arXiv subjects

Alexandra Carpentier

Publications and source records attributed to Alexandra Carpentier.

At least 55 records · Page 3Linked to original sources

arXiv2019

Total variation distance for discretely observed Lévy processes: a Gaussian approximation of the small jumps

It is common practice to treat small jumps of Lévy processes as Wiener noise and thus to approximate its marginals by a Gaussian distribution. However, results that allow to quantify the goodness of this approximation according to a given metric are rare. In this paper, we clarify what happens when the chosen metric is the total variation distance. Such a choice is motivated by its statistical interpretation. If the total variation distance between two statistical models converges to zero, then no tests can be constructed to distinguish the two models which are therefore equivalent, statistically speaking. We elaborate a fine analysis of a Gaussian approximation for the small jumps of Lévy processes with infinite Lévy measure in total variation distance. Non asymptotic bounds for the total variation distance between $n$ discrete observations of small jumps of a Lévy process and the corresponding Gaussian distribution is presented and extensively discussed. As a byproduct, new upper bounds for the total variation distance between discrete observations of Lévy processes are provided. The theory is illustrated by concrete examples.

math.ST↗

arXiv2018

A minimax near-optimal algorithm for adaptive rejection sampling

Rejection Sampling is a fundamental Monte-Carlo method. It is used to sample from distributions admitting a probability density function which can be evaluated exactly at any given point, albeit at a high computational cost. However, without proper tuning, this technique implies a high rejection rate. Several methods have been explored to cope with this problem, based on the principle of adaptively estimating the density by a simpler function, using the information of the previous samples. Most of them either rely on strong assumptions on the form of the density, or do not offer any theoretical performance guarantee. We give the first theoretical lower bound for the problem of adaptive rejection sampling and introduce a new algorithm which guarantees a near-optimal rejection rate in a minimax sense.

stat.ML↗

arXiv2018

Minimax rate of testing in sparse linear regression

We consider the problem of testing the hypothesis that the parameter of linear regression model is 0 against an s-sparse alternative separated from 0 in the l2-distance. We show that, in Gaussian linear regression model with p < n, where p is the dimension of the parameter and n is the sample size, the non-asymptotic minimax rate of testing has the form sqrt((s/n) log(1 + sqrt(p)/s )). We also show that this is the minimax rate of estimation of the l2-norm of the regression parameter.

math.ST↗

arXiv2018

Estimating minimum effect with outlier selection

We introduce one-sided versions of Huber's contamination model, in which corrupted samples tend to take larger values than uncorrupted ones. Two intertwined problems are addressed: estimation of the mean of uncorrupted samples (minimum effect) and selection of corrupted samples (outliers). Regarding the minimum effect estimation, we derive the minimax risks and introduce adaptive estimators to the unknown number of contaminations. Interestingly, the optimal convergence rate highly differs from that in classical Huber's contamination model. Also, our analysis uncovers the effect of particular structural assumptions on the distribution of the contaminated samples. As for the problem of selecting the outliers, we formulate the problem in a multiple testing framework for which the location/scaling of the null hypotheses are unknown. We rigorously prove how estimating the null hypothesis is possible while maintaining a theoretical guarantee on the amount of the falsely selected outliers, both through false discovery rate (FDR) or post hoc bounds. As a by-product, we address a long-standing open issue on FDR control under equi-correlation, which reinforces the interest of removing dependency when making multiple testing.

math.ST↗

arXiv2018

Minimax Euclidean Separation Rates for Testing Convex Hypotheses in $\mathbb{R}^d$

We consider composite-composite testing problems for the expectation in the Gaussian sequence model where the null hypothesis corresponds to a convex subset $\mathcal{C}$ of $\mathbb{R}^d$. We adopt a minimax point of view and our primary objective is to describe the smallest Euclidean distance between the null and alternative hypotheses such that there is a test with small total error probability. In particular, we focus on the dependence of this distance on the dimension $d$ and the sample size/variance parameter $n$ giving rise to the minimax separation rate. In this paper we discuss lower and upper bounds on this rate for different smooth and non- smooth choices for $\mathcal{C}$.

math.ST↗

arXiv2017

An Adaptive Strategy for Active Learning with Smooth Decision Boundary

We present the first adaptive strategy for active learning in the setting of classification with smooth decision boundary. The problem of adaptivity (to unknown distributional parameters) has remained opened since the seminal work of Castro and Nowak (2007), which first established (active learning) rates for this setting. While some recent advances on this problem establish adaptive rates in the case of univariate data, adaptivity in the more practical setting of multivariate data has so far remained elusive. Combining insights from various recent works, we show that, for the multivariate case, a careful reduction to univariate-adaptive strategies yield near-optimal rates without prior knowledge of distributional parameters.

stat.ML↗

arXiv2017

Two-Sample Tests for Large Random Graphs Using Network Statistics

We consider a two-sample hypothesis testing problem, where the distributions are defined on the space of undirected graphs, and one has access to only one observation from each model. A motivating example for this problem is comparing the friendship networks on Facebook and LinkedIn. The practical approach to such problems is to compare the networks based on certain network statistics. In this paper, we present a general principle for two-sample hypothesis testing in such scenarios without making any assumption about the network generation process. The main contribution of the paper is a general formulation of the problem based on concentration of network statistics, and consequently, a consistent two-sample test that arises as the natural solution for this problem. We also show that the proposed test is minimax optimal for certain network statistics.

stat.ME↗

arXiv2017

Constructing confidence sets for the matrix completion problem

In the present note we consider the problem of constructing honest and adaptive confidence sets for the matrix completion problem. For the Bernoulli model with known variance of the noise we provide a realizable method for constructing confidence sets that adapt to the unknown rank of the true matrix.

math.ST↗

arXiv2017

Adaptivity to Noise Parameters in Nonparametric Active Learning

This work addresses various open questions in the theory of active learning for nonparametric classification. Our contributions are both statistical and algorithmic: -We establish new minimax-rates for active learning under common \textit{noise conditions}. These rates display interesting transitions -- due to the interaction between noise \textit{smoothness and margin} -- not present in the passive setting. Some such transitions were previously conjectured, but remained unconfirmed. -We present a generic algorithmic strategy for adaptivity to unknown noise smoothness and margin; our strategy achieves optimal rates in many general situations; furthermore, unlike in previous work, we avoid the need for \textit{adaptive confidence sets}, resulting in strictly milder distributional requirements.

stat.ML↗

arXiv2017

Adaptive estimation of the sparsity in the Gaussian vector model

Consider the Gaussian vector model with mean value θ. We study the twin problems of estimating the number |θ|_0 of non-zero components of θ and testing whether |θ|_0 is smaller than some value. For testing, we establish the minimax separation distances for this model and introduce a minimax adaptive test. Extensions to the case of unknown variance are also discussed. Rewriting the estimation of |θ|_0 as a multiple testing problem of all hypotheses {|θ|_0 <= q}, we both derive a new way of assessing the optimality of a sparsity estimator and we exhibit such an optimal procedure. This general approach provides a roadmap for estimating the complexity of the signal in various statistical models.

math.ST↗

arXiv2017

Adaptive confidence sets for matrix completion

In the present paper we study the problem of existence of honest and adaptive confidence sets for matrix completion. We consider two statistical models: the trace regression model and the Bernoulli model. In the trace regression model, we show that honest confidence sets that adapt to the unknown rank of the matrix exist even when the error variance is unknown. Contrary to this, we prove that in the Bernoulli model, honest and adaptive confidence sets exist only when the error variance is known a priori. In the course of our proofs we obtain bounds for the minimax rates of certain composite hypothesis testing problems arising in low rank inference.

math.ST↗

arXiv2016

Tight (Lower) Bounds for the Fixed Budget Best Arm Identification Bandit Problem

We consider the problem of \textit{best arm identification} with a \textit{fixed budget $T$}, in the $K$-armed stochastic bandit setting, with arms distribution defined on $[0,1]$. We prove that any bandit strategy, for at least one bandit problem characterized by a complexity $H$, will misidentify the best arm with probability lower bounded by $$\exp\Big(-\frac{T}{\log(K)H}\Big),$$ where $H$ is the sum for all sub-optimal arms of the inverse of the squared gaps. Our result disproves formally the general belief - coming from results in the fixed confidence setting - that there must exist an algorithm for this problem whose probability of error is upper bounded by $\exp(-T/H)$. This also proves that some existing strategies based on the Successive Rejection of the arms are optimal - closing therefore the current gap between upper and lower bounds for the fixed budget best arm identification problem.

stat.ML↗

arXiv2016

An optimal algorithm for the Thresholding Bandit Problem

We study a specific \textit{combinatorial pure exploration stochastic bandit problem} where the learner aims at finding the set of arms whose means are above a given threshold, up to a given precision, and \textit{for a fixed time horizon}. We propose a parameter-free algorithm based on an original heuristic, and prove that it is optimal for this problem by deriving matching upper and lower bounds. To the best of our knowledge, this is the first non-trivial pure exploration setting with \textit{fixed budget} for which optimal strategies are constructed.

stat.ML↗

arXiv2016

An iterative hard thresholding estimator for low rank matrix recovery with explicit limiting distribution

We consider the problem of low rank matrix recovery in a stochastically noisy high dimensional setting. We propose a new estimator for the low rank matrix, based on the iterative hard thresholding method, and that is computationally efficient and simple. We prove that our estimator is efficient both in terms of the Frobenius risk, and in terms of the entry-wise risk uniformly over any change of orthonormal basis. This result allows us, in the case where the design is Gaussian, to provide the limiting distribution of the estimator, which is of great interest for constructing tests and confidence sets for low dimensional subsets of entries of the low rank matrix.

math.ST↗

arXiv2016

Learning relationships between data obtained independently

The aim of this paper is to provide a new method for learning the relationships between data that have been obtained independently. Unlike existing methods like matching, the proposed technique does not require any contextual information, provided that the dependency between the variables of interest is monotone. It can therefore be easily combined with matching in order to exploit the advantages of both methods. This technique can be described as a mix between quantile matching, and deconvolution. We provide for it a theoretical and an empirical validation.

stat.ML↗

arXiv2015

Uncertainty Quantification for Matrix Compressed Sensing and Quantum Tomography Problems

We construct minimax optimal non-asymptotic confidence sets for low rank matrix recovery algorithms such as the Matrix Lasso or Dantzig selector. These are employed to devise adaptive sequential sampling procedures that guarantee recovery of the true matrix in Frobenius norm after a data-driven stopping time $\hat n$ for the number of measurements that have to be taken. With high probability, this stopping time is minimax optimal. We detail applications to quantum tomography problems where measurements arise from Pauli observables. We also give a theoretical construction of a confidence set for the density matrix of a quantum state that has optimal diameter in nuclear norm. The non-asymptotic properties of our confidence sets are further investigated in a simulation study.

math.ST↗

arXiv2015

On signal detection and confidence sets for low rank inference problems

We consider the signal detection problem in the Gaussian design trace regression model with low rank alternative hypotheses. We derive the precise (Ingster-type) detection boundary for the Frobenius and the nuclear norm. We then apply these results to show that honest confidence sets for the unknown matrix parameter that adapt to all low rank sub-models in nuclear norm do not exist. This shows that recently obtained positive results in (Carpentier, Eisert, Gross and Nickl, 2015) for confidence sets in low rank recovery problems are essentially optimal.

math.ST↗

arXiv2015

Upper-Confidence-Bound Algorithms for Active Learning in Multi-Armed Bandits

In this paper, we study the problem of estimating uniformly well the mean values of several distributions given a finite budget of samples. If the variance of the distributions were known, one could design an optimal sampling strategy by collecting a number of independent samples per distribution that is proportional to their variance. However, in the more realistic case where the distributions are not known in advance, one needs to design adaptive sampling strategies in order to select which distribution to sample from according to the previously observed samples. We describe two strategies based on pulling the distributions a number of times that is proportional to a high-probability upper-confidence-bound on their variance (built from previous observed samples) and report a finite-sample performance analysis on the excess estimation error compared to the optimal allocation. We show that the performance of these allocation strategies depends not only on the variances but also on the full shape of the distributions.

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