Jean-Baptiste Fermanian

Jean-Baptiste Fermanian, Batiste Le Bars, Aurélien Bellet

Personalized Federated Learning (PFL) enables a collection of agents to collaboratively learn individual models without sharing raw data. We propose a new PFL approach in which each agent optimizes a weighted combination of all agents' empirical risks, with the weights learned from data rather than specified a priori. The novelty of our method lies in formulating the estimation of these collaborative weights as a kernel mean embedding estimation problem with multiple data sources, leveraging tools from multi-task averaging to capture statistical relationships between agents. This perspective yields a fully adaptive procedure that requires no prior knowledge of data heterogeneity and can automatically transition between global and local learning regimes. By recasting the objective as a high-dimensional mean estimation problem, we derive finite-sample guarantees on local excess risks for a broad class of distributions, explicitly quantifying the statistical gains of collaboration. To address communication constraints inherent to federated settings, we also propose a practical implementation based on random Fourier features, which allows one to trade communication cost for statistical efficiency. Numerical experiments validate our theoretical results.

Tiffany Ding, Jean-Baptiste Fermanian, Joseph Salmon

Many real-world classification problems, such as plant identification, have extremely long-tailed class distributions. In order for prediction sets to be useful in such settings, they should (i) provide good class-conditional coverage, ensuring that rare classes are not systematically omitted from the prediction sets, and (ii) be a reasonable size, allowing users to easily verify candidate labels. Unfortunately, existing conformal prediction methods, when applied to the long-tailed setting, force practitioners to make a binary choice between small sets with poor class-conditional coverage or sets with very good class-conditional coverage but that are extremely large. We propose methods with guaranteed marginal coverage that smoothly trade off between set size and class-conditional coverage. First, we introduce a new conformal score function called prevalence-adjusted softmax that targets macro-coverage, a relaxed notion of class-conditional coverage. Second, we propose a new procedure that interpolates between marginal and class-conditional conformal prediction by linearly interpolating their conformal score thresholds. We demonstrate our methods on Pl@ntNet-300K and iNaturalist-2018, two long-tailed image datasets with 1,081 and 8,142 classes, respectively.

Jean-Baptiste Fermanian, Mohamed Hebiri, Joseph Salmon

Conformal prediction methods are statistical tools designed to quantify uncertainty and generate predictive sets with guaranteed coverage probabilities. This work introduces an innovative refinement to these methods for classification tasks, specifically tailored for scenarios where multiple observations (multi-inputs) of a single instance are available at prediction time. Our approach is particularly motivated by applications in citizen science, where multiple images of the same plant or animal are captured by individuals. Our method integrates the information from each observation into conformal prediction, enabling a reduction in the size of the predicted label set while preserving the required class-conditional coverage guarantee. The approach is based on the aggregation of conformal p-values computed from each observation of a multi-input. By exploiting the exact distribution of these p-values, we propose a general aggregation framework using an abstract scoring function, encompassing many classical statistical tools. Knowledge of this distribution also enables refined versions of standard strategies, such as majority voting. We evaluate our method on simulated and real data, with a particular focus on Pl@ntNet, a prominent citizen science platform that facilitates the collection and identification of plant species through user-submitted images.

Jean-Baptiste Fermanian, Pierre Humbert, Gilles Blanchard

We introduce a method based on Conformal Prediction (CP) to quantify the uncertainty of full ranking algorithms. We focus on a specific scenario where $n+m$ items are to be ranked by some ``black box'' algorithm. It is assumed that the relative (ground truth) ranking of $n$ of them is known. The objective is then to quantify the error made by the algorithm on the ranks of the $m$ new items among the total $(n+m)$. In such a setting, the true ranks of the $n$ original items in the total $(n+m)$ depend on the (unknown) true ranks of the $m$ new ones. Consequently, we have no direct access to a calibration set to apply a classical CP method. To address this challenge, we propose to construct distribution-free bounds of the unknown conformity scores using recent results on the distribution of conformal p-values. Using these scores upper bounds, we provide valid prediction sets for the rank of any item. We also control the false coverage proportion, a crucial quantity when dealing with multiple prediction sets. Finally, we empirically show on both synthetic and real data the efficiency of our CP method for state-of-the-art algorithms such as RankNet or LambdaMart.

Gilles Blanchard, Jean-Baptiste Fermanian, Hannah Marienwald

We endeavour to estimate numerous multi-dimensional means of various probability distributions on a common space based on independent samples. Our approach involves forming estimators through convex combinations of empirical means derived from these samples. We introduce two strategies to find appropriate data-dependent convex combination weights: a first one employing a testing procedure to identify neighbouring means with low variance, which results in a closed-form plug-in formula for the weights, and a second one determining weights via minimization of an upper confidence bound on the quadratic risk. Through theoretical analysis, we evaluate the improvement in quadratic risk offered by our methods compared to the empirical means. Our analysis focuses on a dimensional asymptotics perspective, showing that our methods asymptotically approach an oracle (minimax) improvement as the effective dimension of the data increases. We demonstrate the efficacy of our methods in estimating multiple kernel mean embeddings through experiments on both simulated and real-world datasets.

Gilles Blanchard, Jean-Baptiste Fermanian

Let $\mathbf{X} = (X_i)_{1\leq i \leq n}$ be an i.i.d. sample of square-integrable variables in $\mathbb{R}^d$, \GB{with common expectation $μ$ and covariance matrix $Σ$, both unknown.} We consider the problem of testing if $μ$ is $η$-close to zero, i.e. $\|μ\| \leq η$ against $\|μ\| \geq (η+ δ)$; we also tackle the more general two-sample mean closeness (also known as {\em relevant difference}) testing problem. The aim of this paper is to obtain nonasymptotic upper and lower bounds on the minimal separation distance $δ$ such that we can control both the Type I and Type II errors at a given level. The main technical tools are concentration inequalities, first for a suitable estimator of $\|μ\|^2$ used a test statistic, and secondly for estimating the operator and Frobenius norms of $Σ$ coming into the quantiles of said test statistic. These properties are obtained for Gaussian and bounded distributions. A particular attention is given to the dependence in the pseudo-dimension $d_*$ of the distribution, defined as $d_* := \|Σ\|_2^2/\|Σ\|_\infty^2$. In particular, for $η=0$, the minimum separation distance is $Θ( d_*^{\frac{1}{4}}\sqrt{\|Σ\|_\infty/n})$, in contrast with the minimax estimation distance for $μ$, which is $Θ(d_e^{\frac{1}{2}}\sqrt{\|Σ\|_\infty/n})$ (where $d_e:=\|Σ\|_1/\|Σ\|_\infty$). This generalizes a phenomenon spelled out in particular by Baraud (2002).

Hannah Marienwald, Jean-Baptiste Fermanian, Gilles Blanchard

We propose an improved estimator for the multi-task averaging problem, whose goal is the joint estimation of the means of multiple distributions using separate, independent data sets. The naive approach is to take the empirical mean of each data set individually, whereas the proposed method exploits similarities between tasks, without any related information being known in advance. First, for each data set, similar or neighboring means are determined from the data by multiple testing. Then each naive estimator is shrunk towards the local average of its neighbors. We prove theoretically that this approach provides a reduction in mean squared error. This improvement can be significant when the dimension of the input space is large, demonstrating a "blessing of dimensionality" phenomenon. An application of this approach is the estimation of multiple kernel mean embeddings, which plays an important role in many modern applications. The theoretical results are verified on artificial and real world data.