Jean-Philippe Vert

Lawrence Stewart, Francis Bach, Quentin Berthet et al.

Neural networks can be trained to solve regression problems by using gradient-based methods to minimize the square loss. However, practitioners often prefer to reformulate regression as a classification problem, observing that training on the cross entropy loss results in better performance. By focusing on two-layer ReLU networks, which can be fully characterized by measures over their feature space, we explore how the implicit bias induced by gradient-based optimization could partly explain the above phenomenon. We provide theoretical evidence that the regression formulation yields a measure whose support can differ greatly from that for classification, in the case of one-dimensional data. Our proposed optimal supports correspond directly to the features learned by the input layer of the network. The different nature of these supports sheds light on possible optimization difficulties the square loss could encounter during training, and we present empirical results illustrating this phenomenon.

Pierre Marion, Adeline Fermanian, Gérard Biau et al.

Deep ResNets are recognized for achieving state-of-the-art results in complex machine learning tasks. However, the remarkable performance of these architectures relies on a training procedure that needs to be carefully crafted to avoid vanishing or exploding gradients, particularly as the depth $L$ increases. No consensus has been reached on how to mitigate this issue, although a widely discussed strategy consists in scaling the output of each layer by a factor $α_L$. We show in a probabilistic setting that with standard i.i.d.~initializations, the only non-trivial dynamics is for $α_L = \frac{1}{\sqrt{L}}$; other choices lead either to explosion or to identity mapping. This scaling factor corresponds in the continuous-time limit to a neural stochastic differential equation, contrarily to a widespread interpretation that deep ResNets are discretizations of neural ordinary differential equations. By contrast, in the latter regime, stability is obtained with specific correlated initializations and $α_L = \frac{1}{L}$. Our analysis suggests a strong interplay between scaling and regularity of the weights as a function of the layer index. Finally, in a series of experiments, we exhibit a continuous range of regimes driven by these two parameters, which jointly impact performance before and after training.

Adeline Fermanian, Pierre Marion, Jean-Philippe Vert et al.

Building on the interpretation of a recurrent neural network (RNN) as a continuous-time neural differential equation, we show, under appropriate conditions, that the solution of a RNN can be viewed as a linear function of a specific feature set of the input sequence, known as the signature. This connection allows us to frame a RNN as a kernel method in a suitable reproducing kernel Hilbert space. As a consequence, we obtain theoretical guarantees on generalization and stability for a large class of recurrent networks. Our results are illustrated on simulated datasets.

Mathieu Blondel, Quentin Berthet, Marco Cuturi et al.

Automatic differentiation (autodiff) has revolutionized machine learning. It allows to express complex computations by composing elementary ones in creative ways and removes the burden of computing their derivatives by hand. More recently, differentiation of optimization problem solutions has attracted widespread attention with applications such as optimization layers, and in bi-level problems such as hyper-parameter optimization and meta-learning. However, so far, implicit differentiation remained difficult to use for practitioners, as it often required case-by-case tedious mathematical derivations and implementations. In this paper, we propose automatic implicit differentiation, an efficient and modular approach for implicit differentiation of optimization problems. In our approach, the user defines directly in Python a function $F$ capturing the optimality conditions of the problem to be differentiated. Once this is done, we leverage autodiff of $F$ and the implicit function theorem to automatically differentiate the optimization problem. Our approach thus combines the benefits of implicit differentiation and autodiff. It is efficient as it can be added on top of any state-of-the-art solver and modular as the optimality condition specification is decoupled from the implicit differentiation mechanism. We show that seemingly simple principles allow to recover many existing implicit differentiation methods and create new ones easily. We demonstrate the ease of formulating and solving bi-level optimization problems using our framework. We also showcase an application to the sensitivity analysis of molecular dynamics.

Mathieu Blondel, Arthur Mensch, Jean-Philippe Vert

Computing the discrepancy between time series of variable sizes is notoriously challenging. While dynamic time warping (DTW) is popularly used for this purpose, it is not differentiable everywhere and is known to lead to bad local optima when used as a "loss". Soft-DTW addresses these issues, but it is not a positive definite divergence: due to the bias introduced by entropic regularization, it can be negative and it is not minimized when the time series are equal. We propose in this paper a new divergence, dubbed soft-DTW divergence, which aims to correct these issues. We study its properties; in particular, under conditions on the ground cost, we show that it is a valid divergence: it is non-negative and minimized if and only if the two time series are equal. We also propose a new "sharp" variant by further removing entropic bias. We showcase our divergences on time series averaging and demonstrate significant accuracy improvements compared to both DTW and soft-DTW on 84 time series classification datasets.

Luigi Carratino, Moustapha Cissé, Rodolphe Jenatton et al.

Mixup is a data augmentation technique that creates new examples as convex combinations of training points and labels. This simple technique has empirically shown to improve the accuracy of many state-of-the-art models in different settings and applications, but the reasons behind this empirical success remain poorly understood. In this paper we take a substantial step in explaining the theoretical foundations of Mixup, by clarifying its regularization effects. We show that Mixup can be interpreted as standard empirical risk minimization estimator subject to a combination of data transformation and random perturbation of the transformed data. We gain two core insights from this new interpretation. First, the data transformation suggests that, at test time, a model trained with Mixup should also be applied to transformed data, a one-line change in code that we show empirically to improve both accuracy and calibration of the prediction. Second, we show how the random perturbation of the new interpretation of Mixup induces multiple known regularization schemes, including label smoothing and reduction of the Lipschitz constant of the estimator. These schemes interact synergistically with each other, resulting in a self calibrated and effective regularization effect that prevents overfitting and overconfident predictions. We corroborate our theoretical analysis with experiments that support our conclusions.

Marco Cuturi, Olivier Teboul, Quentin Berthet et al.

When the infection prevalence of a disease is low, Dorfman showed 80 years ago that testing groups of people can prove more efficient than testing people individually. Our goal in this paper is to propose new group testing algorithms that can operate in a noisy setting (tests can be mistaken) to decide adaptively (looking at past results) which groups to test next, with the goal to converge to a good detection, as quickly, and with as few tests as possible. We cast this problem as a Bayesian sequential experimental design problem. Using the posterior distribution of infection status vectors for $n$ patients, given observed tests carried out so far, we seek to form groups that have a maximal utility. We consider utilities such as mutual information, but also quantities that have a more direct relevance to testing, such as the AUC of the ROC curve of the test. Practically, the posterior distributions on $\{0,1\}^n$ are approximated by sequential Monte Carlo (SMC) samplers and the utility maximized by a greedy optimizer. Our procedures show in simulations significant improvements over both adaptive and non-adaptive baselines, and are far more efficient than individual tests when disease prevalence is low. Additionally, we show empirically that loopy belief propagation (LBP), widely regarded as the SoTA decoder to decide whether an individual is infected or not given previous tests, can be unreliable and exhibit oscillatory behavior. Our SMC decoder is more reliable, and can improve the performance of other group testing algorithms.

Imke Mayer, Julie Josse, Félix Raimundo et al.

Inferring causal effects of a treatment, intervention or policy from observational data is central to many applications. However, state-of-the-art methods for causal inference seldom consider the possibility that covariates have missing values, which is ubiquitous in many real-world analyses. Missing data greatly complicate causal inference procedures as they require an adapted unconfoundedness hypothesis which can be difficult to justify in practice. We circumvent this issue by considering latent confounders whose distribution is learned through variational autoencoders adapted to missing values. They can be used either as a pre-processing step prior to causal inference but we also suggest to embed them in a multiple imputation strategy to take into account the variability due to missing values. Numerical experiments demonstrate the effectiveness of the proposed methodology especially for non-linear models compared to competitors.

Quentin Berthet, Mathieu Blondel, Olivier Teboul et al.

Machine learning pipelines often rely on optimization procedures to make discrete decisions (e.g., sorting, picking closest neighbors, or shortest paths). Although these discrete decisions are easily computed, they break the back-propagation of computational graphs. In order to expand the scope of learning problems that can be solved in an end-to-end fashion, we propose a systematic method to transform optimizers into operations that are differentiable and never locally constant. Our approach relies on stochastically perturbed optimizers, and can be used readily together with existing solvers. Their derivatives can be evaluated efficiently, and smoothness tuned via the chosen noise amplitude. We also show how this framework can be connected to a family of losses developed in structured prediction, and give theoretical guarantees for their use in learning tasks. We demonstrate experimentally the performance of our approach on various tasks.

Marco Cuturi, Olivier Teboul, Jonathan Niles-Weed et al.

Low rank matrix factorization is a fundamental building block in machine learning, used for instance to summarize gene expression profile data or word-document counts. To be robust to outliers and differences in scale across features, a matrix factorization step is usually preceded by ad-hoc feature normalization steps, such as \texttt{tf-idf} scaling or data whitening. We propose in this work to learn these normalization operators jointly with the factorization itself. More precisely, given a $d\times n$ matrix $X$ of $d$ features measured on $n$ individuals, we propose to learn the parameters of quantile normalization operators that can operate row-wise on the values of $X$ and/or of its factorization $UV$ to improve the quality of the low-rank representation of $X$ itself. This optimization is facilitated by the introduction of a new differentiable quantile normalization operator built using optimal transport, providing new results on top of existing work by (Cuturi et al. 2019). We demonstrate the applicability of these techniques on synthetic and genomics datasets.

Aude Genevay, Gabriel Dulac-Arnold, Jean-Philippe Vert

Clustering is a fundamental unsupervised learning approach. Many clustering algorithms -- such as $k$-means -- rely on the euclidean distance as a similarity measure, which is often not the most relevant metric for high dimensional data such as images. Learning a lower-dimensional embedding that can better reflect the geometry of the dataset is therefore instrumental for performance. We propose a new approach for this task where the embedding is performed by a differentiable model such as a deep neural network. By rewriting the $k$-means clustering algorithm as an optimal transport task, and adding an entropic regularization, we derive a fully differentiable loss function that can be minimized with respect to both the embedding parameters and the cluster parameters via stochastic gradient descent. We show that this new formulation generalizes a recently proposed state-of-the-art method based on soft-$k$-means by adding constraints on the cluster sizes. Empirical evaluations on image classification benchmarks suggest that compared to state-of-the-art methods, our optimal transport-based approach provide better unsupervised accuracy and does not require a pre-training phase.

Beyrem Khalfaoui, Joseph Boyd, Jean-Philippe Vert

Dropout is a regularisation technique in neural network training where unit activations are randomly set to zero with a given probability \emph{independently}. In this work, we propose a generalisation of dropout and other multiplicative noise injection schemes for shallow and deep neural networks, where the random noise applied to different units is not independent but follows a joint distribution that is either fixed or estimated during training. We provide theoretical insights on why such adaptive structured noise injection (ASNI) may be relevant, and empirically confirm that it helps boost the accuracy of simple feedforward and convolutional neural networks, disentangles the hidden layer representations, and leads to sparser representations. Our proposed method is a straightforward modification of the classical dropout and does not require additional computational overhead.

Gabriel Dulac-Arnold, Neil Zeghidour, Marco Cuturi et al.

We propose a learning algorithm capable of learning from label proportions instead of direct data labels. In this scenario, our data are arranged into various bags of a certain size, and only the proportions of each label within a given bag are known. This is a common situation in cases where per-data labeling is lengthy, but a more general label is easily accessible. Several approaches have been proposed to learn in this setting with linear models in the multiclass setting, or with nonlinear models in the binary classification setting. Here we investigate the more general nonlinear multiclass setting, and compare two differentiable loss functions to train end-to-end deep neural networks from bags with label proportions. We illustrate the relevance of our methods on an image classification benchmark, and demonstrate the possibility to learn accurate image classifiers from bags of images.

Marco Cuturi, Olivier Teboul, Jean-Philippe Vert

Sorting an array is a fundamental routine in machine learning, one that is used to compute rank-based statistics, cumulative distribution functions (CDFs), quantiles, or to select closest neighbors and labels. The sorting function is however piece-wise constant (the sorting permutation of a vector does not change if the entries of that vector are infinitesimally perturbed) and therefore has no gradient information to back-propagate. We propose a framework to sort elements that is algorithmically differentiable. We leverage the fact that sorting can be seen as a particular instance of the optimal transport (OT) problem on $\mathbb{R}$, from input values to a predefined array of sorted values (e.g. $1,2,\dots,n$ if the input array has $n$ elements). Building upon this link , we propose generalized CDFs and quantile operators by varying the size and weights of the target presorted array. Because this amounts to using the so-called Kantorovich formulation of OT, we call these quantities K-sorts, K-CDFs and K-quantiles. We recover differentiable algorithms by adding to the OT problem an entropic regularization, and approximate it using a few Sinkhorn iterations. We call these operators S-sorts, S-CDFs and S-quantiles, and use them in various learning settings: we benchmark them against the recently proposed neuralsort [Grover et al. 2019], propose applications to quantile regression and introduce differentiable formulations of the top-k accuracy that deliver state-of-the art performance.

Edouard Pauwels, Francis Bach, Jean-Philippe Vert

Statistical leverage scores emerged as a fundamental tool for matrix sketching and column sampling with applications to low rank approximation, regression, random feature learning and quadrature. Yet, the very nature of this quantity is barely understood. Borrowing ideas from the orthogonal polynomial literature, we introduce the regularized Christoffel function associated to a positive definite kernel. This uncovers a variational formulation for leverage scores for kernel methods and allows to elucidate their relationships with the chosen kernel as well as population density. Our main result quantitatively describes a decreasing relation between leverage score and population density for a broad class of kernels on Euclidean spaces. Numerical simulations support our findings.

Beyrem Khalfaoui, Jean-Philippe Vert

Single-cell RNA sequencing (scRNA-seq) is a fast growing approach to measure the genome-wide transcriptome of many individual cells in parallel, but results in noisy data with many dropout events. Existing methods to learn molecular signatures from bulk transcriptomic data may therefore not be adapted to scRNA-seq data, in order to automatically classify individual cells into predefined classes. We propose a new method called DropLasso to learn a molecular signature from scRNA-seq data. DropLasso extends the dropout regularisation technique, popular in neural network training, to esti- mate sparse linear models. It is well adapted to data corrupted by dropout noise, such as scRNA-seq data, and we clarify how it relates to elastic net regularisation. We provide promising results on simulated and real scRNA-seq data, suggesting that DropLasso may be better adapted than standard regularisa- tions to infer molecular signatures from scRNA-seq data.

Yunlong Jiao, Jean-Philippe Vert

We propose new positive definite kernels for permutations. First we introduce a weighted version of the Kendall kernel, which allows to weight unequally the contributions of different item pairs in the permutations depending on their ranks. Like the Kendall kernel, we show that the weighted version is invariant to relabeling of items and can be computed efficiently in $O(n \ln(n))$ operations, where $n$ is the number of items in the permutation. Second, we propose a supervised approach to learn the weights by jointly optimizing them with the function estimated by a kernel machine. Third, while the Kendall kernel considers pairwise comparison between items, we extend it by considering higher-order comparisons among tuples of items and show that the supervised approach of learning the weights can be systematically generalized to higher-order permutation kernels.

Marine Le Morvan, Jean-Philippe Vert

Learning sparse linear models with two-way interactions is desirable in many application domains such as genomics. l1-regularised linear models are popular to estimate sparse models, yet standard implementations fail to address specifically the quadratic explosion of candidate two-way interactions in high dimensions, and typically do not scale to genetic data with hundreds of thousands of features. Here we present WHInter, a working set algorithm to solve large l1-regularised problems with two-way interactions for binary design matrices. The novelty of WHInter stems from a new bound to efficiently identify working sets while avoiding to scan all features, and on fast computations inspired from solutions to the maximum inner product search problem. We apply WHInter to simulated and real genetic data and show that it is more scalable and two orders of magnitude faster than the state of the art.

Marine Le Morvan, Jean-Philippe Vert

Quantile normalisation is a popular normalisation method for data subject to unwanted variations such as images, speech, or genomic data. It applies a monotonic transformation to the feature values of each sample to ensure that after normalisation, they follow the same target distribution for each sample. Choosing a "good" target distribution remains however largely empirical and heuristic, and is usually done independently of the subsequent analysis of normalised data. We propose instead to couple the quantile normalisation step with the subsequent analysis, and to optimise the target distribution jointly with the other parameters in the analysis. We illustrate this principle on the problem of estimating a linear model over normalised data, and show that it leads to a particular low-rank matrix regression problem that can be solved efficiently. We illustrate the potential of our method, which we term SUQUAN, on simulated data, images and genomic data, where it outperforms standard quantile normalisation.

Kévin Vervier, Pierre Mahé, Jean-Baptiste Veyrieras et al.

Microbial identification is a central issue in microbiology, in particular in the fields of infectious diseases diagnosis and industrial quality control. The concept of species is tightly linked to the concept of biological and clinical classification where the proximity between species is generally measured in terms of evolutionary distances and/or clinical phenotypes. Surprisingly, the information provided by this well-known hierarchical structure is rarely used by machine learning-based automatic microbial identification systems. Structured machine learning methods were recently proposed for taking into account the structure embedded in a hierarchy and using it as additional a priori information, and could therefore allow to improve microbial identification systems. We test and compare several state-of-the-art machine learning methods for microbial identification on a new Matrix-Assisted Laser Desorption/Ionization Time-of-Flight mass spectrometry (MALDI-TOF MS) dataset. We include in the benchmark standard and structured methods, that leverage the knowledge of the underlying hierarchical structure in the learning process. Our results show that although some methods perform better than others, structured methods do not consistently perform better than their "flat" counterparts. We postulate that this is partly due to the fact that standard methods already reach a high level of accuracy in this context, and that they mainly confuse species close to each other in the tree, a case where using the known hierarchy is not helpful.

Kévin Vervier, Pierre Mahé, Maud Tournoud et al.

Metagenomics characterizes the taxonomic diversity of microbial communities by sequencing DNA directly from an environmental sample. One of the main challenges in metagenomics data analysis is the binning step, where each sequenced read is assigned to a taxonomic clade. Due to the large volume of metagenomics datasets, binning methods need fast and accurate algorithms that can operate with reasonable computing requirements. While standard alignment-based methods provide state-of-the-art performance, compositional approaches that assign a taxonomic class to a DNA read based on the k-mers it contains have the potential to provide faster solutions. In this work, we investigate the potential of modern, large-scale machine learning implementations for taxonomic affectation of next-generation sequencing reads based on their k-mers profile. We show that machine learning-based compositional approaches benefit from increasing the number of fragments sampled from reference genome to tune their parameters, up to a coverage of about 10, and from increasing the k-mer size to about 12. Tuning these models involves training a machine learning model on about 10 8 samples in 10 7 dimensions, which is out of reach of standard soft-wares but can be done efficiently with modern implementations for large-scale machine learning. The resulting models are competitive in terms of accuracy with well-established alignment tools for problems involving a small to moderate number of candidate species, and for reasonable amounts of sequencing errors. We show, however, that compositional approaches are still limited in their ability to deal with problems involving a greater number of species, and more sensitive to sequencing errors. We finally confirm that compositional approach achieve faster prediction times, with a gain of 3 to 15 times with respect to the BWA-MEM short read mapper, depending on the number of candidate species and the level of sequencing noise.

Emile Richard, Guillaume Obozinski, Jean-Philippe Vert

Based on a new atomic norm, we propose a new convex formulation for sparse matrix factorization problems in which the number of nonzero elements of the factors is assumed fixed and known. The formulation counts sparse PCA with multiple factors, subspace clustering and low-rank sparse bilinear regression as potential applications. We compute slow rates and an upper bound on the statistical dimension of the suggested norm for rank 1 matrices, showing that its statistical dimension is an order of magnitude smaller than the usual $\ell\_1$-norm, trace norm and their combinations. Even though our convex formulation is in theory hard and does not lead to provably polynomial time algorithmic schemes, we propose an active set algorithm leveraging the structure of the convex problem to solve it and show promising numerical results.

Erwan Scornet, Gérard Biau, Jean-Philippe Vert

Random forests are a learning algorithm proposed by Breiman [Mach. Learn. 45 (2001) 5--32] that combines several randomized decision trees and aggregates their predictions by averaging. Despite its wide usage and outstanding practical performance, little is known about the mathematical properties of the procedure. This disparity between theory and practice originates in the difficulty to simultaneously analyze both the randomization process and the highly data-dependent tree structure. In the present paper, we take a step forward in forest exploration by proving a consistency result for Breiman's [Mach. Learn. 45 (2001) 5--32] original algorithm in the context of additive regression models. Our analysis also sheds an interesting light on how random forests can nicely adapt to sparsity. 1. Introduction. Random forests are an ensemble learning method for classification and regression that constructs a number of randomized decision trees during the training phase and predicts by averaging the results. Since its publication in the seminal paper of Breiman (2001), the procedure has become a major data analysis tool, that performs well in practice in comparison with many standard methods. What has greatly contributed to the popularity of forests is the fact that they can be applied to a wide range of prediction problems and have few parameters to tune. Aside from being simple to use, the method is generally recognized for its accuracy and its ability to deal with small sample sizes, high-dimensional feature spaces and complex data structures. The random forest methodology has been successfully involved in many practical problems, including air quality prediction (winning code of the EMC data science global hackathon in 2012, see http://www.kaggle.com/c/dsg-hackathon), chemoinformatics [Svetnik et al. (2003)], ecology [Prasad, Iverson and Liaw (2006), Cutler et al. (2007)], 3D

Anne-Claire Haury, Fantine Mordelet, Paola Vera-Licona et al.

Inferring the structure of gene regulatory networks (GRN) from gene expression data has many applications, from the elucidation of complex biological processes to the identification of potential drug targets. It is however a notoriously difficult problem, for which the many existing methods reach limited accuracy. In this paper, we formulate GRN inference as a sparse regression problem and investigate the performance of a popular feature selection method, least angle regression (LARS) combined with stability selection. We introduce a novel, robust and accurate scoring technique for stability selection, which improves the performance of feature selection with LARS. The resulting method, which we call TIGRESS (Trustful Inference of Gene REgulation using Stability Selection), was ranked among the top methods in the DREAM5 gene network reconstruction challenge. We investigate in depth the influence of the various parameters of the method and show that a fine parameter tuning can lead to significant improvements and state-of-the-art performance for GRN inference. TIGRESS reaches state-of-the-art performance on benchmark data. This study confirms the potential of feature selection techniques for GRN inference. Code and data are available on http://cbio.ensmp.fr/~ahaury. Running TIGRESS online is possible on GenePattern: http://www.broadinstitute.org/cancer/software/genepattern/.