censored. We provide performance lower bounds as well as two simple but efficient algorithms based on the UCB and KLUCB frameworks. The latter algorithm, which is preferable when conversion rates are low, is based on a Poissonization argument, of independent interest in other settings where aggregation of Bernoulli observations with different success probabilities is required.

Paul Lagrée, Claire Vernade, Olivier Cappé

Sequentially learning to place items in multi-position displays or lists is a task that can be cast into the multiple-play semi-bandit setting. However, a major concern in this context is when the system cannot decide whether the user feedback for each item is actually exploitable. Indeed, much of the content may have been simply ignored by the user. The present work proposes to exploit available information regarding the display position bias under the so-called Position-based click model (PBM). We first discuss how this model differs from the Cascade model and its variants considered in several recent works on multiple-play bandits. We then provide a novel regret lower bound for this model as well as computationally efficient algorithms that display good empirical and theoretical performance.

Hossein Vahabi, Paul Lagrée, Claire Vernade et al.

In many web applications, a recommendation is not a single item suggested to a user but a list of possibly interesting contents that may be ranked in some contexts. The combinatorial bandit problem has been studied quite extensively these last two years and many theoretical results now exist : lower bounds on the regret or asymptotically optimal algorithms. However, because of the variety of situations that can be considered, results are designed to solve the problem for a specific reward structure such as the Cascade Model. The present work focuses on the problem of ranking items when the user is allowed to click on several items while scanning the list from top to bottom.

Emilie Kaufmann, Olivier Cappé, Aurélien Garivier

The stochastic multi-armed bandit model is a simple abstraction that has proven useful in many different contexts in statistics and machine learning. Whereas the achievable limit in terms of regret minimization is now well known, our aim is to contribute to a better understanding of the performance in terms of identifying the m best arms. We introduce generic notions of complexity for the two dominant frameworks considered in the literature: fixed-budget and fixed-confidence settings. In the fixed-confidence setting, we provide the first known distribution-dependent lower bound on the complexity that involves information-theoretic quantities and holds when m is larger than 1 under general assumptions. In the specific case of two armed-bandits, we derive refined lower bounds in both the fixed-confidence and fixed-budget settings, along with matching algorithms for Gaussian and Bernoulli bandit models. These results show in particular that the complexity of the fixed-budget setting may be smaller than the complexity of the fixed-confidence setting, contradicting the familiar behavior observed when testing fully specified alternatives. In addition, we also provide improved sequential stopping rules that have guaranteed error probabilities and shorter average running times. The proofs rely on two technical results that are of independent interest : a deviation lemma for self-normalized sums (Lemma 19) and a novel change of measure inequality for bandit models (Lemma 1).

Emilie Kaufmann, Olivier Cappé, Aurélien Garivier

A/B testing refers to the task of determining the best option among two alternatives that yield random outcomes. We provide distribution-dependent lower bounds for the performance of A/B testing that improve over the results currently available both in the fixed-confidence (or delta-PAC) and fixed-budget settings. When the distribution of the outcomes are Gaussian, we prove that the complexity of the fixed-confidence and fixed-budget settings are equivalent, and that uniform sampling of both alternatives is optimal only in the case of equal variances. In the common variance case, we also provide a stopping rule that terminates faster than existing fixed-confidence algorithms. In the case of Bernoulli distributions, we show that the complexity of fixed-budget setting is smaller than that of fixed-confidence setting and that uniform sampling of both alternatives -though not optimal- is advisable in practice when combined with an appropriate stopping criterion.