Jean-Michel Loubes

Eustasio del Barrio, Alberto Gonzalez-Sanz, Jean-Michel Loubes et al.

We prove a central limit theorem for the entropic transportation cost between subgaussian probability measures, centered at the population cost. This is the first result which allows for asymptotically valid inference for entropic optimal transport between measures which are not necessarily discrete. In the compactly supported case, we complement these results with new, faster, convergence rates for the expected entropic transportation cost between empirical measures. Our proof is based on strengthening convergence results for dual solutions to the entropic optimal transport problem.

Valentin Lafargue, Ariel Guerra-Adames, Emmanuelle Claeys et al.

Large language models (LLMs) are increasingly deployed in applications with societal impact, raising concerns about the cultural biases they encode. We probe these representations by evaluating whether LLMs can perform author profiling from song lyrics in a zero-shot setting, inferring singers' gender and ethnicity without task-specific fine-tuning. Across several open-source models evaluated on more than 10,000 lyrics, we find that LLMs achieve non-trivial profiling performance but demonstrate systematic cultural alignment: most models default toward North American ethnicity, while DeepSeek-1.5B aligns more strongly with Asian ethnicity. This finding emerges from both the models' prediction distributions and an analysis of their generated rationales. To quantify these disparities, we introduce two fairness metrics, Modality Accuracy Divergence (MAD) and Recall Divergence (RD), and show that Ministral-8B displays the strongest ethnicity bias among the evaluated models, whereas Gemma-12B shows the most balanced behavior. Our code is available on GitHub (https://github.com/ValentinLafargue/CulturalProbingLLM).

François Bachoc, Louis Béthune, Alberto Gonzalez-Sanz et al.

We present a novel kernel over the space of probability measures based on the dual formulation of optimal regularized transport. We propose an Hilbertian embedding of the space of probabilities using their Sinkhorn potentials, which are solutions of the dual entropic relaxed optimal transport between the probabilities and a reference measure $\mathcal{U}$. We prove that this construction enables to obtain a valid kernel, by using the Hilbert norms. We prove that the kernel enjoys theoretical properties such as universality and some invariances, while still being computationally feasible. Moreover we provide theoretical guarantees on the behaviour of a Gaussian process based on this kernel. The empirical performances are compared with other traditional choices of kernels for processes indexed on distributions.

Laurent Risser, Agustin Picard, Lucas Hervier et al.

The problem of algorithmic bias in machine learning has gained a lot of attention in recent years due to its concrete and potentially hazardous implications in society. In much the same manner, biases can also alter modern industrial and safety-critical applications where machine learning are based on high dimensional inputs such as images. This issue has however been mostly left out of the spotlight in the machine learning literature. Contrarily to societal applications where a set of proxy variables can be provided by the common sense or by regulations to draw the attention on potential risks, industrial and safety-critical applications are most of the times sailing blind. The variables related to undesired biases can indeed be indirectly represented in the input data, or can be unknown, thus making them harder to tackle. This raises serious and well-founded concerns towards the commercial deployment of AI-based solutions, especially in a context where new regulations clearly address the issues opened by undesired biases in AI. Consequently, we propose here to make an overview of recent advances in this area, firstly by presenting how such biases can demonstrate themselves, then by exploring different ways to bring them to light, and by probing different possibilities to mitigate them. We finally present a practical remote sensing use-case of industrial Fairness.

Fanny Jourdan, Titon Tshiongo Kaninku, Nicholas Asher et al.

Automatic recommendation systems based on deep neural networks have become extremely popular during the last decade. Some of these systems can however be used for applications which are ranked as High Risk by the European Commission in the A.I. act, as for instance for online job candidate recommendation. When used in the European Union, commercial AI systems for this purpose will then be required to have to proper statistical properties with regard to potential discrimination they could engender. This motivated our contribution, where we present a novel optimal transport strategy to mitigate undesirable algorithmic biases in multi-class neural-network classification. Our stratey is model agnostic and can be used on any multi-class classification neural-network model. To anticipate the certification of recommendation systems using textual data, we then used it on the Bios dataset, for which the learning task consists in predicting the occupation of female and male individuals, based on their LinkedIn biography. Results show that it can reduce undesired algorithmic biases in this context to lower levels than a standard strategy.

William Todo, Beatrice Laurent, Jean-Michel Loubes et al.

In this work, we explore dimensionality reduction techniques for univariate and multivariate time series data. We especially conduct a comparison between wavelet decomposition and convolutional variational autoencoders for dimension reduction. We show that variational autoencoders are a good option for reducing the dimension of high dimensional data like ECG. We make these comparisons on a real world, publicly available, ECG dataset that has lots of variability and use the reconstruction error as the metric. We then explore the robustness of these models with noisy data whether for training or inference. These tests are intended to reflect the problems that exist in real-world time series data and the VAE was robust to both tests.

Fanny Jourdan, Laurent Risser, Jean-Michel Loubes et al.

This paper presents novel experiments shedding light on the shortcomings of current metrics for assessing biases of gender discrimination made by machine learning algorithms on textual data. We focus on the Bios dataset, and our learning task is to predict the occupation of individuals, based on their biography. Such prediction tasks are common in commercial Natural Language Processing (NLP) applications such as automatic job recommendations. We address an important limitation of theoretical discussions dealing with group-wise fairness metrics: they focus on large datasets, although the norm in many industrial NLP applications is to use small to reasonably large linguistic datasets for which the main practical constraint is to get a good prediction accuracy. We then question how reliable are different popular measures of bias when the size of the training set is simply sufficient to learn reasonably accurate predictions. Our experiments sample the Bios dataset and learn more than 200 models on different sample sizes. This allows us to statistically study our results and to confirm that common gender bias indices provide diverging and sometimes unreliable results when applied to relatively small training and test samples. This highlights the crucial importance of variance calculations for providing sound results in this field.

Timothée Chauvin, Clément Lalanne, Erwan Le Merrer et al.

Remote change detection in LLMs is a difficult problem. Existing methods are either too expensive for deployment at scale, or require initial white-box access to model weights or grey-box access to log probabilities. We aim to achieve both low cost and strict black-box operation, observing only output tokens. Our approach hinges on specific inputs we call Border Inputs, for which there exists more than one output top token. From a statistical perspective, optimal change detection depends on the model's Jacobian and the Fisher information of the output distribution. Analyzing these quantities in low-temperature regimes shows that border inputs enable powerful change detection tests. Building on this insight, we propose the Black-Box Border Input Tracking (B3IT) scheme. Extensive in-vivo and in-vitro experiments show that border inputs are easily found for non-reasoning tested endpoints, and achieve performance on par with the best available grey-box approaches. B3IT reduces costs by $30\times$ compared to existing methods, while operating in a strict black-box setting.

Baptiste Ferrere, Nicolas Bousquet, Fabrice Gamboa et al.

The functional ANOVA, or Hoeffding decomposition, provides a principled framework for interpretability by decomposing a model prediction into main effects and higher-order interactions. For independent inputs, this classical decomposition is explicit. It is closely connected to SHAP values, generalized additive models, and orthogonal polynomial expansions, and therefore constitutes a fundamental tool for additive explainability. In the more general and realistic dependent setting, however, obtaining a tractable representation and estimating the decomposition from data remain challenging. In this work, we address this problem for continuous inputs. By combining Hilbert space methods with the generalized functional ANOVA, we build an explicit decomposition Riesz Basis allowing to easily compute the decomposition. Our formulation recovers the classical independent case and its associated orthogonal decomposition. Building on this representation, we propose a simple but mighty algorithm to estimate the decomposition from a data sample in a model-agnostic setting and we compare it empirically with several state-of-the-art explanation methods, demonstrating the power of the approach.

Natasa Krco, Thibault Laugel, Vincent Grari et al.

Most research on fair machine learning has prioritized optimizing criteria such as Demographic Parity and Equalized Odds. Despite these efforts, there remains a limited understanding of how different bias mitigation strategies affect individual predictions and whether they introduce arbitrariness into the debiasing process. This paper addresses these gaps by exploring whether models that achieve comparable fairness and accuracy metrics impact the same individuals and mitigate bias in a consistent manner. We introduce the FRAME (FaiRness Arbitrariness and Multiplicity Evaluation) framework, which evaluates bias mitigation through five dimensions: Impact Size (how many people were affected), Change Direction (positive versus negative changes), Decision Rates (impact on models' acceptance rates), Affected Subpopulations (who was affected), and Neglected Subpopulations (where unfairness persists). This framework is intended to help practitioners understand the impacts of debiasing processes and make better-informed decisions regarding model selection. Applying FRAME to various bias mitigation approaches across key datasets allows us to exhibit significant differences in the behaviors of debiasing methods. These findings highlight the limitations of current fairness criteria and the inherent arbitrariness in the debiasing process.

Baptiste Ferrere, Nicolas Bousquet, Fabrice Gamboa et al.

Functional ANOVA offers a principled framework for interpretability by decomposing a model's prediction into main effects and higher-order interactions. For independent features, this decomposition is well-defined, strongly linked with SHAP values, and serves as a cornerstone of additive explainability. However, the lack of an explicit closed-form expression for general dependent distributions has forced practitioners to rely on costly sampling-based approximations. We completely resolve this limitation for categorical inputs. By bridging functional analysis with the extension of discrete Fourier analysis, we derive a closed-form decomposition without any assumption. Our formulation is computationally very efficient. It seamlessly recovers the classical independent case and extends to arbitrary dependence structures, including distributions with non-rectangular support. Furthermore, leveraging the intrinsic link between SHAP and ANOVA under independence, our framework yields a natural generalization of SHAP values for the general categorical setting.

Adriana Laurindo Monteiro, Jean-Michel Loubes

The massive use of Machine Learning (ML) tools in industry comes with critical challenges, such as the lack of explainable models and the use of black-box algorithms. We address this issue by applying Optimal Transport theory in the analysis of responses of ML models to variations in the distribution of input variables. We find the closest distribution, in the Wasserstein sense, that satisfies a given constraintt and examine its impact on model behavior. Furthermore, we establish convergence results for this projected distribution and demonstrate our approach using examples and real-world datasets in both regression and classification settings.

Mahdi Tavassoli Kejani, Fadi Dornaika, Charlotte Laclau et al.

In recent years, Graph Neural Networks (GNNs) have achieved remarkable success in tasks such as node classification, link prediction, and graph representation learning. However, they remain susceptible to biases that can arise not only from node attributes but also from the graph structure itself. Addressing fairness in GNNs has therefore emerged as a critical research challenge. In this work, we propose a novel model for training fairness-aware GNNs by improving the counterfactual augmented fair graph neural network framework (CAF). Specifically, our approach introduces a two-phase training strategy: in the first phase, we edit the graph to increase homophily ratio with respect to class labels while reducing homophily ratio with respect to sensitive attribute labels; in the second phase, we integrate a modified supervised contrastive loss and environmental loss into the optimization process, enabling the model to jointly improve predictive performance and fairness. Experiments on five real-world datasets demonstrate that our model outperforms CAF and several state-of-the-art graph-based learning methods in both classification accuracy and fairness metrics.

Mahdi Tavassoli Kejani, Fadi Dornaika, Jean-Michel Loubes

Graph Neural Networks (GNNs) have demonstrated exceptional efficacy in relational learning tasks, including node classification and link prediction. However, their application raises significant fairness concerns, as GNNs can perpetuate and even amplify societal biases against protected groups defined by sensitive attributes such as race or gender. These biases are often inherent in the node features, structural topology, and message-passing mechanisms of the graph itself. A critical limitation of existing fairness-aware GNN methods is their reliance on the strong assumption that sensitive attributes are fully available for all nodes during training--a condition that poses a practical impediment due to privacy concerns and data collection constraints. To address this gap, we propose a novel, model-agnostic fairness regularization framework designed for the realistic scenario where sensitive attributes are only partially available. Our approach formalizes a fairness-aware objective function that integrates both equal opportunity and statistical parity as differentiable regularization terms. Through a comprehensive empirical evaluation across five real-world benchmark datasets, we demonstrate that the proposed method significantly mitigates bias across key fairness metrics while maintaining competitive node classification performance. Results show that our framework consistently outperforms baseline models in achieving a favorable fairness-accuracy trade-off, with minimal degradation in predictive accuracy. The datasets and source code will be publicly released at https://github.com/mtavassoli/GNN-FC.

Eustasio del Barrio, Hristo Inouzhe, Jean-Michel Loubes

We consider the problem of diversity enhancing clustering, i.e, developing clustering methods which produce clusters that favour diversity with respect to a set of protected attributes such as race, sex, age, etc. In the context of fair clustering, diversity plays a major role when fairness is understood as demographic parity. To promote diversity, we introduce perturbations to the distance in the unprotected attributes that account for protected attributes in a way that resembles attraction-repulsion of charged particles in Physics. These perturbations are defined through dissimilarities with a tractable interpretation. Cluster analysis based on attraction-repulsion dissimilarities penalizes homogeneity of the clusters with respect to the protected attributes and leads to an improvement in diversity. An advantage of our approach, which falls into a pre-processing set-up, is its compatibility with a wide variety of clustering methods and whit non-Euclidean data. We illustrate the use of our procedures with both synthetic and real data and provide discussion about the relation between diversity, fairness, and cluster structure. Our procedures are implemented in an R package freely available at https://github.com/HristoInouzhe/AttractionRepulsionClustering.

Renan D. B. Brotto, Jean-Michel Loubes, Laurent Risser et al.

We tackle the problem of bias mitigation of algorithmic decisions in a setting where both the output of the algorithm and the sensitive variable are continuous. Most of prior work deals with discrete sensitive variables, meaning that the biases are measured for subgroups of persons defined by a label, leaving out important algorithmic bias cases, where the sensitive variable is continuous. Typical examples are unfair decisions made with respect to the age or the financial status. In our work, we then propose a bias mitigation strategy for continuous sensitive variables, based on the notion of endogeneity which comes from the field of econometrics. In addition to solve this new problem, our bias mitigation strategy is a weakly supervised learning method which requires that a small portion of the data can be measured in a fair manner. It is model agnostic, in the sense that it does not make any hypothesis on the prediction model. It also makes use of a reasonably large amount of input observations and their corresponding predictions. Only a small fraction of the true output predictions should be known. This therefore limits the need for expert interventions. Results obtained on synthetic data show the effectiveness of our approach for examples as close as possible to real-life applications in econometrics.

Veronika Shilova, Emmanuel Malherbe, Giovanni Palma et al.

Within a legal framework, fairness in datasets and models is typically assessed by dividing observations into predefined groups and then computing fairness measures (e.g., Disparate Impact or Equality of Odds with respect to gender). However, when sensitive attributes such as skin color are continuous, dividing into default groups may overlook or obscure the discrimination experienced by certain minority subpopulations. To address this limitation, we propose a fairness-based grouping approach for continuous (possibly multidimensional) sensitive attributes. By grouping data according to observed levels of discrimination, our method identifies the partition that maximizes a novel criterion based on inter-group variance in discrimination, thereby isolating the most critical subgroups. We validate the proposed approach using multiple synthetic datasets and demonstrate its robustness under changing population distributions - revealing how discrimination is manifested within the space of sensitive attributes. Furthermore, we examine a specialized setting of monotonic fairness for the case of skin color. Our empirical results on both CelebA and FFHQ, leveraging the skin tone as predicted by an industrial proprietary algorithm, show that the proposed segmentation uncovers more nuanced patterns of discrimination than previously reported, and that these findings remain stable across datasets for a given model. Finally, we leverage our grouping model for debiasing purpose, aiming at predicting fair scores with group-by-group post-processing. The results demonstrate that our approach improves fairness while having minimal impact on accuracy, thus confirming our partition method and opening the door for industrial deployment.

François Bachoc, Jérôme Bolte, Ryan Boustany et al.

Despite growing empirical evidence of bias amplification in machine learning, its theoretical foundations remain poorly understood. We develop a formal framework for majority-minority learning tasks, showing how standard training can favor majority groups and produce stereotypical predictors that neglect minority-specific features. Assuming population and variance imbalance, our analysis reveals three key findings: (i) the close proximity between ``full-data'' and stereotypical predictors, (ii) the dominance of a region where training the entire model tends to merely learn the majority traits, and (iii) a lower bound on the additional training required. Our results are illustrated through experiments in deep learning for tabular and image classification tasks.

Elena M. De-Diego, Adrián Perez-Suay, Paula Gordaliza et al.

We revisit the problem of fair representation learning by proposing Fair Partial Least Squares (PLS) components. PLS is widely used in statistics to efficiently reduce the dimension of the data by providing representation tailored for the prediction. We propose a novel method to incorporate fairness constraints in the construction of PLS components. This new algorithm provides a feasible way to construct such features both in the linear and the non linear case using kernel embeddings. The efficiency of our method is evaluated on different datasets, and we prove its superiority with respect to standard fair PCA method.

Emmanuelle Claeys, Elena Kerjean, Jean-Michel Loubes

We introduce a sequential reinforcement learning framework for imitation learning designed to model heterogeneous cognitive strategies in pollinators. Focusing on honeybees, our approach leverages trajectory similarity to capture and forecast behavior across individuals that rely on distinct strategies: some exploiting numerical cues, others drawing on memory, or being influenced by environmental factors such as weather. Through empirical evaluation, we show that state-of-the-art imitation learning methods often fail in this setting: when expert policies shift across memory windows or deviate from optimality, these models overlook both fast and slow learning behaviors and cannot faithfully reproduce key decision patterns. Moreover, they offer limited interpretability, hindering biological insight. Our contribution addresses these challenges by (i) introducing a model that minimizes predictive loss while identifying the effective memory horizon most consistent with behavioral data, and (ii) ensuring full interpretability to enable biologists to analyze underlying decision-making strategies and finally (iii) providing a mathematical framework linking bee policy search with bandit formulations under varying exploration-exploitation dynamics, and releasing a novel dataset of 80 tracked bees observed under diverse weather conditions. This benchmark facilitates research on pollinator cognition and supports ecological governance by improving simulations of insect behavior in agroecosystems. Our findings shed new light on the learning strategies and memory interplay shaping pollinator decision-making.

Baptiste Ferrere, Nicolas Bousquet, Fabrice Gamboa et al.

Understanding the behavior of predictive models with random inputs can be achieved through functional decompositions into sub-models that capture interpretable effects of input groups. Building on recent advances in uncertainty quantification, the existence and uniqueness of a generalized Hoeffding decomposition have been established for correlated input variables, using oblique projections onto suitable functional subspaces. This work focuses on the case of Bernoulli inputs and provides a complete analytical characterization of the decomposition. We show that, in this discrete setting, the associated subspaces are one-dimensional and that the decomposition admits a closed-form representation. One of the main contributions of this study is to generalize the classical Fourier--Walsh--Hadamard decomposition for pseudo-Boolean functions to the correlated case, yielding an oblique version when the underlying distribution is not a product measure, and recovering the standard orthogonal form when independence holds. This explicit structure offers a fully interpretable framework, clarifying the contribution of each input combination and theoretically enabling model reverse engineering. From this formulation, explicit sensitivity measures-such as Sobol' indices and Shapley effects-can be directly derived. Numerical experiments illustrate the practical interest of the approach for decision-support problems involving binary features. The paper concludes with perspectives on extending the methodology to high-dimensional settings and to models involving inputs with finite, non-binary support.

Valentin Lafargue, Adriana Laurindo Monteiro, Emmanuelle Claeys et al.

Proving the compliance of AI algorithms has become an important challenge with the growing deployment of such algorithms for real-life applications. Inspecting possible biased behaviors is mandatory to satisfy the constraints of the regulations of the EU Artificial Intelligence's Act. Regulation-driven audits increasingly rely on global fairness metrics, with Disparate Impact being the most widely used. Yet such global measures depend highly on the distribution of the sample on which the measures are computed. We investigate first how to manipulate data samples to artificially satisfy fairness criteria, creating minimally perturbed datasets that remain statistically indistinguishable from the original distribution while satisfying prescribed fairness constraints. Then we study how to detect such manipulation. Our analysis (i) introduces mathematically sound methods for modifying empirical distributions under fairness constraints using entropic or optimal transport projections, (ii) examines how an auditee could potentially circumvent fairness inspections, and (iii) offers recommendations to help auditors detect such data manipulations. These results are validated through experiments on classical tabular datasets in bias detection.

David Rodríguez-Vítores, Clément Lalanne, Jean-Michel Loubes

In this work, we introduce a novel framework for privately optimizing objectives that rely on Wasserstein distances between data-dependent empirical measures. Our main theoretical contribution is, based on an explicit formulation of the Wasserstein gradient in a fully discrete setting, a control on the sensitivity of this gradient to individual data points, allowing strong privacy guarantees at minimal utility cost. Building on these insights, we develop a deep learning approach that incorporates gradient and activations clipping, originally designed for DP training of problems with a finite-sum structure. We further demonstrate that privacy accounting methods extend to Wasserstein-based objectives, facilitating large-scale private training. Empirical results confirm that our framework effectively balances accuracy and privacy, offering a theoretically sound solution for privacy-preserving machine learning tasks relying on optimal transport distances such as Wasserstein distance or sliced-Wasserstein distance.

Mahdi Tavassoli Kejani, Fadi Dornaika, Jean-Michel Loubes

In recent years, Graph Neural Networks (GNNs) have made significant advancements, particularly in tasks such as node classification, link prediction, and graph representation. However, challenges arise from biases that can be hidden not only in the node attributes but also in the connections between entities. Therefore, ensuring fairness in graph neural network learning has become a critical problem. To address this issue, we propose a novel model for training fairness-aware GNN, which enhances the Counterfactual Augmented Fair Graph Neural Network Framework (CAF). Our approach integrates Supervised Contrastive Loss and Environmental Loss to enhance both accuracy and fairness. Experimental validation on three real datasets demonstrates the superiority of our proposed model over CAF and several other existing graph-based learning methods.

Samuele Centorrino, Jean-Pierre Florens, Jean-Michel Loubes

A supervised machine learning algorithm determines a model from a learning sample that will be used to predict new observations. To this end, it aggregates individual characteristics of the observations of the learning sample. But this information aggregation does not consider any potential selection on unobservables and any status-quo biases which may be contained in the training sample. The latter bias has raised concerns around the so-called \textit{fairness} of machine learning algorithms, especially towards disadvantaged groups. In this chapter, we review the issue of fairness in machine learning through the lenses of structural econometrics models in which the unknown index is the solution of a functional equation and issues of endogeneity are explicitly accounted for. We model fairness as a linear operator whose null space contains the set of strictly {\it fair} indexes. A {\it fair} solution is obtained by projecting the unconstrained index into the null space of this operator or by directly finding the closest solution of the functional equation into this null space. We also acknowledge that policymakers may incur a cost when moving away from the status quo. Achieving \textit{approximate fairness} is obtained by introducing a fairness penalty in the learning procedure and balancing more or less heavily the influence between the status quo and a full fair solution.

Alberto González-Sanz, Lucas de Lara, Louis Béthune et al.

This paper introduces the first statistically consistent estimator of the optimal transport map between two probability distributions, based on neural networks. Building on theoretical and practical advances in the field of Lipschitz neural networks, we define a Lipschitz-constrained generative adversarial network penalized by the quadratic transportation cost. Then, we demonstrate that, under regularity assumptions, the obtained generator converges uniformly to the optimal transport map as the sample size increases to infinity. Furthermore, we show through a number of numerical experiments that the learnt mapping has promising performances. In contrast to previous work tackling either statistical guarantees or practicality, we provide an expressive and feasible estimator which paves way for optimal transport applications where the asymptotic behaviour must be certified.

Lucas de Lara, Alberto González-Sanz, Nicholas Asher et al.

Counterfactual frameworks have grown popular in machine learning for both explaining algorithmic decisions but also defining individual notions of fairness, more intuitive than typical group fairness conditions. However, state-of-the-art models to compute counterfactuals are either unrealistic or unfeasible. In particular, while Pearl's causal inference provides appealing rules to calculate counterfactuals, it relies on a model that is unknown and hard to discover in practice. We address the problem of designing realistic and feasible counterfactuals in the absence of a causal model. We define transport-based counterfactual models as collections of joint probability distributions between observable distributions, and show their connection to causal counterfactuals. More specifically, we argue that optimal-transport theory defines relevant transport-based counterfactual models, as they are numerically feasible, statistically-faithful, and can coincide under some assumptions with causal counterfactual models. Finally, these models make counterfactual approaches to fairness feasible, and we illustrate their practicality and efficiency on fair learning. With this paper, we aim at laying out the theoretical foundations for a new, implementable approach to counterfactual thinking.

Hédi Hadiji, Sébastien Gerchinovitz, Jean-Michel Loubes et al.

We consider the bandit-based framework for diversity-preserving recommendations introduced by Celis et al. (2019), who approached it in the case of a polytope mainly by a reduction to the setting of linear bandits. We design a UCB algorithm using the specific structure of the setting and show that it enjoys a bounded distribution-dependent regret in the natural cases when the optimal mixed actions put some probability mass on all actions (i.e., when diversity is desirable). The regret lower bounds provided show that otherwise, at least when the model is mean-unbounded, a $\ln T$ regret is suffered. We also discuss an example beyond the special case of polytopes.

Mathieu Serrurier, Franck Mamalet, Alberto González-Sanz et al.

Adversarial examples have pointed out Deep Neural Networks vulnerability to small local noise. It has been shown that constraining their Lipschitz constant should enhance robustness, but make them harder to learn with classical loss functions. We propose a new framework for binary classification, based on optimal transport, which integrates this Lipschitz constraint as a theoretical requirement. We propose to learn 1-Lipschitz networks using a new loss that is an hinge regularized version of the Kantorovich-Rubinstein dual formulation for the Wasserstein distance estimation. This loss function has a direct interpretation in terms of adversarial robustness together with certifiable robustness bound. We also prove that this hinge regularized version is still the dual formulation of an optimal transportation problem, and has a solution. We also establish several geometrical properties of this optimal solution, and extend the approach to multi-class problems. Experiments show that the proposed approach provides the expected guarantees in terms of robustness without any significant accuracy drop. The adversarial examples, on the proposed models, visibly and meaningfully change the input providing an explanation for the classification.

Eustasio del Barrio, Jean-Michel Loubes

We propose to tackle the problem of understanding the effect of regularization in Sinkhorn algotihms. In the case of Gaussian distributions we provide a closed form for the regularized optimal transport which enables to provide a better understanding of the effect of the regularization from a statistical framework.

Eustasio del Barrio, Paula Gordaliza, Jean-Michel Loubes

A review of the main fairness definitions and fair learning methodologies proposed in the literature over the last years is presented from a mathematical point of view. Following our independence-based approach, we consider how to build fair algorithms and the consequences on the degradation of their performance compared to the possibly unfair case. This corresponds to the price for fairness given by the criteria $\textit{statistical parity}$ or $\textit{equality of odds}$. Novel results giving the expressions of the optimal fair classifier and the optimal fair predictor (under a linear regression gaussian model) in the sense of $\textit{equality of odds}$ are presented.

Thibaut Le Gouic, Jean-Michel Loubes, Philippe Rigollet

In the context of regression, we consider the fundamental question of making an estimator fair while preserving its prediction accuracy as much as possible. To that end, we define its projection to fairness as its closest fair estimator in a sense that reflects prediction accuracy. Our methodology leverages tools from optimal transport to construct efficiently the projection to fairness of any given estimator as a simple post-processing step. Moreover, our approach precisely quantifies the cost of fairness, measured in terms of prediction accuracy.

Philippe Besse, Eustasio del Barrio, Paula Gordaliza et al.

Applications based on Machine Learning models have now become an indispensable part of the everyday life and the professional world. A critical question then recently arised among the population: Do algorithmic decisions convey any type of discrimination against specific groups of population or minorities? In this paper, we show the importance of understanding how a bias can be introduced into automatic decisions. We first present a mathematical framework for the fair learning problem, specifically in the binary classification setting. We then propose to quantify the presence of bias by using the standard Disparate Impact index on the real and well-known Adult income data set. Finally, we check the performance of different approaches aiming to reduce the bias in binary classification outcomes. Importantly, we show that some intuitive methods are ineffective. This sheds light on the fact trying to make fair machine learning models may be a particularly challenging task, in particular when the training observations contain a bias.

Joseph Lam-Weil, Béatrice Laurent, Jean-Michel Loubes

Finding anonymization mechanisms to protect personal data is at the heart of recent machine learning research. Here, we consider the consequences of local differential privacy constraints on goodness-of-fit testing, i.e. the statistical problem assessing whether sample points are generated from a fixed density $f_0$, or not. The observations are kept hidden and replaced by a stochastic transformation satisfying the local differential privacy constraint. In this setting, we propose a testing procedure which is based on an estimation of the quadratic distance between the density $f$ of the unobserved samples and $f_0$. We establish an upper bound on the separation distance associated with this test, and a matching lower bound on the minimax separation rates of testing under non-interactive privacy in the case that $f_0$ is uniform, in discrete and continuous settings. To the best of our knowledge, we provide the first minimax optimal test and associated private transformation under a local differential privacy constraint over Besov balls in the continuous setting, quantifying the price to pay for data privacy. We also present a test that is adaptive to the smoothness parameter of the unknown density and remains minimax optimal up to a logarithmic factor. Finally, we note that our results can be translated to the discrete case, where the treatment of probability vectors is shown to be equivalent to that of piecewise constant densities in our setting. That is why we work with a unified setting for both the continuous and the discrete cases.

Laurent Risser, Alberto Gonzalez Sanz, Quentin Vincenot et al.

The increasingly common use of neural network classifiers in industrial and social applications of image analysis has allowed impressive progress these last years. Such methods are however sensitive to algorithmic bias, i.e. to an under- or an over-representation of positive predictions or to higher prediction errors in specific subgroups of images. We then introduce in this paper a new method to temper the algorithmic bias in Neural-Network based classifiers. Our method is Neural-Network architecture agnostic and scales well to massive training sets of images. It indeed only overloads the loss function with a Wasserstein-2 based regularization term for which we back-propagate the impact of specific output predictions using a new model, based on the Gateaux derivatives of the predictions distribution. This model is algorithmically reasonable and makes it possible to use our regularized loss with standard stochastic gradient-descent strategies. Its good behavior is assessed on the reference Adult census, MNIST, CelebA datasets.

Eustasio del Barrio, Hristo Inouzhe, Jean-Michel Loubes et al.

Data obtained from Flow Cytometry present pronounced variability due to biological and technical reasons. Biological variability is a well-known phenomenon produced by measurements on different individuals, with different characteristics such as illness, age, sex, etc. The use of different settings for measurement, the variation of the conditions during experiments and the different types of flow cytometers are some of the technical causes of variability. This mixture of sources of variability makes the use of supervised machine learning for identification of cell populations difficult. The present work is conceived as a combination of strategies to facilitate the task of supervised gating. We propose $optimalFlowTemplates$, based on a similarity distance and $\text{Wasserstein barycenters}$, which clusters cytometries and produces prototype cytometries for the different groups. We show that supervised learning, restricted to the new groups, performs better than the same techniques applied to the whole collection. We also present $optimalFlowClassification$, which uses a database of gated cytometries and optimalFlowTemplates to assign cell types to a new cytometry. We show that this procedure can outperform state of the art techniques in the proposed datasets. Our code is freely available as $optimalFlow$ a Bioconductor R package at https://bioconductor.org/packages/optimalFlow. optimalFlowTemplates+optimalFlowClassification addresses the problem of using supervised learning while accounting for biological and technical variability. Our methodology provides a robust automated gating workflow that handles the intrinsic variability of flow cytometry data well. Our main innovation is the methodology itself and the optimal-transport techniques that we apply to flow cytometry analysis.

François Bachoc, Fabrice Gamboa, Max Halford et al.

In this paper, we present a new explainability formalism designed to shed light on how each input variable of a test set impacts the predictions of machine learning models. Hence, we propose a group explainability formalism for trained machine learning decision rules, based on their response to the variability of the input variables distribution. In order to emphasize the impact of each input variable, this formalism uses an information theory framework that quantifies the influence of all input-output observations based on entropic projections. This is thus the first unified and model agnostic formalism enabling data scientists to interpret the dependence between the input variables, their impact on the prediction errors, and their influence on the output predictions. Convergence rates of the entropic projections are provided in the large sample case. Most importantly, we prove that computing an explanation in our framework has a low algorithmic complexity, making it scalable to real-life large datasets. We illustrate our strategy by explaining complex decision rules learned by using XGBoost, Random Forest or Deep Neural Network classifiers on various datasets such as Adult Income, MNIST, CelebA, Boston Housing, Iris, as well as synthetic ones. We finally make clear its differences with the explainability strategies LIME and SHAP, that are based on single observations. Results can be reproduced by using the freely distributed Python toolbox https://gems-ai.aniti.fr/.

Philippe Besse, Celine Castets-Renard, Aurelien Garivier et al.

Combining big data and machine learning algorithms, the power of automatic decision tools induces as much hope as fear. Many recently enacted European legislation (GDPR) and French laws attempt to regulate the use of these tools. Leaving aside the well-identified problems of data confidentiality and impediments to competition, we focus on the risks of discrimination, the problems of transparency and the quality of algorithmic decisions. The detailed perspective of the legal texts, faced with the complexity and opacity of the learning algorithms, reveals the need for important technological disruptions for the detection or reduction of the discrimination risk, and for addressing the right to obtain an explanation of the auto- matic decision. Since trust of the developers and above all of the users (citizens, litigants, customers) is essential, algorithms exploiting personal data must be deployed in a strict ethical framework. In conclusion, to answer this need, we list some ways of controls to be developed: institutional control, ethical charter, external audit attached to the issue of a label.

Philippe Besse, Eustasio del Barrio, Paula Gordaliza et al.

We provide the asymptotic distribution of the major indexes used in the statistical literature to quantify disparate treatment in machine learning. We aim at promoting the use of confidence intervals when testing the so-called group disparate impact. We illustrate on some examples the importance of using confidence intervals and not a single value.

Camille Champion, Anne-Claire Brunet, Jean-Michel Loubes et al.

In this paper, we present a new R package COREclust dedicated to the detection of representative variables in high dimensional spaces with a potentially limited number of observations. Variable sets detection is based on an original graph clustering strategy denoted CORE-clustering algorithm that detects CORE-clusters, i.e. variable sets having a user defined size range and in which each variable is very similar to at least another variable. Representative variables are then robustely estimate as the CORE-cluster centers. This strategy is entirely coded in C++ and wrapped by R using the Rcpp package. A particular effort has been dedicated to keep its algorithmic cost reasonable so that it can be used on large datasets. After motivating our work, we will explain the CORE-clustering algorithm as well as a greedy extension of this algorithm. We will then present how to use it and results obtained on synthetic and real data.

François Bachoc, Baptiste Broto, Fabrice Gamboa et al.

In the framework of the supervised learning of a real function defined on a space X , the so called Kriging method stands on a real Gaussian field defined on X. The Euclidean case is well known and has been widely studied. In this paper, we explore the less classical case where X is the non commutative finite group of permutations. In this setting, we propose and study an harmonic analysis of the covariance operators that enables to consider Gaussian processes models and forecasting issues. Our theory is motivated by statistical ranking problems.

Clémentine Barreyre, Béatrice Laurent, Jean-Michel Loubes et al.

We propose a novel procedure for outlier detection in functional data, in a semi-supervised framework. As the data is functional, we consider the coefficients obtained after projecting the observations onto orthonormal bases (wavelet, PCA). A multiple testing procedure based on the two-sample test is defined in order to highlight the levels of the coefficients on which the outliers appear as significantly different to the normal data. The selected coefficients are then called features for the outlier detection, on which we compute the Local Outlier Factor to highlight the outliers. This procedure to select the features is applied on simulated data that mimic the behaviour of space telemetries, and compared with existing dimension reduction techniques.

Thomas Epelbaum, Fabrice Gamboa, Jean-Michel Loubes et al.

In this paper, we propose deep learning architectures (FNN, CNN and LSTM) to forecast a regression model for time dependent data. These algorithm's are designed to handle Floating Car Data (FCD) historic speeds to predict road traffic data. For this we aggregate the speeds into the network inputs in an innovative way. We compare the RMSE thus obtained with the results of a simpler physical model, and show that the latter achieves better RMSE accuracy. We also propose a new indicator, which evaluates the algorithms improvement when compared to a benchmark prediction. We conclude by questioning the interest of using deep learning methods for this specific regression task.

François Bachoc, Fabrice Gamboa, Jean-Michel Loubes et al.

Monge-Kantorovich distances, otherwise known as Wasserstein distances, have received a growing attention in statistics and machine learning as a powerful discrepancy measure for probability distributions. In this paper, we focus on forecasting a Gaussian process indexed by probability distributions. For this, we provide a family of positive definite kernels built using transportation based distances. We provide a probabilistic understanding of these kernels and characterize the corresponding stochastic processes. We prove that the Gaussian processes indexed by distributions corresponding to these kernels can be efficiently forecast, opening new perspectives in Gaussian process modeling.

Philippe Besse, Brendan Guillouet, Jean-Michel Loubes

Management and analysis of big data are systematically associated with a data distributed architecture in the Hadoop and now Spark frameworks. This article offers an introduction for statisticians to these technologies by comparing the performance obtained by the direct use of three reference environments: R, Python Scikit-learn, Spark MLlib on three public use cases: character recognition, recommending films, categorizing products. As main result, it appears that, if Spark is very efficient for data munging and recommendation by collaborative filtering (non-negative factorization), current implementations of conventional learning methods (logistic regression, random forests) in MLlib or SparkML do not ou poorly compete habitual use of these methods (R, Python Scikit-learn) in an integrated or undistributed architecture

Philippe C. Besse, Brendan Guillouet, Jean-Michel Loubes et al.

In this paper we propose a new method to predict the final destination of vehicle trips based on their initial partial trajectories. We first review how we obtained clustering of trajectories that describes user behaviour. Then, we explain how we model main traffic flow patterns by a mixture of 2d Gaussian distributions. This yielded a density based clustering of locations, which produces a data driven grid of similar points within each pattern. We present how this model can be used to predict the final destination of a new trajectory based on their first locations using a two step procedure: We first assign the new trajectory to the clusters it mot likely belongs. Secondly, we use characteristics from trajectories inside these clusters to predict the final destination. Finally, we present experimental results of our methods for classification of trajectories and final destination prediction on datasets of timestamped GPS-Location of taxi trips. We test our methods on two different datasets, to assess the capacity of our method to adapt automatically to different subsets.

Philippe Besse, Brendan Guillouet, Jean-Michel Loubes et al.

In this paper we tackle the issue of clustering trajectories of geolocalized observations. Using clustering technics based on the choice of a distance between the observations, we first provide a comprehensive review of the different distances used in the literature to compare trajectories. Then based on the limitations of these methods, we introduce a new distance : Symmetrized Segment-Path Distance (SSPD). We finally compare this new distance to the others according to their corresponding clustering results obtained using both hierarchical clustering and affinity propagation methods.

Guillaume Allain, Fabrice Gamboa, Philippe Goudal et al.

We provide a model to understand how adverse weather conditions modify traffic flow dynamic. We first prove that the microscopic Free Flow Speed of the vehicles is changed and then provide a rule to model this change. For this, we consider a thresholded linear model, corresponding to an application of a MARS model to road trafficking. This model adapts itself locally to the whole road network and provides accurate unbiased forecasted speed using live or short term forecasted weather data information.