arXiv subjects

Stéphane Gaïffas

Publications and source records attributed to Stéphane Gaïffas.

34 records · Page 2Linked to original sources

arXiv2018

ConvSCCS: convolutional self-controlled case series model for lagged adverse event detection

With the increased availability of large databases of electronic health records (EHRs) comes the chance of enhancing health risks screening. Most post-marketing detections of adverse drug reaction (ADR) rely on physicians' spontaneous reports, leading to under reporting. To take up this challenge, we develop a scalable model to estimate the effect of multiple longitudinal features (drug exposures) on a rare longitudinal outcome. Our procedure is based on a conditional Poisson model also known as self-controlled case series (SCCS). We model the intensity of outcomes using a convolution between exposures and step functions, that are penalized using a combination of group-Lasso and total-variation. This approach does not require the specification of precise risk periods, and allows to study in the same model several exposures at the same time. We illustrate the fact that this approach improves the state-of-the-art for the estimation of the relative risks both on simulations and on a cohort of diabetic patients, extracted from the large French national health insurance database (SNIIRAM), a SQL database built around medical reimbursements of more than 65 million people. This work has been done in the context of a research partnership between Ecole Polytechnique and CNAMTS (in charge of SNIIRAM).

stat.AP↗

arXiv2017

High-dimensional robust regression and outliers detection with SLOPE

The problems of outliers detection and robust regression in a high-dimensional setting are fundamental in statistics, and have numerous applications. Following a recent set of works providing methods for simultaneous robust regression and outliers detection, we consider in this paper a model of linear regression with individual intercepts, in a high-dimensional setting. We introduce a new procedure for simultaneous estimation of the linear regression coefficients and intercepts, using two dedicated sorted-$\ell_1$ penalizations, also called SLOPE. We develop a complete theory for this problem: first, we provide sharp upper bounds on the statistical estimation error of both the vector of individual intercepts and regression coefficients. Second, we give an asymptotic control on the False Discovery Rate (FDR) and statistical power for support selection of the individual intercepts. As a consequence, this paper is the first to introduce a procedure with guaranteed FDR and statistical power control for outliers detection under the mean-shift model. Numerical illustrations, with a comparison to recent alternative approaches, are provided on both simulated and several real-world datasets. Experiments are conducted using an open-source software written in Python and C++.

stat.ML↗

arXiv2017

C-mix: a high dimensional mixture model for censored durations, with applications to genetic data

We introduce a mixture model for censored durations (C-mix), and develop maximum likelihood inference for the joint estimation of the time distributions and latent regression parameters of the model. We consider a high-dimensional setting, with datasets containing a large number of biomedical covariates. We therefore penalize the negative log-likelihood by the Elastic-Net, which leads to a sparse parameterization of the model. Inference is achieved using an efficient Quasi-Newton Expectation Maximization (QNEM) algorithm, for which we provide convergence properties. We then propose a score by assessing the patients risk of early adverse event. The statistical performance of the method is examined on an extensive Monte Carlo simulation study, and finally illustrated on three genetic datasets with high-dimensional covariates. We show that our approach outperforms the state-of-the-art, namely both the CURE and Cox proportional hazards models for this task, both in terms of C-index and AUC(t).

stat.ML↗

arXiv2017

Universal consistency and minimax rates for online Mondrian Forests

We establish the consistency of an algorithm of Mondrian Forests, a randomized classification algorithm that can be implemented online. First, we amend the original Mondrian Forest algorithm, that considers a fixed lifetime parameter. Indeed, the fact that this parameter is fixed hinders the statistical consistency of the original procedure. Our modified Mondrian Forest algorithm grows trees with increasing lifetime parameters $λ_n$, and uses an alternative updating rule, allowing to work also in an online fashion. Second, we provide a theoretical analysis establishing simple conditions for consistency. Our theoretical analysis also exhibits a surprising fact: our algorithm achieves the minimax rate (optimal rate) for the estimation of a Lipschitz regression function, which is a strong extension of previous results to an arbitrary dimension.

stat.ML↗

arXiv2017

Sparse inference of the drift of a high-dimensional Ornstein-Uhlenbeck process

Given the observation of a high-dimensional Ornstein-Uhlenbeck (OU) process in continuous time, we proceed to the inference of the drift parameter under a row-sparsity assumption. Towards that aim, we consider the negative log-likelihood of the process, penalized by an $\ell^1$-penalization (Lasso and Adaptive Lasso). We provide both non-asymptotic and asymptotic results for this procedure, by means of a sharp oracle inequality, and a limit theorem in the long-time asymptotics, including asymptotic consistency for variable selection. As a by-product, we point out the fact that for the Ornstein-Uhlenbeck process, one does not need an assumption of restricted eigenvalue type in order to derive fast rates for the Lasso, while it is well-known to be mandatory for linear regression for instance. Numerical results illustrate the benefits of this penalized procedure compared to standard maximum likelihood approaches both on simulations and real-world financial data.

stat.ML↗

arXiv2017

Uncovering Causality from Multivariate Hawkes Integrated Cumulants

We design a new nonparametric method that allows one to estimate the matrix of integrated kernels of a multivariate Hawkes process. This matrix not only encodes the mutual influences of each nodes of the process, but also disentangles the causality relationships between them. Our approach is the first that leads to an estimation of this matrix without any parametric modeling and estimation of the kernels themselves. A consequence is that it can give an estimation of causality relationships between nodes (or users), based on their activity timestamps (on a social network for instance), without knowing or estimating the shape of the activities lifetime. For that purpose, we introduce a moment matching method that fits the third-order integrated cumulants of the process. We show on numerical experiments that our approach is indeed very robust to the shape of the kernels, and gives appealing results on the MemeTracker database.

stat.ML↗

arXiv2016

SGD with Variance Reduction beyond Empirical Risk Minimization

We introduce a doubly stochastic proximal gradient algorithm for optimizing a finite average of smooth convex functions, whose gradients depend on numerically expensive expectations. Our main motivation is the acceleration of the optimization of the regularized Cox partial-likelihood (the core model used in survival analysis), but our algorithm can be used in different settings as well. The proposed algorithm is doubly stochastic in the sense that gradient steps are done using stochastic gradient descent (SGD) with variance reduction, where the inner expectations are approximated by a Monte-Carlo Markov-Chain (MCMC) algorithm. We derive conditions on the MCMC number of iterations guaranteeing convergence, and obtain a linear rate of convergence under strong convexity and a sublinear rate without this assumption. We illustrate the fact that our algorithm improves the state-of-the-art solver for regularized Cox partial-likelihood on several datasets from survival analysis.

stat.ML↗

arXiv2016

Concentration for matrix martingales in continuous time and microscopic activity of social networks

This paper gives new concentration inequalities for the spectral norm of a wide class of matrix martingales in continuous time. These results extend previously established Freedman and Bernstein inequalities for series of random matrices to the class of continuous time processes. Our analysis relies on a new supermartingale property of the trace exponential proved within the framework of stochastic calculus. We provide also several examples that illustrate the fact that our results allow us to recover easily several formerly obtained sharp bounds for discrete time matrix martingales.

math.PR↗

arXiv2015

Mean-field inference of Hawkes point processes

We propose a fast and efficient estimation method that is able to accurately recover the parameters of a d-dimensional Hawkes point-process from a set of observations. We exploit a mean-field approximation that is valid when the fluctuations of the stochastic intensity are small. We show that this is notably the case in situations when interactions are sufficiently weak, when the dimension of the system is high or when the fluctuations are self-averaging due to the large number of past events they involve. In such a regime the estimation of a Hawkes process can be mapped on a least-squares problem for which we provide an analytic solution. Though this estimator is biased, we show that its precision can be comparable to the one of the Maximum Likelihood Estimator while its computation speed is shown to be improved considerably. We give a theoretical control on the accuracy of our new approach and illustrate its efficiency using synthetic datasets, in order to assess the statistical estimation error of the parameters.

cs.LG↗

arXiv2015

Learning the intensity of time events with change-points

We consider the problem of learning the inhomogeneous intensity of a counting process, under a sparse segmentation assumption. We introduce a weighted total-variation penalization, using data-driven weights that correctly scale the penalization along the observation interval. We prove that this leads to a sharp tuning of the convex relaxation of the segmentation prior, by stating oracle inequalities with fast rates of convergence, and consistency for change-points detection. This provides first theoretical guarantees for segmentation with a convex proxy beyond the standard i.i.d signal + white noise setting. We introduce a fast algorithm to solve this convex problem. Numerical experiments illustrate our approach on simulated and on a high-frequency genomics dataset.

math.ST↗

arXiv2011

Weighted algorithms for compressed sensing and matrix completion

This paper is about iteratively reweighted basis-pursuit algorithms for compressed sensing and matrix completion problems. In a first part, we give a theoretical explanation of the fact that reweighted basis pursuit can improve a lot upon basis pursuit for exact recovery in compressed sensing. We exhibit a condition that links the accuracy of the weights to the RIP and incoherency constants, which ensures exact recovery. In a second part, we introduce a new algorithm for matrix completion, based on the idea of iterative reweighting. Since a weighted nuclear "norm" is typically non-convex, it cannot be used easily as an objective function. So, we define a new estimator based on a fixed-point equation. We give empirical evidences of the fact that this new algorithm leads to strong improvements over nuclear norm minimization on simulated and real matrix completion problems.

cs.IT↗

arXiv2010

Nonparametric regression with martingale increment errors

We consider the problem of adaptive estimation of the regression function in a framework where we replace ergodicity assumptions (such as independence or mixing) by another structural assumption on the model. Namely, we propose adaptive upper bounds for kernel estimators with data-driven bandwidth (Lepski's selection rule) in a regression model where the noise is an increment of martingale. It includes, as very particular cases, the usual i.i.d. regression and auto-regressive models. The cornerstone tool for this study is a new result for self-normalized martingales, called ``stability'', which is of independent interest. In a first part, we only use the martingale increment structure of the noise. We give an adaptive upper bound using a random rate, that involves the occupation time near the estimation point. Thanks to this approach, the theoretical study of the statistical procedure is disconnected from usual ergodicity properties like mixing. Then, in a second part, we make a link with the usual minimax theory of deterministic rates. Under a beta-mixing assumption on the covariates process, we prove that the random rate considered in the first part is equivalent, with large probability, to a deterministic rate which is the usual minimax adaptive one.

math.ST↗

arXiv2010

Sharp oracle inequalities for the prediction of a high-dimensional matrix

We observe $(X_i,Y_i)_{i=1}^n$ where the $Y_i$'s are real valued outputs and the $X_i$'s are $m\times T$ matrices. We observe a new entry $X$ and we want to predict the output $Y$ associated with it. We focus on the high-dimensional setting, where $m T \gg n$. This includes the matrix completion problem with noise, as well as other problems. We consider linear prediction procedures based on different penalizations, involving a mixture of several norms: the nuclear norm, the Frobenius norm and the $\ell_1$-norm. For these procedures, we prove sharp oracle inequalities, using a statistical learning theory point of view. A surprising fact in our results is that the rates of convergence do not depend on $m$ and $T$ directly. The analysis is conducted without the usually considered incoherency condition on the unknown matrix or restricted isometry condition on the sampling operator. Moreover, our results are the first to give for this problem an analysis of penalization (such nuclear norm penalization) as a regularization algorithm: our oracle inequalities prove that these procedures have a prediction accuracy close to the deterministic oracle one, given that the reguralization parameters are well-chosen.

math.ST↗

arXiv2009

Hyper-sparse optimal aggregation

In this paper, we consider the problem of "hyper-sparse aggregation". Namely, given a dictionary $F = \{f_1, ..., f_M \}$ of functions, we look for an optimal aggregation algorithm that writes $\tilde f = \sum_{j=1}^M θ_j f_j$ with as many zero coefficients $θ_j$ as possible. This problem is of particular interest when $F$ contains many irrelevant functions that should not appear in $\tilde{f}$. We provide an exact oracle inequality for $\tilde f$, where only two coefficients are non-zero, that entails $\tilde f$ to be an optimal aggregation algorithm. Since selectors are suboptimal aggregation procedures, this proves that 2 is the minimal number of elements of $F$ required for the construction of an optimal aggregation procedures in every situations. A simulated example of this algorithm is proposed on a dictionary obtained using LARS, for the problem of selection of the regularization parameter of the LASSO. We also give an example of use of aggregation to achieve minimax adaptation over anisotropic Besov spaces, which was not previously known in minimax theory (in regression on a random design).

stat.ML↗

arXiv2009

Learning and adaptive estimation for marker-dependent counting processes

We consider the problem of statistical learning for the intensity of a counting process with covariates. In this context, we introduce an empirical risk, and prove risk bounds for the corresponding empirical risk minimizers. Then, we give an oracle inequality for the popular algorithm of aggregation with exponential weights. This provides a way of constructing estimators that are adaptive to the smoothness and to the structure of the intensity. We prove that these estimators are adaptive over anisotropic Besov balls. The probabilistic tools are maximal inequalities using the generic chaining mechanism, which was introduced by Talagrand (2006), together with Bernstein's inequality for the underlying martingales.

math.ST↗

arXiv2007

Optimal rates and adaptation in the single-index model using aggregation

We want to recover the regression function in the single-index model. Using an aggregation algorithm with local polynomial estimators, we answer in particular to the second part of Question~2 from Stone (1982) on the optimal convergence rate. The procedure constructed here has strong adaptation properties: it adapts both to the smoothness of the link function and to the unknown index. Moreover, the procedure locally adapts to the distribution of the design. We propose new upper bounds for the local polynomial estimator (which are results of independent interest) that allows a fairly general design. The behavior of this algorithm is studied through numerical simulations. In particular, we show empirically that it improves strongly over empirical risk minimization.

math.ST↗