arXiv subjects

Julien Fageot

Publications and source records attributed to Julien Fageot.

At least 19 recordsLinked to original sources

arXiv2026

The Benefits of Diversity: Combining Comparisons and Ratings for Efficient Scoring

Should humans be asked to evaluate entities individually or comparatively? This question has been the subject of long debates. In this work, we show that, interestingly, combining both forms of preference elicitation can outperform the focus on a single kind. More specifically, we introduce SCoRa (Scoring from Comparisons and Ratings), a unified probabilistic model that allows to learn from both signals. We prove that the MAP estimator of SCoRa is well-behaved. It verifies monotonicity and robustness guarantees. We then empirically show that SCoRa recovers accurate scores, even under model mismatch. Most interestingly, we identify a realistic setting where combining comparisons and ratings outperforms using either one alone, and when the accurate ordering of top entities is critical. Given the de facto availability of signals of multiple forms, SCoRa additionally offers a versatile foundation for preference learning.

cs.LG↗

arXiv2026

Byzantine Machine Learning: MultiKrum and an optimal notion of robustness

Aggregation rules are the cornerstone of distributed (or federated) learning in the presence of adversaries, under the so-called Byzantine threat model. They are also interesting mathematical objects from the point of view of robust mean estimation. The Krum aggregation rule has been extensively studied, and endowed with formal robustness and convergence guarantees. Yet, MultiKrum, a natural extension of Krum, is often preferred in practice for its superior empirical performance, even though no theoretical guarantees were available until now. In this work, we provide the first proof that MultiKrum is a robust aggregation rule, and bound its robustness coefficient. To do so, we introduce $\kappa^\star$, the optimal *robustness coefficient* of an aggregation rule, which quantifies the accuracy of mean estimation in the presence of adversaries in a tighter manner compared with previously adopted notions of robustness. We then construct an upper and a lower bound on MultiKrum's robustness coefficient. As a by-product, we also improve on the best-known bounds on Krum's robustness coefficient. We show that MultiKrum's bounds are never worse than Krum's, and better in realistic regimes. We illustrate this analysis by an experimental investigation on the quality of the lower bound.

stat.ML↗

arXiv2025

Decoupled Solution for Composite Sparse-plus-Smooth Inverse Problems

We consider composite linear inverse problems where the signal to recover is modeled as a sum of two functions. We study a variational framework formulated as an optimization problem over the pairs of components using two regularization terms, each of them acting on a different part of the solution. The specificity of our work is to study the case where one component is regularized with an atomic norm over a Banach space, which is known to promote sparse reconstruction, while the other is regularized with a quadratic norm over a Hilbert space, which promotes smooth solution. We show how this composite optimization problem can be reduced to an optimization problem over the Banach space component only up to a linear problem. This reveals a decoupling between the two components, allowing for a new composite representer theorem. It naturally induces a decoupled numerical procedure to solve the composite optimization problem. We exemplify our main result with a composite deconvolution problem of Dirac recovery over a smooth background. In this setting, we illustrate the relevance of a composite model and show a significant temporal gain on signal reconstruction, which results from our decoupled algorithmic approach.

math.OC↗

arXiv2025

Generalizing while preserving monotonicity in comparison-based preference learning models

If you tell a learning model that you prefer an alternative $a$ over another alternative $b$, then you probably expect the model to be monotone, that is, the valuation of $a$ increases, and that of $b$ decreases. Yet, perhaps surprisingly, many widely deployed comparison-based preference learning models, including large language models, fail to have this guarantee. Until now, the only comparison-based preference learning algorithms that were proved to be monotone are the Generalized Bradley-Terry models. Yet, these models are unable to generalize to uncompared data. In this paper, we advance the understanding of the set of models with generalization ability that are monotone. Namely, we propose a new class of Linear Generalized Bradley-Terry models with Diffusion Priors, and identify sufficient conditions on alternatives' embeddings that guarantee monotonicity. Our experiments show that this monotonicity is far from being a general guarantee, and that our new class of generalizing models improves accuracy, especially when the dataset is limited.

math.ST↗

arXiv2025

On Monotonicity in AI Alignment

Comparison-based preference learning has become central to the alignment of AI models with human preferences. However, these methods may behave counterintuitively. After empirically observing that, when accounting for a preference for response $y$ over $z$, the model may actually decrease the probability (and reward) of generating $y$ (an observation also made by others), this paper investigates the root causes of (non) monotonicity, for a general comparison-based preference learning framework that subsumes Direct Preference Optimization (DPO), Generalized Preference Optimization (GPO) and Generalized Bradley-Terry (GBT). Under mild assumptions, we prove that such methods still satisfy what we call local pairwise monotonicity. We also provide a bouquet of formalizations of monotonicity, and identify sufficient conditions for their guarantee, thereby providing a toolbox to evaluate how prone learning models are to monotonicity violations. These results clarify the limitations of current methods and provide guidance for developing more trustworthy preference learning algorithms.

math.ST↗

arXiv2025

Variational Seasonal-Trend Decomposition with Sparse Continuous-Domain Regularization

We consider the inverse problem of recovering a continuous-domain function from a finite number of noisy linear measurements. The unknown signal is modeled as the sum of a slowly varying trend and a periodic or quasi-periodic seasonal component. We formulate a variational framework for their joint recovery by introducing convex regularizations based on generalized total variation, which promote sparsity in spline-like representations. Our analysis is conducted in an infinite-dimensional setting and leads to a representer theorem showing that minimizers are splines in both components. To make the approach numerically feasible, we introduce a family of discrete approximations and prove their convergence to the original problem in the sense of $\Gamma$-convergence. This further ensures the uniform convergence of discrete solutions to their continuous counterparts. The proposed framework offers a principled approach to seasonal-trend decomposition in the presence of noise and limited measurements, with theoretical guarantees on both representation and discretization.

math.FA↗

arXiv2024

PolyCLEAN: Atomic Optimization for Super-Resolution Imaging and Uncertainty Estimation in Radio Interferometry

Aims: We address two issues for the adoption of convex optimization in radio interferometric imaging. First, a method for a fine resolution setup is proposed which scales naturally in terms of memory usage and reconstruction speed. Second, a new tool to localize a region of uncertainty is developed, paving the way for quantitative imaging in radio interferometry. Methods: The classical $\ell_1$ penalty is used to turn the inverse problem into a sparsity-promoting optimization. For efficient implementation, the so-called Frank-Wolfe algorithm is used together with a \textit{polyatomic} refinement. The algorithm naturally produces sparse images at each iteration, leveraged to reduce memory and computational requirements. In that regard, PolyCLEAN reproduces the numerical behavior of CLEAN while guaranteeing that it solves the minimization problem of interest. Additionally, we introduce the concept of the \textit{dual certificate image}, which appears as a numerical byproduct of the Frank-Wolfe algorithm. This image is proposed as a tool for uncertainty quantification on the location of the recovered sources. Results: PolyCLEAN demonstrates good scalability performance, in particular for fine-resolution grids. On simulations, the Python-based implementation is competitive with the fast numerically-optimized CLEAN solver. This acceleration does not affect image reconstruction quality: PolyCLEAN images are consistent with CLEAN-obtained ones for both point sources and diffuse emission recovery. We also highlight PolyCLEAN reconstruction capabilities on observed radio measurements. Conclusions: PolyCLEAN can be considered as an alternative to CLEAN in the radio interferometric imaging pipeline, as it enables the use of Bayesian priors without impacting the scalability and numerical performance of the imaging method.

astro-ph.IM↗

arXiv2024

A Decoupled Approach for Composite Sparse-plus-Smooth Penalized Optimization

We consider a linear inverse problem whose solution is expressed as a sum of two components: one smooth and the other sparse. This problem is addressed by minimizing an objective function with a least squares data-fidelity term and a different regularization term applied to each of the components. Sparsity is promoted with an $\ell_1$ norm, while the smooth component is penalized with an $\ell_2$ norm. We characterize the solution set of this composite optimization problem by stating a Representer Theorem. Consequently, we identify that solving the optimization problem can be decoupled by first identifying the sparse solution as a solution of a modified single-variable problem and then deducing the smooth component. We illustrate that this decoupled solving method can lead to significant computational speedups in applications, considering the problem of Dirac recovery over a smooth background with two-dimensional partial Fourier measurements.

eess.SP↗

arXiv2023

Generalized Bradley-Terry Models for Score Estimation from Paired Comparisons

Many applications, e.g. in content recommendation, sports, or recruitment, leverage the comparisons of alternatives to score those alternatives. The classical Bradley-Terry model and its variants have been widely used to do so. The historical model considers binary comparisons (victory or defeat) between alternatives, while more recent developments allow finer comparisons to be taken into account. In this article, we introduce a probabilistic model encompassing a broad variety of paired comparisons that can take discrete or continuous values. We do so by considering a well-behaved subset of the exponential family, which we call the family of generalized Bradley-Terry (GBT) models, as it includes the classical Bradley-Terry model and many of its variants. Remarkably, we prove that all GBT models are guaranteed to yield a strictly convex negative log-likelihood. Moreover, assuming a Gaussian prior on alternatives' scores, we prove that the maximum a posteriori (MAP) of GBT models, whose existence, uniqueness and fast computation are thus guaranteed, varies monotonically with respect to comparisons (the more A beats B, the better the score of A) and is Lipschitz-resilient with respect to each new comparison (a single new comparison can only have a bounded effect on all the estimated scores). These desirable properties make GBT models appealing for practical use. We illustrate some features of GBT models on simulations.

stat.ME↗

arXiv2022

On the Uniqueness of Solutions for the Basis Pursuit in the Continuum

This paper studies the continuous-domain inverse problem of recovering Radon measures on the one-dimensional torus from low-frequency Fourier coefficients, where Kc is the cutoff frequency. Our approach consists in minimizing the total-variation norm among all Radon measures that are consistent with the observations. We call this problem the basis pursuit in the continuum (BPC). We characterize the solution set of (BPC) in terms of uniqueness and describe its sparse solutions which are sums of few signed Dirac masses. The characterization is determined by the spectrum of a Toeplitz and Hermitian-symmetric matrix that solely depends on the observations. More precisely, we prove that (BPC) has a unique solution if and only if this matrix is neither positive definite nor negative definite. If it has both a positive and negative eigenvalue, then the unique solution is the sum of at most 2Kc Dirac masses, with at least one positive and one negative weight. If this matrix is positive (respectively negative) semi-definite and rank deficient, then the unique solution is composed of a number of Dirac masses equal to the rank of the matrix, all of which have nonnegative (respectively nonpositive) weights. Finally, in cases where (BPC) has multiple solutions, we demonstrate that there are infinitely many solutions composed of Kc+1 Dirac masses, with nonnegative (respectively nonpositive) weights if the matrix is positive (respectively negative) definite.

math.OC↗

arXiv2022

Solidago: A Modular Collaborative Scoring Pipeline

This paper presents Solidago, an end-to-end modular pipeline to allow any community of users to collaboratively score any number of entities. Solidago proposes a six-module decomposition. First, it uses pretrust and peer-to-peer vouches to assign trust scores to users. Second, based on participation, trust scores are turned into voting rights per user per entity. Third, for each user, a preference model is learned from the user's evaluation data. Fourth, users' models are put on a similar scale. Fifth, these models are securely aggregated. Sixth, models are post-processed to yield human-readable global scores. We also propose default implementations of the six modules, including a novel trust propagation algorithm, and adaptations of state-of-the-art scaling and aggregation solutions. Our pipeline has been successfully deployed on the open-source platform tournesol.app. We thereby lay an appealing foundation for the collaborative, effective, scalable, fair, interpretable and secure scoring of any set of entities.

cs.SI↗

arXiv2022

Functional Estimation of Anisotropic Covariance and Autocovariance Operators on the Sphere

We propose nonparametric estimators for the second-order central moments of possibly anisotropic spherical random fields, within a functional data analysis context. We consider a measurement framework where each random field among an identically distributed collection of spherical random fields is sampled at a few random directions, possibly subject to measurement error. The collection of random fields could be i.i.d. or serially dependent. Though similar setups have already been explored for random functions defined on the unit interval, the nonparametric estimators proposed in the literature often rely on local polynomials, which do not readily extend to the (product) spherical setting. We therefore formulate our estimation procedure as a variational problem involving a generalized Tikhonov regularization term. The latter favours smooth covariance/autocovariance functions, where the smoothness is specified by means of suitable Sobolev-like pseudo-differential operators. Using the machinery of reproducing kernel Hilbert spaces, we establish representer theorems that fully characterize the form of our estimators. We determine their uniform rates of convergence as the number of random fields diverges, both for the dense (increasing number of spatial samples) and sparse (bounded number of spatial samples) regimes. We moreover demonstrate the computational feasibility and practical merits of our estimation procedure in a simulation setting, assuming a fixed number of samples per random field. Our numerical estimation procedure leverages the sparsity and second-order Kronecker structure of our setup to reduce the computational and memory requirements by approximately three orders of magnitude compared to a naive implementation would require.

math.ST↗

arXiv2022

Complex-Order Scale-Invariant Operators and Self-Similar Processes

Derivatives and integration operators are well-studied examples of linear operators that commute with scaling up to a fixed multiplicative factor; i.e., they are scale-invariant. Fractional order derivatives (integration operators) also belong to this family. In this paper, we extend the fractional operators to complex-order operators by constructing them in the Fourier domain. We analyze these operators in details with a special emphasis on the decay properties of the outputs. We further use these operators to introduce a family of complex-valued stable processes that are self-similar with complex-valued Hurst indices. These processes are expressed via the characteristic functionals over the Schwartz space of functions. Besides the self-similarity and stationarity, we study the regularity (in terms of Sobolev spaces) of the proposed processes.

math.FA↗

arXiv2022

Une version polyatomique de l'algorithme Frank-Wolfe pour résoudre le problème LASSO en grandes dimensions

Nous nous intéressons à la reconstruction parcimonieuse d'images à l'aide du problème d'optimisation régularisé LASSO. Dans de nombreuses applications pratiques, les grandes dimensions des objets à reconstruire limitent, voire empêchent, l'utilisation des méthodes de résolution proximales classiques. C'est le cas par exemple en radioastronomie. Nous détaillons dans cet article le fonctionnement de l'algorithme \textit{Frank-Wolfe Polyatomique}, spécialement développé pour résoudre le problème LASSO dans ces contextes exigeants. Nous démontrons sa supériorité par rapport aux méthodes proximales dans des situations en grande dimension avec des mesures de Fourier, lors de la résolution de problèmes simulés inspirés de la radio-interférométrie. -- We consider the problem of recovering sparse images by means of the penalised optimisation problem LASSO. For various practical applications, it is impossible to rely on the proximal solvers commonly used for that purpose due to the size of the objects to recover, as it is the case for radio astronomy. In this article we explain the mechanisms of the \textit{Polyatomic Frank-Wolfe algorithm}, specifically designed to minimise the LASSO problem in such challenging contexts. We demonstrate in simulated problems inspired from radio-interferometry the preeminence of this algorithm over the proximal methods for high dimensional images with Fourier measurements.

eess.SP↗

arXiv2022

TV-based Spline Reconstruction with Fourier Measurements: Uniqueness and Convergence of Grid-Based Methods

We study the problem of recovering piecewise-polynomial periodic functions from their low-frequency information. This means that we only have access to possibly corrupted versions of the Fourier samples of the ground truth up to a maximum cutoff frequency $K_c$. The reconstruction task is specified as an optimization problem with total-variation (TV) regularization (in the sense of measures) involving the $M$-th order derivative regularization operator $\mathrm{L} = \mathrm{D}^M$. The order $M \geq 1$ determines the degree of the reconstructed piecewise polynomial spline, whereas the TV regularization norm, which is known to promote sparsity, guarantees a small number of pieces. We show that the solution of our optimization problem is always unique, which, to the best of our knowledge, is a first for TV-based problems. Moreover, we show that this solution is a periodic spline matched to the regularization operator $\mathrm{L}$ whose number of knots is upper-bounded by $2 K_c$. We then consider the grid-based discretization of our optimization problem in the space of uniform $\mathrm{L}$-splines. On the theoretical side, we show that any sequence of solutions of the discretized problem converges uniformly to the unique solution of the gridless problem as the grid size vanishes. Finally, on the algorithmic side, we propose a B-spline-based algorithm to solve the grid-based problem, and we demonstrate its numerical feasibility experimentally. On both of these aspects, we leverage the uniqueness of the solution of the original problem.

math.OC↗

arXiv2022

A Fast and Scalable Polyatomic Frank-Wolfe Algorithm for the LASSO

We propose a fast and scalable Polyatomic Frank-Wolfe (P-FW) algorithm for the resolution of high-dimensional LASSO regression problems. The latter improves upon traditional Frank-Wolfe methods by considering generalized greedy steps with polyatomic (i.e. linear combinations of multiple atoms) update directions, hence allowing for a more efficient exploration of the search space. To preserve sparsity of the intermediate iterates, we re-optimize the LASSO problem over the set of selected atoms at each iteration. For efficiency reasons, the accuracy of this re-optimization step is relatively low for early iterations and gradually increases with the iteration count. We provide convergence guarantees for our algorithm and validate it in simulated compressed sensing setups. Our experiments reveal that P-FW outperforms state-of-the-art methods in terms of runtime, both for FW methods and optimal first-order proximal gradient methods such as the Fast Iterative Soft-Thresholding Algorithm (FISTA).

eess.SP↗

arXiv2022

On Tempered Discrete and Lévy White Noises

We study the growth properties of the family of i.i.d. random sequences, also known as discrete white noises, and of their continuous-domain generalization, the family of Lévy white noises. More precisely, we characterize the members of both families which are tempered in terms of their moment properties. We recover the characterization of tempered Lévy white noises obtained by Robert Dalang and Thomas Humeau and provide a new proof of there fundamental result. Our approach is based on a fruitful connection between the discrete and continuous-domain white noises.

math.PR↗

arXiv2021

Sparsest Piecewise-Linear Regression of One-Dimensional Data

We study the problem of one-dimensional regression of data points with total-variation (TV) regularization (in the sense of measures) on the second derivative, which is known to promote piecewise-linear solutions with few knots. While there are efficient algorithms for determining such adaptive splines, the difficulty with TV regularization is that the solution is generally non-unique, an aspect that is often ignored in practice. In this paper, we present a systematic analysis that results in a complete description of the solution set with a clear distinction between the cases where the solution is unique and those, much more frequent, where it is not. For the latter scenario, we identify the sparsest solutions, i.e., those with the minimum number of knots, and we derive a formula to compute the minimum number of knots based solely on the data points. To achieve this, we first consider the problem of exact interpolation which leads to an easier theoretical analysis. Next, we relax the exact interpolation requirement to a regression setting, and we consider a penalized optimization problem with a strictly convex data-fidelity cost function. We show that the underlying penalized problem can be reformulated as a constrained problem, and thus that all our previous results still apply. Based on our theoretical analysis, we propose a simple and fast two-step algorithm, agnostic to uniqueness, to reach a sparsest solution of this penalized problem.

math.OC↗