Online Learning with Feedback Graphs: The True Shape of Regret

Sequential learning with feedback graphs is a natural extension of the multi-armed bandit problem where the problem is equipped with an underlying graph structure that provides additional information - playing an action reveals the losses of all the neighbors of the action. This problem was introduced by \citet{mannor2011} and received considerable attention in recent years. It is generally stated in the literature that the minimax regret rate for this problem is of order $\sqrt{\alpha T}$, where $\alpha$ is the independence number of the graph, and $T$ is the time horizon. However, this is proven only when the number of rounds $T$ is larger than $\alpha^3$, which poses a significant restriction for the usability of this result in large graphs. In this paper, we define a new quantity $R^*$, called the \emph{problem complexity}, and prove that the minimax regret is proportional to $R^*$ for any graph and time horizon $T$. Introducing an intricate exploration strategy, we define the \mainAlgorithm algorithm that achieves the minimax optimal regret bound and becomes the first provably optimal algorithm for this setting, even if $T$ is smaller than $\alpha^3$.

Active Ranking of Experts Based on their Performances in Many Tasks

We consider the problem of ranking n experts based on their performances on d tasks. We make a monotonicity assumption stating that for each pair of experts, one outperforms the other on all tasks. We consider the sequential setting where in each round, the learner has access to noisy evaluations of actively chosen pair of expert-task, given the information available up to the actual round. Given a confidence parameter $\delta$ $\in$ (0, 1), we provide strategies allowing to recover the correct ranking of experts and develop a bound on the total number of queries made by our algorithm that hold with probability at least 1 -- $\delta$. We show that our strategy is adaptive to the complexity of the problem (our bounds are instance dependent), and develop matching lower bounds up to a poly-logarithmic factor. Finally, we adapt our strategy to the relaxed problem of best expert identification and provide numerical simulation consistent with our theoretical results.

The price of unfairness in linear bandits with biased feedback

Artificial intelligence is increasingly used in a wide range of decision making scenarios with higher and higher stakes. At the same time, recent work has highlighted that these algorithms can be dangerously biased, and that their results often need to be corrected to avoid leading to unfair decisions. In this paper, we study the problem of sequential decision making with biased linear bandit feedback. At each round, a player selects an action described by a covariate and by a sensitive attribute. She receives a reward corresponding to the covariates of the action that she has chosen, but only observe a biased evaluation of this reward, where the bias depends on the sensitive attribute. To tackle this problem, we design a Fair Phased Elimination algorithm. We establish an upper bound on its worst-case regret, showing that it is smaller than C$\kappa$ 1/3 * log(T) 1/3 T 2/3 , where C is a numerical constant and $\kappa$ * an explicit geometrical constant characterizing the difficulty of bias estimation. The worst case regret is higher than the dT 1/2 log(T) regret rate obtained under unbiased feedback. We show that this rate cannot be improved for all instances : we obtain lower bounds on the worst-case regret for some sets of actions showing that this rate is tight up to a sub-logarithmic factor. We also obtain gap-dependent upper bounds on the regret, and establish matching lower bounds for some problem instance. Interestingly, the gap-dependent rates reveal the existence of non-trivial instances where the problem is no more difficult than its unbiased counterpart.

Problem Dependent View on Structured Thresholding Bandit Problems

We investigate the problem dependent regime in the stochastic Thresholding Bandit problem (TBP) under several shape constraints. In the TBP, the objective of the learner is to output, at the end of a sequential game, the set of arms whose means are above a given threshold. The vanilla, unstructured, case is already well studied in the literature. Taking $K$ as the number of arms, we consider the case where (i) the sequence of arm's means $(\mu_k)_{k=1}^K$ is monotonically increasing (MTBP) and (ii) the case where $(\mu_k)_{k=1}^K$ is concave (CTBP). We consider both cases in the problem dependent regime and study the probability of error - i.e. the probability to mis-classify at least one arm. In the fixed budget setting, we provide upper and lower bounds for the probability of error in both the concave and monotone settings, as well as associated algorithms. In both settings the bounds match in the problem dependent regime up to universal constants in the exponential.

Bandits with many optimal arms

We consider a stochastic bandit problem with a possibly infinite number of arms. We write $p^*$ for the proportion of optimal arms and $\Delta$ for the minimal mean-gap between optimal and sub-optimal arms. We characterize the optimal learning rates both in the cumulative regret setting, and in the best-arm identification setting in terms of the problem parameters $T$ (the budget), $p^*$ and $\Delta$. For the objective of minimizing the cumulative regret, we provide a lower bound of order $\Omega(\log(T)/(p^*\Delta))$ and a UCB-style algorithm with matching upper bound up to a factor of $\log(1/\Delta)$. Our algorithm needs $p^*$ to calibrate its parameters, and we prove that this knowledge is necessary, since adapting to $p^*$ in this setting is impossible. For best-arm identification we also provide a lower bound of order $\Omega(\exp(-cT\Delta^2p^*))$ on the probability of outputting a sub-optimal arm where $c>0$ is an absolute constant. We also provide an elimination algorithm with an upper bound matching the lower bound up to a factor of order $\log(1/\Delta)$ in the exponential, and that does not need $p^*$ or $\Delta$ as parameter.

Generalized non-stationary bandits

In this paper, we study a non-stationary stochastic bandit problem, which generalizes the switching bandit problem. On top of the switching bandit problem (\textbf{Case a}), we are interested in three concrete examples: (\textbf{b}) the means of the arms are local polynomials, (\textbf{c}) the means of the arms are locally smooth, and (\textbf{d}) the gaps of the arms have a bounded number of inflexion points and where the highest arm mean cannot vary too much in a short range. These three settings are very different, but have in common the following: (i) the number of similarly-sized level sets of the logarithm of the gaps can be controlled, and (ii) the highest mean has a limited number of abrupt changes, and otherwise has limited variations. We propose a single algorithm in this general setting, that in particular solves in an efficient and unified way the four problems (a)-(d) mentioned.

The Elliptical Potential Lemma Revisited

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.

Stochastic bandits with arm-dependent delays

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.

The Influence of Shape Constraints on the Thresholding Bandit Problem

We investigate the stochastic Thresholding Bandit problem (TBP) under several shape constraints. On top of (i) the vanilla, unstructured TBP, we consider the case where (ii) the sequence of arm's means $(\mu_k)_k$ is monotonically increasing MTBP, (iii) the case where $(\mu_k)_k$ is unimodal UTBP and (iv) the case where $(\mu_k)_k$ is concave CTBP. In the TBP problem the aim is to output, at the end of the sequential game, the set of arms whose means are above a given threshold. The regret is the highest gap between a misclassified arm and the threshold. In the fixed budget setting, we provide problem independent minimax rates for the expected regret in all settings, as well as associated algorithms. We prove that the minimax rates for the regret are (i) $\sqrt{\log(K)K/T}$ for TBP, (ii) $\sqrt{\log(K)/T}$ for MTBP, (iii) $\sqrt{K/T}$ for UTBP and (iv) $\sqrt{\log\log K/T}$ for CTBP, where $K$ is the number of arms and $T$ is the budget. These rates demonstrate that the dependence on $K$ of the minimax regret varies significantly depending on the shape constraint. This highlights the fact that the shape constraints modify fundamentally the nature of the TBP.