arXiv subjects

Alexandra Carpentier

Publications and source records attributed to Alexandra Carpentier.

At least 37 records · Page 2Linked to original sources

arXiv2023

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 $\sqrt{αT}$, where $α$ 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 $α^3$, 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 $α^3$.

cs.LG↗

arXiv2023

Optimal Permutation Estimation in Crowd-Sourcing problems

Motivated by crowd-sourcing applications, we consider a model where we have partial observations from a bivariate isotonic n x d matrix with an unknown permutation $π$ * acting on its rows. Focusing on the twin problems of recovering the permutation $π$ * and estimating the unknown matrix, we introduce a polynomial-time procedure achieving the minimax risk for these two problems, this for all possible values of n, d, and all possible sampling efforts. Along the way, we establish that, in some regimes, recovering the unknown permutation $π$ * is considerably simpler than estimating the matrix.

math.ST↗

arXiv2023

Goodness-of-Fit Testing for Hölder-Continuous Densities: Sharp Local Minimax Rates

We consider the goodness-of fit testing problem for Hölder smooth densities over $\mathbb{R}^d$: given $n$ iid observations with unknown density $p$ and given a known density $p_0$, we investigate how large $ρ$ should be to distinguish, with high probability, the case $p=p_0$ from the composite alternative of all Hölder-smooth densities $p$ such that $\|p-p_0\|_t \geq ρ$ where $t \in [1,2]$. The densities are assumed to be defined over $\mathbb{R}^d$ and to have Hölder smoothness parameter $α>0$. In the present work, we solve the case $α\leq 1$ and handle the case $α>1$ using an additional technical restriction on the densities. We identify matching upper and lower bounds on the local minimax rates of testing, given explicitly in terms of $p_0$. We propose novel test statistics which we believe could be of independent interest. We also establish the first definition of an explicit cutoff $u_B$ allowing us to split $\mathbb{R}^d$ into a bulk part (defined as the subset of $\mathbb{R}^d$ where $p_0$ takes only values greater than or equal to $u_B$) and a tail part (defined as the complementary of the bulk), each part involving fundamentally different contributions to the local minimax rates of testing.

math.ST↗

arXiv2022

Optimal multiple change-point detection for high-dimensional data

This manuscript makes two contributions to the field of change-point detection. In a generalchange-point setting, we provide a generic algorithm for aggregating local homogeneity testsinto an estimator of change-points in a time series. Interestingly, we establish that the errorrates of the collection of tests directly translate into detection properties of the change-pointestimator. This generic scheme is then applied to various problems including covariance change-point detection, nonparametric change-point detection and sparse multivariate mean change-point detection. For the latter, we derive minimax optimal rates that are adaptive to theunknown sparsity and to the distance between change-points when the noise is Gaussian. Forsub-Gaussian noise, we introduce a variant that is optimal in almost all sparsity regimes.

math.ST↗

arXiv2022

The price of unfairness in linear bandits with biased feedback

In this paper, we study the problem of fair 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. The perceived reward is a linear combination of the covariates of the chosen action, but the player only observes a biased evaluation of this reward, depending on the sensitive attribute. To characterize the difficulty of this problem, we design a phased elimination algorithm that corrects the unfair evaluations, and establish upper bounds on its regret. We show that the worst-case regret is smaller than $\mathcal{O}(κ_*^{1/3}\log(T)^{1/3}T^{2/3})$, where $κ_*$ is an explicit geometrical constant characterizing the difficulty of bias estimation. We prove lower bounds on the worst-case regret for some sets of actions showing that this rate is tight up to a possible sub-logarithmic factor. We also derive gap-dependent upper bounds on the regret, and matching lower bounds for some problem instance.Interestingly, these results reveal a transition between a regime where the problem is as difficult as its unbiased counterpart, and a regime where it can be much harder.

math.ST↗

arXiv2021

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 $Δ$ 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 $Δ$. For the objective of minimizing the cumulative regret, we provide a lower bound of order $Ω(\log(T)/(p^*Δ))$ and a UCB-style algorithm with matching upper bound up to a factor of $\log(1/Δ)$. 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 $Ω(\exp(-cTΔ^2 p^*))$ 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 $\log(T)$ in the exponential, and that does not need $p^*$ or $Δ$ as parameter. Our results apply directly to the three related problems of competing against the $j$-th best arm, identifying an $ε$ good arm, and finding an arm with mean larger than a quantile of a known order.

cs.LG↗

arXiv2021

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 $(μ_k)_{k=1}^K$ is monotonically increasing (MTBP) and (ii) the case where $(μ_k)_{k=1}^K$ 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.

stat.ML↗

arXiv2021

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 $(μ_k)_k$ is monotonically increasing MTBP, (iii) the case where $(μ_k)_k$ is unimodal UTBP and (iv) the case where $(μ_k)_k$ 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.

cs.LG↗

arXiv2021

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.

stat.ML↗

arXiv2021

Local minimax rates for closeness testing of discrete distributions

We consider the closeness testing problem for discrete distributions. The goal is to distinguish whether two samples are drawn from the same unspecified distribution, or whether their respective distributions are separated in $L_1$-norm. In this paper, we focus on adapting the rate to the shape of the underlying distributions, i.e. we consider \textit{a local minimax setting}. We provide, to the best of our knowledge, the first local minimax rate for the separation distance up to logarithmic factors, together with a test that achieves it. In view of the rate, closeness testing turns out to be substantially harder than the related one-sample testing problem over a wide range of cases.

math.ST↗

arXiv2020

Estimation of the $l_2$-norm and testing in sparse linear regression with unknown variance

We consider the related problems of estimating the $l_2$-norm and the squared $l_2$-norm in sparse linear regression with unknown variance, as well as the problem of testing the hypothesis that the regression parameter is null under sparse alternatives with $l_2$ separation. We establish the minimax optimal rates of estimation (respectively, testing) in these three problems.

math.ST↗

arXiv2020

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.

stat.ML↗

arXiv2020

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.

stat.ML↗

arXiv2020

Rotting bandits are not harder than stochastic ones

In stochastic multi-armed bandits, the reward distribution of each arm is assumed to be stationary. This assumption is often violated in practice (e.g., in recommendation systems), where the reward of an arm may change whenever is selected, i.e., rested bandit setting. In this paper, we consider the non-parametric rotting bandit setting, where rewards can only decrease. We introduce the filtering on expanding window average (FEWA) algorithm that constructs moving averages of increasing windows to identify arms that are more likely to return high rewards when pulled once more. We prove that for an unknown horizon $T$, and without any knowledge on the decreasing behavior of the $K$ arms, FEWA achieves problem-dependent regret bound of $\widetilde{\mathcal{O}}(\log{(KT)}),$ and a problem-independent one of $\widetilde{\mathcal{O}}(\sqrt{KT})$. Our result substantially improves over the algorithm of Levine et al. (2017), which suffers regret $\widetilde{\mathcal{O}}(K^{1/3}T^{2/3})$. FEWA also matches known bounds for the stochastic bandit setting, thus showing that the rotting bandits are not harder. Finally, we report simulations confirming the theoretical improvements of FEWA.

stat.ML↗

arXiv2020

Optimal Sparsity Testing in Linear regression Model

We consider the problem of sparsity testing in the high-dimensional linear regression model. The problem is to test whether the number of non-zero components (aka the sparsity) of the regression parameter $θ^*$ is less than or equal to $k_0$. We pinpoint the minimax separation distances for this problem, which amounts to quantifying how far a $k_1$-sparse vector $θ^*$ has to be from the set of $k_0$-sparse vectors so that a test is able to reject the null hypothesis with high probability. Two scenarios are considered. In the independent scenario, the covariates are i.i.d. normally distributed and the noise level is known. In the general scenario, both the covariance matrix of the covariates and the noise level are unknown. Although the minimax separation distances differ in these two scenarios, both of them actually depend on $k_0$ and $k_1$ illustrating that for this composite-composite testing problem both the size of the null and of the alternative hypotheses play a key role.

math.ST↗

arXiv2020

Linear Bandits with Stochastic Delayed Feedback

Stochastic linear bandits are a natural and well-studied model for structured exploration/exploitation problems and are widely used in applications such as online marketing and recommendation. One of the main challenges faced by practitioners hoping to apply existing algorithms is that usually the feedback is randomly delayed and delays are only partially observable. For example, while a purchase is usually observable some time after the display, the decision of not buying is never explicitly sent to the system. In other words, the learner only observes delayed positive events. We formalize this problem as a novel stochastic delayed linear bandit and propose ${\tt OTFLinUCB}$ and ${\tt OTFLinTS}$, two computationally efficient algorithms able to integrate new information as it becomes available and to deal with the permanently censored feedback. We prove optimal $\tilde O(\smash{d\sqrt{T}})$ bounds on the regret of the first algorithm and study the dependency on delay-dependent parameters. Our model, assumptions and results are validated by experiments on simulated and real data.

stat.ML↗

arXiv2019

Two-sample Hypothesis Testing for Inhomogeneous Random Graphs

The study of networks leads to a wide range of high dimensional inference problems. In many practical applications, one needs to draw inference from one or few large sparse networks. The present paper studies hypothesis testing of graphs in this high-dimensional regime, where the goal is to test between two populations of inhomogeneous random graphs defined on the same set of $n$ vertices. The size of each population $m$ is much smaller than $n$, and can even be a constant as small as 1. The critical question in this context is whether the problem is solvable for small $m$. We answer this question from a minimax testing perspective. Let $P,Q$ be the population adjacencies of two sparse inhomogeneous random graph models, and $d$ be a suitably defined distance function. Given a population of $m$ graphs from each model, we derive minimax separation rates for the problem of testing $P=Q$ against $d(P,Q)>ρ$. We observe that if $m$ is small, then the minimax separation is too large for some popular choices of $d$, including total variation distance between corresponding distributions. This implies that some models that are widely separated in $d$ cannot be distinguished for small $m$, and hence, the testing problem is generally not solvable in these cases. We also show that if $m>1$, then the minimax separation is relatively small if $d$ is the Frobenius norm or operator norm distance between $P$ and $Q$. For $m=1$, only the latter distance provides small minimax separation. Thus, for these distances, the problem is solvable for small $m$. We also present near-optimal two-sample tests in both cases, where tests are adaptive with respect to sparsity level of the graphs.

stat.ME↗

arXiv2019

Restless dependent bandits with fading memory

We study the stochastic multi-armed bandit problem in the case when the arm samples are dependent over time and generated from so-called weak $\cC$-mixing processes. We establish a $\cC-$Mix Improved UCB agorithm and provide both problem-dependent and independent regret analysis in two different scenarios. In the first, so-called fast-mixing scenario, we show that pseudo-regret enjoys the same upper bound (up to a factor) as for i.i.d. observations; whereas in the second, slow mixing scenario, we discover a surprising effect, that the regret upper bound is similar to the independent case, with an incremental {\em additive} term which does not depend on the number of arms. The analysis of slow mixing scenario is supported with a minmax lower bound, which (up to a $\log(T)$ factor) matches the obtained upper bound.

stat.ML↗