In the strategic multi-armed bandit setting, when arms possess perfect information about the player’s behavior, they can establish an equilibrium where: 1. they retain almost all of their value, 2. they leave the player with a substantial (linear) regret. This study illustrates that, even if complete information is not publicly available to all arms but is shared among them, it is possible to achieve a similar equilibrium. The primary challenge lies in designing a communication protocol that incentivizes the arms to communicate truthfully.

Motivated by online display advertising, this work considers repeated second-price auctions, where agents sample their value from an unknown distribution with cumulative distribution function $F$. In each auction $t$, a decision-maker bound by limited observations selects $n_t$ agents from a coalition of $N$ to compete for a prize with $p$ other agents, aiming to maximize the cumulative reward of the coalition across all auctions. The problem is framed as an $N$-armed structured bandit, each number of player sent being an arm $n$, with expected reward $r(n)$ fully characterized by $F$ and $p+n$. We present two algorithms, Local-Greedy (LG) and Greedy-Grid (GG), both achieving *constant* problem-dependent regret. This relies on three key ingredients: **1.** an estimator of $r(n)$ from feedback collected from any arm $k$, **2.** concentration bounds of these estimates for $k$ within an estimation neighborhood of $n$ and **3.** the unimodality property of $r$ under standard assumptions on $F$. Additionally, GG exhibits problem-independent guarantees on top of best problem-dependent guarantees. However, by avoiding to rely on confidence intervals, LG practically outperforms GG, as well as standard unimodal bandit algorithms such as OSUB or multi-armed bandit algorithms.

We examine multi-armed bandit problems featuring strategic arms under debt-free reporting. In this context, each arm is characterized by a bounded support reward distribution and strategically aims to maximize its own utility by retaining a portion of the observed reward, potentially disclosing only a fraction of it to the player. This scenario unfolds as a game over $T$ rounds, leading to a competition of objectives between the player, aiming to minimize regret, and the arms, motivated by the desire to maximize their individual utilities. To address these dynamics, we propose an algorithm that establishes an equilibrium wherein each arm behaves truthfully and discloses as much of its rewards as possible. Utilizing this algorithm, the player can attain the second-highest average (true) reward among arms, with a cumulative regret bounded by $O(\log(T)/\Delta)$ (problem-dependent) or $O(\sqrt{T\log(T)})$ (worst-case).

As the issue of robustness in AI systems becomes vital, statistical learning techniques that are reliable even in presence of partly contaminated data have to be developed. Preference data, in the form of (complete) rankings in the simplest situations, are no exception and the demand for appropriate concepts and tools is all the more pressing given that technologies fed by or producing this type of data ($\textit{e.g.}$ search engines, recommending systems) are now massively deployed. However, the lack of vector space structure for the set of rankings ($\textit{i.e.}$ the symmetric group $\mathfrak{S}_n$) and the complex nature of statistics considered in ranking data analysis make the formulation of robustness objectives in this domain challenging. In this paper, we introduce notions of robustness, together with dedicated statistical methods, for $\textit{Consensus Ranking}$ the flagship problem in ranking data analysis, aiming at summarizing a probability distribution on $\mathfrak{S}_n$ by a $\textit{median}$ ranking. Precisely, we propose specific extensions of the popular concept of *breakdown point*, tailored to consensus ranking, and address the related computational issues. Beyond the theoretical contributions, the relevance of the approach proposed is supported by an experimental study.

Logistic Bandits have recently undergone careful scrutiny by virtue of their combined theoretical and practical relevance. This research effort delivered statistically efficient algorithms, improving the regret of previous strategies by exponentially large factors. Such algorithms are however strikingly costly as they require $\Omega(t)$ operations at each round. On the other hand, a different line of research focused on computational efficiency ($\mathcal{O}(1)$ per-round cost), but at the cost of letting go of the aforementioned exponential improvements. Obtaining the best of both world is unfortunately not a matter of marrying both approaches. Instead we introduce a new learning procedure for Logistic Bandits. It yields confidence sets which sufficient statistics can be easily maintained online without sacrificing statistical tightness. Combined with efficient planning mechanisms we design fast algorithms which regret performance still match the problem-dependent lower-bound of Abeille et al (2021). To the best of our knowledge, those are the first Logistic Bandit algorithms that simultaneously enjoy statistical and computational efficiency.

Logistic Bandits have recently attracted substantial attention, by providing an uncluttered yet challenging framework for understanding the impact of non-linearity in parametrized bandits. It was shown by Faury et al. (2020) that the learning-theoretic difficulties of Logistic Bandits can be embodied by a large (sometimes prohibitively) problem-dependent constant $\kappa$, characterizing the magnitude of the reward’s non-linearity. In this paper we introduce an algorithm for which we provide a refined analysis. This allows for a better characterization of the effect of non-linearity and yields improved problem-dependent guarantees. In most favorable cases this leads to a regret upper-bound scaling as $\tilde{\mathcal{O}}(d\sqrt{T/\kappa})$, which dramatically improves over the $\tilde{\mathcal{O}}(d\sqrt{T}+\kappa)$ state-of-the-art guarantees. We prove that this rate is \emph{minimax-optimal} by deriving a $\Omega(d\sqrt{T/\kappa})$ problem-dependent lower-bound. Our analysis identifies two regimes (permanent and transitory) of the regret, which ultimately re-conciliates (Faury et al., 2020) with the Bayesian approach of Dong et al. (2019). In contrast to previous works, we find that in the permanent regime non-linearity can dramatically ease the exploration-exploitation trade-off. While it also impacts the length of the transitory phase in a problem-dependent fashion, we show that this impact is mild in most reasonable configurations.

In this note we identify several mistakes appearing in the existing literature on non-stationary parametric bandits. More precisely, we study Generalized Linear Bandits (GLBs) in drifting environments, where the level of non-stationarity is characterized by a general metric known as the variation-budget. Existing methods to solve such problems typically involve forgetting mechanisms, which allow for a fine balance between the learning and tracking requirements of the problem. We uncover two significant mistakes in their theoretical analysis. The first arises when bounding the tracking error suffered by forgetting mechanisms. The second emerges when considering non-linear reward models, which requires extra care to balance the learning and tracking guarantees. We introduce a geometrical assumption on the arm set, sufficient to overcome the aforementioned technical gaps and recover minimax-optimality. We also share preliminary attempts at fixing those gaps under general configurations. Unfortunately, our solution yields degraded rates (w.r.t to the horizon), which raises new open questions regarding the optimality of forgetting mechanisms in non-stationary parametric bandits.

Finding an optimal matching in a weighted graph is a standard combinatorial problem. We consider its semi-bandit version where either a pair or a full matching is sampled sequentially. We prove that it is possible to leverage a rank-1 assumption on the adjacency matrix to reduce the sample complexity and the regret of off-the-shelf algorithms up to reaching a linear dependency in the number of vertices (up to to poly-log terms).

We consider the practical and classical setting where the seller is using an exploration stage to learn the value distributions of the bidders before running a revenue-maximizing auction in a exploitation phase. In this two-stage process, we exhibit practical, simple and robust strategies with large utility uplifts for the bidders. We quantify precisely the seller revenue against non-discounted buyers, complementing recent studies that had focused on impatient/heavily discounted buyers. We also prove the robustness of these shading strategies to sample approximation error of the seller, to bidder’s approximation error of the competition and to possible change of the mechanisms.

In display advertising, a small group of sellers and bidders face each other in up to $10^{12}$ auctions a day. In this context, revenue maximisation via monopoly price learning is a high-value problem for sellers. By nature, these auctions are online and produce a very high frequency stream of data. This results in a computational strain that requires algorithms be real-time. Unfortunately, existing methods inherited from the batch setting suffer $O(\sqrt{t})$ time/memory complexity at each update, prohibiting their use. In this paper, we provide the first algorithm for online learning of monopoly prices in online auctions whose update is constant in time and memory.

The generalized linear bandit framework has attracted a lot of attention in recent years by extending the well-understood linear setting and allowing to model richer reward structures. It notably covers the logistic model, widely used when rewards are binary. For logistic bandits, the frequentist regret guarantees of existing algorithms are $\tilde{\mathcal{O}}(\kappa \sqrt{T})$, where $\kappa$ is a problem-dependent constant. Unfortunately, $\kappa$ can be arbitrarily large as it scales exponentially with the size of the decision set. This may lead to significantly loose regret bounds and poor empirical performance. In this work, we study the logistic bandit with a focus on the prohibitive dependencies introduced by $\kappa$. We propose a new optimistic algorithm based on a finer examination of the non-linearities of the reward function. We show that it enjoys a $\tilde{\mathcal{O}}(\sqrt{T})$ regret with no dependency in $\kappa$, but for a second order term. Our analysis is based on a new tail-inequality for self-normalized martingales, of independent interest.

Ranking and selection tasks appear in different contexts with specific desiderata, such as the maximizaton of average relevance on the top of the list, the requirement of diverse rankings, or, relatedly, the focus on providing at least one relevant items to as many users as possible. This paper addresses the question of which of these tasks are asymptotically solved by sorting by decreasing order of expected utility, for some suitable notion of utility, or, equivalently, \emph{when is square loss regression consistent for ranking \emph{via} score-and-sort?}. We provide an answer to this question in the form of a structural characterization of ranking losses for which a suitable regression is consistent. This result has two fundamental corollaries. First, whenever there exists a consistent approach based on convex risk minimization, there also is a consistent approach based on regression. Second, when regression is not consistent, there are data distributions for which consistent surrogate approaches necessarily have non-trivial local minima, and optimal scoring function are necessarily discontinuous, even when the underlying data distribution is regular. In addition to providing a better understanding of surrogate approaches for ranking, these results illustrate the intrinsic difficulty of solving general ranking problems with the score-and-sort approach.

State of the art online learning procedures focus either on selecting the best alternative (“best arm identification”) or on minimizing the cost (the “regret”). We merge these two objectives by providing the theoretical analysis of cost minimizing algorithms that are also $\delta$-PAC (with a proven guaranteed bound on the decision time), hence fulfilling at the same time regret minimization and best arm identification. This analysis sheds light on the common observation that ill-callibrated UCB-algorithms minimize regret while still identifying quickly the best arm. We also extend these results to the non-iid case faced by many practitioners. This provides a technique to make cost versus decision time compromise when doing adaptive tests with applications ranging from website A/B testing to clinical trials.

We address the problem of algorithmic fairness: ensuring that the outcome of a classifier is not biased towards certain values of sensitive variables such as age, race or gender. As common fairness metrics can be expressed as measures of (conditional) independence between variables, we propose to use the Rényi maximum correlation coefficient to generalize fairness measurement to continuous variables. We exploit Witsenhausen’s characterization of the Rényi correlation coefficient to propose a differentiable implementation linked to $f$-divergences. This allows us to generalize fairness-aware learning to continuous variables by using a penalty that upper bounds this coefficient. Theses allows fairness to be extented to variables such as mixed ethnic groups or financial status without thresholds effects. This penalty can be estimated on mini-batches allowing to use deep nets. Experiments show favorable comparisons to state of the art on binary variables and prove the ability to protect continuous ones

Online A/B testing evaluates the impact of a new technology by running it in a real production environment and testing its performance on a subset of the users of the platform. It is a well-known practice to run a preliminary offline evaluation on historical data to iterate faster on new ideas, and to detect poor policies in order to avoid losing money or breaking the system. For such offline evaluations, we are interested in methods that can compute offline an estimate of the potential uplift of performance generated by a new technology. Offline performance can be measured using estimators known as counterfactual or off-policy estimators. Traditional counterfactual estimators, such as capped importance sampling or normalised importance sampling, exhibit unsatisfying bias-variance compromises when experimenting on personalized product recommendation systems. To overcome this issue, we model the bias incurred by these estimators rather than bound it in the worst case, which leads us to propose a new counterfactual estimator. We provide a benchmark of the different estimators showing their correlation with business metrics observed by running online A/B tests on a large-scale commercial recommender system.

We study surrogate losses for learning to rank, in a framework where the rankings are induced by scores and the task is to learn the scoring function. We focus on the calibration of surrogate losses with respect to a ranking evaluation metric, where the calibration is equivalent to the guarantee that near-optimal values of the sur- rogate risk imply near-optimal values of the risk defined by the evaluation metric. We prove that if a surrogate loss is a convex function of the scores, then it is not calibrated with respect to two evaluation metrics widely used for search engine evaluation, namely the Average Precision and the Expected Reciprocal Rank. We also show that such convex surrogate losses cannot be calibrated with respect to the Pairwise Disagreement, an evaluation metric used when learning from pair- wise preferences. Our results cast lights on the intrinsic difficulty of some ranking problems, as well as on the limitations of learning-to-rank algorithms based on the minimization of a convex surrogate risk.