arXiv subjects

Pierre Alquier

Publications and source records attributed to Pierre Alquier.

At least 55 records · Page 3Linked to original sources

arXiv2018

Simultaneous Dimension Reduction and Clustering via the NMF-EM Algorithm

Mixture models are among the most popular tools for clustering. However, when the dimension and the number of clusters is large, the estimation of the clusters become challenging, as well as their interpretation. Restriction on the parameters can be used to reduce the dimension. An example is given by mixture of factor analyzers for Gaussian mixtures. The extension of MFA to non-Gaussian mixtures is not straightforward. We propose a new constraint for parameters in non-Gaussian mixture model: the $K$ components parameters are combinations of elements from a small dictionary, say $H$ elements, with $H \ll K$. Including a nonnegative matrix factorization (NMF) in the EM algorithm allows us to simultaneously estimate the dictionary and the parameters of the mixture. We propose the acronym NMF-EM for this algorithm, implemented in the R package {\tt nmfem}. This original approach is motivated by passengers clustering from ticketing data: we apply NMF-EM to data from two Transdev public transport networks. In this case, the words are easily interpreted as typical slots in a timetable.

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arXiv2018

Informed Sub-Sampling MCMC: Approximate Bayesian Inference for Large Datasets

This paper introduces a framework for speeding up Bayesian inference conducted in presence of large datasets. We design a Markov chain whose transition kernel uses an (unknown) fraction of (fixed size) of the available data that is randomly refreshed throughout the algorithm. Inspired by the Approximate Bayesian Computation (ABC) literature, the subsampling process is guided by the fidelity to the observed data, as measured by summary statistics. The resulting algorithm, Informed Sub-Sampling MCMC (ISS-MCMC), is a generic and flexible approach which, contrary to existing scalable methodologies, preserves the simplicity of the Metropolis-Hastings algorithm. Even though exactness is lost, i.e. the chain distribution approximates the posterior, we study and quantify theoretically this bias and show on a diverse set of examples that it yields excellent performances when the computational budget is limited. If available and cheap to compute, we show that setting the summary statistics as the maximum likelihood estimator is supported by theoretical arguments.

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arXiv2017

Estimation bounds and sharp oracle inequalities of regularized procedures with Lipschitz loss functions

We obtain estimation error rates and sharp oracle inequalities for regularization procedures of the form \begin{equation*} \hat f \in argmin_{f\in F}\left(\frac{1}{N}\sum_{i=1}^N\ell(f(X_i), Y_i)+λ\|f\|\right) \end{equation*} when $\|\cdot\|$ is any norm, $F$ is a convex class of functions and $\ell$ is a Lipschitz loss function satisfying a Bernstein condition over $F$. We explore both the bounded and subgaussian stochastic frameworks for the distribution of the $f(X_i)$'s, with no assumption on the distribution of the $Y_i$'s. The general results rely on two main objects: a complexity function, and a sparsity equation, that depend on the specific setting in hand (loss $\ell$ and norm $\|\cdot\|$). As a proof of concept, we obtain minimax rates of convergence in the following problems: 1) matrix completion with any Lipschitz loss function, including the hinge and logistic loss for the so-called 1-bit matrix completion instance of the problem, and quantile losses for the general case, which enables to estimate any quantile on the entries of the matrix; 2) logistic LASSO and variants such as the logistic SLOPE; 3) kernel methods, where the loss is the hinge loss, and the regularization function is the RKHS norm.

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arXiv2016

Regret Bounds for Lifelong Learning

We consider the problem of transfer learning in an online setting. Different tasks are presented sequentially and processed by a within-task algorithm. We propose a lifelong learning strategy which refines the underlying data representation used by the within-task algorithm, thereby transferring information from one task to the next. We show that when the within-task algorithm comes with some regret bound, our strategy inherits this good property. Our bounds are in expectation for a general loss function, and uniform for a convex loss. We discuss applications to dictionary learning and finite set of predictors. In the latter case, we improve previous $O(1/\sqrt{m})$ bounds to $O(1/m)$ where $m$ is the per task sample size.

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arXiv2016

Pseudo-Bayesian Quantum Tomography with Rank-adaptation

Quantum state tomography, an important task in quantum information processing, aims at reconstructing a state from prepared measurement data. Bayesian methods are recognized to be one of the good and reliable choice in estimating quantum states~\cite{blume2010optimal}. Several numerical works showed that Bayesian estimations are comparable to, and even better than other methods in the problem of $1$-qubit state recovery. However, the problem of choosing prior distribution in the general case of $n$ qubits is not straightforward. More importantly, the statistical performance of Bayesian type estimators have not been studied from a theoretical perspective yet. In this paper, we propose a novel prior for quantum states (density matrices), and we define pseudo-Bayesian estimators of the density matrix. Then, using PAC-Bayesian theorems, we derive rates of convergence for the posterior mean. The numerical performance of these estimators are tested on simulated and real datasets.

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arXiv2016

1-bit Matrix Completion: PAC-Bayesian Analysis of a Variational Approximation

Due to challenging applications such as collaborative filtering, the matrix completion problem has been widely studied in the past few years. Different approaches rely on different structure assumptions on the matrix in hand. Here, we focus on the completion of a (possibly) low-rank matrix with binary entries, the so-called 1-bit matrix completion problem. Our approach relies on tools from machine learning theory: empirical risk minimization and its convex relaxations. We propose an algorithm to compute a variational approximation of the pseudo-posterior. Thanks to the convex relaxation, the corresponding minimization problem is bi-convex, and thus the method behaves well in practice. We also study the performance of this variational approximation through PAC-Bayesian learning bounds. On the contrary to previous works that focused on upper bounds on the estimation error of M with various matrix norms, we are able to derive from this analysis a PAC bound on the prediction error of our algorithm. We focus essentially on convex relaxation through the hinge loss, for which we present the complete analysis, a complete simulation study and a test on the MovieLens data set. However, we also discuss a variational approximation to deal with the logistic loss.

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arXiv2015

Light and Widely Applicable MCMC: Approximate Bayesian Inference for Large Datasets

Light and Widely Applicable (LWA-) MCMC is a novel approximation of the Metropolis-Hastings kernel targeting a posterior distribution defined on a large number of observations. Inspired by Approximate Bayesian Computation, we design a Markov chain whose transition makes use of an unknown but fixed, fraction of the available data, where the random choice of sub-sample is guided by the fidelity of this sub-sample to the observed data, as measured by summary (or sufficient) statistics. LWA-MCMC is a generic and flexible approach, as illustrated by the diverse set of examples which we explore. In each case LWA-MCMC yields excellent performance and in some cases a dramatic improvement compared to existing methodologies.

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arXiv2015

On the properties of variational approximations of Gibbs posteriors

The PAC-Bayesian approach is a powerful set of techniques to derive non- asymptotic risk bounds for random estimators. The corresponding optimal distribution of estimators, usually called the Gibbs posterior, is unfortunately intractable. One may sample from it using Markov chain Monte Carlo, but this is often too slow for big datasets. We consider instead variational approximations of the Gibbs posterior, which are fast to compute. We undertake a general study of the properties of such approximations. Our main finding is that such a variational approximation has often the same rate of convergence as the original PAC-Bayesian procedure it approximates. We specialise our results to several learning tasks (classification, ranking, matrix completion),discuss how to implement a variational approximation in each case, and illustrate the good properties of said approximation on real datasets.

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arXiv2015

A Bayesian Approach for Noisy Matrix Completion: Optimal Rate under General Sampling Distribution

Bayesian methods for low-rank matrix completion with noise have been shown to be very efficient computationally. While the behaviour of penalized minimization methods is well understood both from the theoretical and computational points of view in this problem, the theoretical optimality of Bayesian estimators have not been explored yet. In this paper, we propose a Bayesian estimator for matrix completion under general sampling distribution. We also provide an oracle inequality for this estimator. This inequality proves that, whatever the rank of the matrix to be estimated, our estimator reaches the minimax-optimal rate of convergence (up to a logarithmic factor). We end the paper with a short simulation study.

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arXiv2014

Bayesian matrix completion: prior specification

Low-rank matrix estimation from incomplete measurements recently received increased attention due to the emergence of several challenging applications, such as recommender systems; see in particular the famous Netflix challenge. While the behaviour of algorithms based on nuclear norm minimization is now well understood, an as yet unexplored avenue of research is the behaviour of Bayesian algorithms in this context. In this paper, we briefly review the priors used in the Bayesian literature for matrix completion. A standard approach is to assign an inverse gamma prior to the singular values of a certain singular value decomposition of the matrix of interest; this prior is conjugate. However, we show that two other types of priors (again for the singular values) may be conjugate for this model: a gamma prior, and a discrete prior. Conjugacy is very convenient, as it makes it possible to implement either Gibbs sampling or Variational Bayes. Interestingly enough, the maximum a posteriori for these different priors is related to the nuclear norm minimization problems. We also compare all these priors on simulated datasets, and on the classical MovieLens and Netflix datasets.

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arXiv2014

PAC-Bayesian AUC classification and scoring

We develop a scoring and classification procedure based on the PAC-Bayesian approach and the AUC (Area Under Curve) criterion. We focus initially on the class of linear score functions. We derive PAC-Bayesian non-asymptotic bounds for two types of prior for the score parameters: a Gaussian prior, and a spike-and-slab prior; the latter makes it possible to perform feature selection. One important advantage of our approach is that it is amenable to powerful Bayesian computational tools. We derive in particular a Sequential Monte Carlo algorithm, as an efficient method which may be used as a gold standard, and an Expectation-Propagation algorithm, as a much faster but approximate method. We also extend our method to a class of non-linear score functions, essentially leading to a nonparametric procedure, by considering a Gaussian process prior.

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arXiv2013

Rank penalized estimation of a quantum system

We introduce a new method to reconstruct the density matrix $ρ$ of a system of $n$-qubits and estimate its rank $d$ from data obtained by quantum state tomography measurements repeated $m$ times. The procedure consists in minimizing the risk of a linear estimator $\hatρ$ of $ρ$ penalized by given rank (from 1 to $2^n$), where $\hatρ$ is previously obtained by the moment method. We obtain simultaneously an estimator of the rank and the resulting density matrix associated to this rank. We establish an upper bound for the error of penalized estimator, evaluated with the Frobenius norm, which is of order $dn(4/3)^n /m$ and consistency for the estimator of the rank. The proposed methodology is computationaly efficient and is illustrated with some example states and real experimental data sets.

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arXiv2013

PAC-Bayesian Estimation and Prediction in Sparse Additive Models

The present paper is about estimation and prediction in high-dimensional additive models under a sparsity assumption ($p\gg n$ paradigm). A PAC-Bayesian strategy is investigated, delivering oracle inequalities in probability. The implementation is performed through recent outcomes in high-dimensional MCMC algorithms, and the performance of our method is assessed on simulated data.

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arXiv2012

Prediction of time series by statistical learning: general losses and fast rates

We establish rates of convergences in time series forecasting using the statistical learning approach based on oracle inequalities. A series of papers extends the oracle inequalities obtained for iid observations to time series under weak dependence conditions. Given a family of predictors and $n$ observations, oracle inequalities state that a predictor forecasts the series as well as the best predictor in the family up to a remainder term $Δ_n$. Using the PAC-Bayesian approach, we establish under weak dependence conditions oracle inequalities with optimal rates of convergence. We extend previous results for the absolute loss function to any Lipschitz loss function with rates $Δ_n\sim\sqrt{c(Θ)/ n}$ where $c(Θ)$ measures the complexity of the model. We apply the method for quantile loss functions to forecast the french GDP. Under additional conditions on the loss functions (satisfied by the quadratic loss function) and on the time series, we refine the rates of convergence to $Δ_n \sim c(Θ)/n$. We achieve for the first time these fast rates for uniformly mixing processes. These rates are known to be optimal in the iid case and for individual sequences. In particular, we generalize the results of Dalalyan and Tsybakov on sparse regression estimation to the case of autoregression.

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arXiv2012

Prediction of quantiles by statistical learning and application to GDP forecasting

In this paper, we tackle the problem of prediction and confidence intervals for time series using a statistical learning approach and quantile loss functions. In a first time, we show that the Gibbs estimator (also known as Exponentially Weighted aggregate) is able to predict as well as the best predictor in a given family for a wide set of loss functions. In particular, using the quantile loss function of Koenker and Bassett (1978), this allows to build confidence intervals. We apply these results to the problem of prediction and confidence regions for the French Gross Domestic Product (GDP) growth, with promising results.

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arXiv2012

Model selection for weakly dependent time series forecasting

Observing a stationary time series, we propose a two-step procedure for the prediction of the next value of the time series. The first step follows machine learning theory paradigm and consists in determining a set of possible predictors as randomized estimators in (possibly numerous) different predictive models. The second step follows the model selection paradigm and consists in choosing one predictor with good properties among all the predictors of the first steps. We study our procedure for two different types of bservations: causal Bernoulli shifts and bounded weakly dependent processes. In both cases, we give oracle inequalities: the risk of the chosen predictor is close to the best prediction risk in all predictive models that we consider. We apply our procedure for predictive models such as linear predictors, neural networks predictors and non-parametric autoregressive.

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arXiv2012

Fast rates in learning with dependent observations

In this paper we tackle the problem of fast rates in time series forecasting from a statistical learning perspective. In a serie of papers (e.g. Meir 2000, Modha and Masry 1998, Alquier and Wintenberger 2012) it is shown that the main tools used in learning theory with iid observations can be extended to the prediction of time series. The main message of these papers is that, given a family of predictors, we are able to build a new predictor that predicts the series as well as the best predictor in the family, up to a remainder of order $1/\sqrt{n}$. It is known that this rate cannot be improved in general. In this paper, we show that in the particular case of the least square loss, and under a strong assumption on the time series (phi-mixing) the remainder is actually of order $1/n$. Thus, the optimal rate for iid variables, see e.g. Tsybakov 2003, and individual sequences, see \cite{lugosi} is, for the first time, achieved for uniformly mixing processes. We also show that our method is optimal for aggregating sparse linear combinations of predictors.

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arXiv2011

Sparse single-index model

Let $(\bX, Y)$ be a random pair taking values in $\mathbb R^p \times \mathbb R$. In the so-called single-index model, one has $Y=f^{\star}(θ^{\star T}\bX)+\bW$, where $f^{\star}$ is an unknown univariate measurable function, $θ^{\star}$ is an unknown vector in $\mathbb R^d$, and $W$ denotes a random noise satisfying $\mathbb E[\bW|\bX]=0$. The single-index model is known to offer a flexible way to model a variety of high-dimensional real-world phenomena. However, despite its relative simplicity, this dimension reduction scheme is faced with severe complications as soon as the underlying dimension becomes larger than the number of observations ("$p$ larger than $n$" paradigm). To circumvent this difficulty, we consider the single-index model estimation problem from a sparsity perspective using a PAC-Bayesian approach. On the theoretical side, we offer a sharp oracle inequality, which is more powerful than the best known oracle inequalities for other common procedures of single-index recovery. The proposed method is implemented by means of the reversible jump Markov chain Monte Carlo technique and its performance is compared with that of standard procedures.

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