arXiv subjects

Stephan Clémençon

Publications and source records attributed to Stephan Clémençon.

42 records · Page 3Linked to original sources

arXiv2018

Mass Volume Curves and Anomaly Ranking

This paper aims at formulating the issue of ranking multivariate unlabeled observations depending on their degree of abnormality as an unsupervised statistical learning task. In the 1-d situation, this problem is usually tackled by means of tail estimation techniques: univariate observations are viewed as all the more `abnormal' as they are located far in the tail(s) of the underlying probability distribution. It would be desirable as well to dispose of a scalar valued `scoring' function allowing for comparing the degree of abnormality of multivariate observations. Here we formulate the issue of scoring anomalies as a M-estimation problem by means of a novel functional performance criterion, referred to as the Mass Volume curve (MV curve in short), whose optimal elements are strictly increasing transforms of the density almost everywhere on the support of the density. We first study the statistical estimation of the MV curve of a given scoring function and we provide a strategy to build confidence regions using a smoothed bootstrap approach. Optimization of this functional criterion over the set of piecewise constant scoring functions is next tackled. This boils down to estimating a sequence of empirical minimum volume sets whose levels are chosen adaptively from the data, so as to adjust to the variations of the optimal MV curve, while controling the bias of its approximation by a stepwise curve. Generalization bounds are then established for the difference in sup norm between the MV curve of the empirical scoring function thus obtained and the optimal MV curve.

stat.ML↗

arXiv2018

Profitable Bandits

Originally motivated by default risk management applications, this paper investigates a novel problem, referred to as the profitable bandit problem here. At each step, an agent chooses a subset of the K possible actions. For each action chosen, she then receives the sum of a random number of rewards. Her objective is to maximize her cumulated earnings. We adapt and study three well-known strategies in this purpose, that were proved to be most efficient in other settings: kl-UCB, Bayes-UCB and Thompson Sampling. For each of them, we prove a finite time regret bound which, together with a lower bound we obtain as well, establishes asymptotic optimality. Our goal is also to compare these three strategies from a theoretical and empirical perspective both at the same time. We give simple, self-contained proofs that emphasize their similarities, as well as their differences. While both Bayesian strategies are automatically adapted to the geometry of information, the numerical experiments carried out show a slight advantage for Thompson Sampling in practice.

stat.ML↗

arXiv2018

Ranking Data with Continuous Labels through Oriented Recursive Partitions

We formulate a supervised learning problem, referred to as continuous ranking, where a continuous real-valued label Y is assigned to an observable r.v. X taking its values in a feature space $\mathcal{X}$ and the goal is to order all possible observations x in $\mathcal{X}$ by means of a scoring function $s:\mathcal{X}\rightarrow \mathbb{R}$ so that s(X) and Y tend to increase or decrease together with highest probability. This problem generalizes bi/multi-partite ranking to a certain extent and the task of finding optimal scoring functions s(x) can be naturally cast as optimization of a dedicated functional criterion, called the IROC curve here, or as maximization of the Kendall $τ$ related to the pair (s(X), Y ). From the theoretical side, we describe the optimal elements of this problem and provide statistical guarantees for empirical Kendall $τ$ maximization under appropriate conditions for the class of scoring function candidates. We also propose a recursive statistical learning algorithm tailored to empirical IROC curve optimization and producing a piecewise constant scoring function that is fully described by an oriented binary tree. Preliminary numerical experiments highlight the difference in nature between regression and continuous ranking and provide strong empirical evidence of the performance of empirical optimizers of the criteria proposed.

stat.ML↗

arXiv2017

Ranking Median Regression: Learning to Order through Local Consensus

This article is devoted to the problem of predicting the value taken by a random permutation $Σ$, describing the preferences of an individual over a set of numbered items $\{1,\; \ldots,\; n\}$ say, based on the observation of an input/explanatory r.v. $X$ e.g. characteristics of the individual), when error is measured by the Kendall $τ$ distance. In the probabilistic formulation of the 'Learning to Order' problem we propose, which extends the framework for statistical Kemeny ranking aggregation developped in \citet{CKS17}, this boils down to recovering conditional Kemeny medians of $Σ$ given $X$ from i.i.d. training examples $(X_1, Σ_1),\; \ldots,\; (X_N, Σ_N)$. For this reason, this statistical learning problem is referred to as \textit{ranking median regression} here. Our contribution is twofold. We first propose a probabilistic theory of ranking median regression: the set of optimal elements is characterized, the performance of empirical risk minimizers is investigated in this context and situations where fast learning rates can be achieved are also exhibited. Next we introduce the concept of local consensus/median, in order to derive efficient methods for ranking median regression. The major advantage of this local learning approach lies in its close connection with the widely studied Kemeny aggregation problem. From an algorithmic perspective, this permits to build predictive rules for ranking median regression by implementing efficient techniques for (approximate) Kemeny median computations at a local level in a tractable manner. In particular, versions of $k$-nearest neighbor and tree-based methods, tailored to ranking median regression, are investigated. Accuracy of piecewise constant ranking median regression rules is studied under a specific smoothness assumption for $Σ$'s conditional distribution given $X$.

math.ST↗

arXiv2017

Max K-armed bandit: On the ExtremeHunter algorithm and beyond

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.

stat.ML↗

arXiv2016

Sharp exponential inequalities in survey sampling: conditional Poisson sampling schemes

This paper is devoted to establishing exponential bounds for the probabilities of deviation of a sample sum from its expectation, when the variables involved in the summation are obtained by sampling in a finite population according to a rejective scheme, generalizing sampling without replacement, and by using an appropriate normalization. In contrast to Poisson sampling, classical deviation inequalities in the i.i.d. setting do not straightforwardly apply to sample sums related to rejective schemes, due to the inherent dependence structure of the sampled points. We show here how to overcome this difficulty, by combining the formulation of rejective sampling as Poisson sampling conditioned upon the sample size with the Escher transformation. In particular, the Bennett/Bernstein type bounds established highlight the effect of the asymptotic variance of the (properly standardized) sample weighted sum and are shown to be much more accurate than those based on the negative association property shared by the terms involved in the summation. Beyond its interest in itself, such a result for rejective sampling is crucial, insofar as it can be extended to many other sampling schemes, namely those that can be accurately approximated by rejective plans in the sense of the total variation distance.

math.ST↗