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Samuel Vaiter

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18 papers

2 author rows

JMLR Journal 2026 Journal Article

CHANI: Correlation-based Hawkes Aggregation of Neurons with bio-Inspiration

Sophie Jaffard
Samuel Vaiter
Patricia Reynaud-Bouret

The present work aims at proving mathematically that a neural network inspired by biology can learn a classification task thanks to local transformations only. In this purpose, we propose a spiking neural network named CHANI (Correlation-based Hawkes Aggregation of Neurons with bio-Inspiration), whose neurons activity is modeled by Hawkes processes. Synaptic weights are updated thanks to an expert aggregation algorithm, providing a local and simple learning rule. We were able to prove that our network can learn on average and asymptotically. Moreover, we demonstrated that it automatically produces neuronal assemblies in the sense that the network can encode several classes and that a same neuron in the intermediate layers might be activated by more than one class, and we provided numerical simulations on synthetic datasets. This theoretical approach contrasts with the traditional empirical validation of biologically inspired networks and paves the way for understanding how local learning rules enable neurons to form assemblies able to represent complex concepts. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2026. ( edit, beta )

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NeurIPS Conference 2025 Conference Paper

Differentiable Generalized Sliced Wasserstein Plans

Laetitia Chapel
Romain Tavenard
Samuel Vaiter

Optimal Transport (OT) has attracted significant interest in the machine learning community, not only for its ability to define meaningful distances between probability distributions -- such as the Wasserstein distance -- but also for its formulation of OT plans. Its computational complexity remains a bottleneck, though, and slicing techniques have been developed to scale OT to large datasets. Recently, a novel slicing scheme, dubbed min-SWGG, lifts a single one-dimensional plan back to the original multidimensional space, finally selecting the slice that yields the lowest Wasserstein distance as an approximation of the full OT plan. Despite its computational and theoretical advantages, min-SWGG inherits typical limitations of slicing methods: (i) the number of required slices grows exponentially with the data dimension, and (ii) it is constrained to linear projections. Here, we reformulate min-SWGG as a bilevel optimization problem and propose a differentiable approximation scheme to efficiently identify the optimal slice, even in high-dimensional settings. We furthermore define its generalized extension for accommodating data living on manifolds. Finally, we demonstrate the practical value of our approach in various applications, including gradient flows on manifolds and high-dimensional spaces, as well as a novel sliced OT-based conditional flow matching for image generation -- where fast computation of transport plans is essential.

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NeurIPS Conference 2025 Conference Paper

From Shortcut to Induction Head: How Data Diversity Shapes Algorithm Selection in Transformers

Ryotaro Kawata
Yujin Song
Alberto Bietti
Naoki Nishikawa
Taiji Suzuki
Samuel Vaiter
Denny Wu

Transformers can implement both generalizable algorithms (e. g. , induction heads) and simple positional shortcuts (e. g. , memorizing fixed output positions). In this work, we study how the choice of pretraining data distribution steers a shallow transformer toward one behavior or the other. Focusing on a minimal trigger-output prediction task -- copying the token immediately following a special trigger upon its second occurrence -- we present a rigorous analysis of gradient-based training of a single-layer transformer. In both the infinite and finite sample regimes, we prove a transition in the learned mechanism: if input sequences exhibit sufficient diversity, measured by a low “max-sum” ratio of trigger-to-trigger distances, the trained model implements an induction head and generalizes to unseen contexts; by contrast, when this ratio is large, the model resorts to a positional shortcut and fails to generalize out-of-distribution (OOD). We also reveal a trade-off between the pretraining context length and OOD generalization, and derive the optimal pretraining distribution that minimizes computational cost per sample. Finally, we validate our theoretical predictions with controlled synthetic experiments, demonstrating that broadening context distributions robustly induces induction heads and enables OOD generalization. Our results shed light on the algorithmic biases of pretrained transformers and offer conceptual guidelines for data-driven control of their learned behaviors.

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NeurIPS Conference 2025 Conference Paper

Learning Theory for Kernel Bilevel Optimization

Fares El Khoury
Edouard Pauwels
Samuel Vaiter
Michael Arbel

Bilevel optimization has emerged as a technique for addressing a wide range of machine learning problems that involve an outer objective implicitly determined by the minimizer of an inner problem. While prior works have primarily focused on the parametric setting, a learning-theoretic foundation for bilevel optimization in the nonparametric case remains relatively unexplored. In this paper, we take a first step toward bridging this gap by studying Kernel Bilevel Optimization (KBO), where the inner objective is optimized over a reproducing kernel Hilbert space. This setting enables rich function approximation while providing a foundation for rigorous theoretical analysis. In this context, we derive novel finite-sample generalization bounds for KBO, leveraging tools from empirical process theory. These bounds further allow us to assess the statistical accuracy of gradient-based methods applied to the empirical discretization of KBO. We numerically illustrate our theoretical findings on a synthetic instrumental variable regression task.

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JMLR Journal 2024 Journal Article

Convergence of Message-Passing Graph Neural Networks with Generic Aggregation on Large Random Graphs

Matthieu Cordonnier
Nicolas Keriven
Nicolas Tremblay
Samuel Vaiter

We study the convergence of message-passing graph neural networks on random graph models toward their continuous counterparts as the number of nodes tends to infinity. Until now, this convergence was only known for architectures with aggregation functions in the form of normalized means, or, equivalently, of an application of classical operators like the adjacency matrix or the graph Laplacian. We extend such results to a large class of aggregation functions, that encompasses all classically used message-passing graph neural networks, such as attention-based message-passing, max convolutional message-passing, (degree-normalized) convolutional message-passing, or moment-based aggregation message-passing. Under mild assumptions, we give non-asymptotic bounds with high probability to quantify this convergence. Our main result is based on the McDiarmid inequality. Interestingly, this result does not apply to the case where the aggregation is a coordinate-wise maximum. We treat this case separately and obtain a different convergence rate. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2024. ( edit, beta )

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NeurIPS Conference 2024 Conference Paper

Derivatives of Stochastic Gradient Descent in parametric optimization

Franck Iutzeler
Edouard Pauwels
Samuel Vaiter

We consider stochastic optimization problems where the objective depends on some parameter, as commonly found in hyperparameter optimization for instance. We investigate the behavior of the derivatives of the iterates of Stochastic Gradient Descent (SGD) with respect to that parameter and show that they are driven by an inexact SGD recursion on a different objective function, perturbed by the convergence of the original SGD. This enables us to establish that the derivatives of SGD converge to the derivative of the solution mapping in terms of mean squared error whenever the objective is strongly convex. Specifically, we demonstrate that with constant step-sizes, these derivatives stabilize within a noise ball centered at the solution derivative, and that with vanishing step-sizes they exhibit $O(\log(k)^2 / k)$ convergence rates. Additionally, we prove exponential convergence in the interpolation regime. Our theoretical findings are illustrated by numerical experiments on synthetic tasks.

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TMLR Journal 2024 Journal Article

Gradient Scarcity in Graph Learning with Bilevel Optimization

Hashem Ghanem
Samuel Vaiter
Nicolas Keriven

Gradient scarcity emerges when learning graphs by minimizing a loss on a subset of nodes under the semi-supervised setting. It consists in edges between unlabeled nodes that are far from the labeled ones receiving zero gradients. The phenomenon was first described when jointly optimizing the graph and the parameters of a shallow Graph Neural Network (GNN) using a single loss function. In this work, we give a precise mathematical characterization of this phenomenon, and prove that it also emerges in bilevel optimization. While for GNNs gradient scarcity occurs due to their finite receptive field, we show that it also occurs with the Laplacian regularization as gradients decrease exponentially in amplitude with distance to labeled nodes, despite the infinite receptive field of this model. We study several solutions to this issue including latent graph learning using a Graph-to-Graph model (G2G), graph regularization to impose a prior structure on the graph, and reducing the graph diameter by optimizing for a larger set of edges. Our empirical results validate our analysis and show that this issue also occurs with the Approximate Personalized Propagation of Neural Predictions (APPNP), which approximates a model of infinite receptive field.

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ICML Conference 2023 Conference Paper

On the Robustness of Text Vectorizers

Rémi Catellier
Samuel Vaiter
Damien Garreau

A fundamental issue in machine learning is the robustness of the model with respect to changes in the input. In natural language processing, models typically contain a first embedding layer, transforming a sequence of tokens into vector representations. While the robustness with respect to changes of continuous inputs is well-understood, the situation is less clear when considering discrete changes, for instance replacing a word by another in an input sentence. Our work formally proves that popular embedding schemes, such as concatenation, TF-IDF, and Paragraph Vector (a. k. a. doc2vec), exhibit robustness in the Hölder or Lipschitz sense with respect to the Hamming distance. We provide quantitative bounds for these schemes and demonstrate how the constants involved are affected by the length of the document. These findings are exemplified through a series of numerical examples.

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NeurIPS Conference 2023 Conference Paper

One-step differentiation of iterative algorithms

Jerome Bolte
Edouard Pauwels
Samuel Vaiter

In appropriate frameworks, automatic differentiation is transparent to the user, at the cost of being a significant computational burden when the number of operations is large. For iterative algorithms, implicit differentiation alleviates this issue but requires custom implementation of Jacobian evaluation. In this paper, we study one-step differentiation, also known as Jacobian-free backpropagation, a method as easy as automatic differentiation and as performant as implicit differentiation for fast algorithms (e. g. superlinear optimization methods). We provide a complete theoretical approximation analysis with specific examples (Newton's method, gradient descent) along with its consequences in bilevel optimization. Several numerical examples illustrate the well-foundness of the one-step estimator.

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NeurIPS Conference 2023 Conference Paper

What functions can Graph Neural Networks compute on random graphs? The role of Positional Encoding

Nicolas Keriven
Samuel Vaiter

We aim to deepen the theoretical understanding of Graph Neural Networks (GNNs) on large graphs, with a focus on their expressive power. Existing analyses relate this notion to the graph isomorphism problem, which is mostly relevant for graphs of small sizes, or studied graph classification or regression tasks, while prediction tasks on \emph{nodes} are far more relevant on large graphs. Recently, several works showed that, on very general random graphs models, GNNs converge to certains functions as the number of nodes grows. In this paper, we provide a more complete and intuitive description of the function space generated by equivariant GNNs for node-tasks, through general notions of convergence that encompass several previous examples. We emphasize the role of input node features, and study the impact of \emph{node Positional Encodings} (PEs), a recent line of work that has been shown to yield state-of-the-art results in practice. Through the study of several examples of PEs on large random graphs, we extend previously known universality results to significantly more general models. Our theoretical results hint at some normalization tricks, which is shown numerically to have a positive impact on GNN generalization on synthetic and real data. Our proofs contain new concentration inequalities of independent interest.

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NeurIPS Conference 2022 Conference Paper

A framework for bilevel optimization that enables stochastic and global variance reduction algorithms

Mathieu Dagréou
Pierre Ablin
Samuel Vaiter
Thomas Moreau

Bilevel optimization, the problem of minimizing a value function which involves the arg-minimum of another function, appears in many areas of machine learning. In a large scale empirical risk minimization setting where the number of samples is huge, it is crucial to develop stochastic methods, which only use a few samples at a time to progress. However, computing the gradient of the value function involves solving a linear system, which makes it difficult to derive unbiased stochastic estimates. To overcome this problem we introduce a novel framework, in which the solution of the inner problem, the solution of the linear system, and the main variable evolve at the same time. These directions are written as a sum, making it straightforward to derive unbiased estimates. The simplicity of our approach allows us to develop global variance reduction algorithms, where the dynamics of all variables is subject to variance reduction. We demonstrate that SABA, an adaptation of the celebrated SAGA algorithm in our framework, has $O(\frac1T)$ convergence rate, and that it achieves linear convergence under Polyak-Lojasciewicz assumption. This is the first stochastic algorithm for bilevel optimization that verifies either of these properties. Numerical experiments validate the usefulness of our method.

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NeurIPS Conference 2022 Conference Paper

Automatic differentiation of nonsmooth iterative algorithms

Jerome Bolte
Edouard Pauwels
Samuel Vaiter

Differentiation along algorithms, i. e. , piggyback propagation of derivatives, is now routinely used to differentiate iterative solvers in differentiable programming. Asymptotics is well understood for many smooth problems but the nondifferentiable case is hardly considered. Is there a limiting object for nonsmooth piggyback automatic differentiation (AD)? Does it have any variational meaning and can it be used effectively in machine learning? Is there a connection with classical derivative? All these questions are addressed under appropriate contractivity conditions in the framework of conservative derivatives which has proved useful in understanding nonsmooth AD. For nonsmooth piggyback iterations, we characterize the attractor set of nonsmooth piggyback iterations as a set-valued fixed point which remains in the conservative framework. This has various consequences and in particular almost everywhere convergence of classical derivatives. Our results are illustrated on parametric convex optimization problems with forward-backward, Douglas-Rachford and Alternating Direction of Multiplier algorithms as well as the Heavy-Ball method.

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NeurIPS Conference 2022 Conference Paper

Benchopt: Reproducible, efficient and collaborative optimization benchmarks

Thomas Moreau
Mathurin Massias
Alexandre Gramfort
Pierre Ablin
Pierre-Antoine Bannier
Benjamin Charlier
Mathieu Dagréou
Tom Dupre la Tour

Numerical validation is at the core of machine learning research as it allows us to assess the actual impact of new methods, and to confirm the agreement between theory and practice. Yet, the rapid development of the field poses several challenges: researchers are confronted with a profusion of methods to compare, limited transparency and consensus on best practices, as well as tedious re-implementation work. As a result, validation is often very partial, which can lead to wrong conclusions that slow down the progress of research. We propose Benchopt, a collaborative framework to automatize, publish and reproduce optimization benchmarks in machine learning across programming languages and hardware architectures. Benchopt simplifies benchmarking for the community by providing an off-the-shelf tool for running, sharing and extending experiments. To demonstrate its broad usability, we showcase benchmarks on three standard ML tasks: $\ell_2$-regularized logistic regression, Lasso and ResNet18 training for image classification. These benchmarks highlight key practical findings that give a more nuanced view of state-of-the-art for these problems, showing that for practical evaluation, the devil is in the details.

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JMLR Journal 2022 Journal Article

Implicit Differentiation for Fast Hyperparameter Selection in Non-Smooth Convex Learning

Quentin Bertrand
Quentin Klopfenstein
Mathurin Massias
Mathieu Blondel
Samuel Vaiter
Alexandre Gramfort
Joseph Salmon

Finding the optimal hyperparameters of a model can be cast as a bilevel optimization problem, typically solved using zero-order techniques. In this work we study first-order methods when the inner optimization problem is convex but non-smooth. We show that the forward-mode differentiation of proximal gradient descent and proximal coordinate descent yield sequences of Jacobians converging toward the exact Jacobian. Using implicit differentiation, we show it is possible to leverage the non-smoothness of the inner problem to speed up the computation. Finally, we provide a bound on the error made on the hypergradient when the inner optimization problem is solved approximately. Results on regression and classification problems reveal computational benefits for hyperparameter optimization, especially when multiple hyperparameters are required. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2022. ( edit, beta )

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NeurIPS Conference 2021 Conference Paper

On the Universality of Graph Neural Networks on Large Random Graphs

Nicolas Keriven
Alberto Bietti
Samuel Vaiter

We study the approximation power of Graph Neural Networks (GNNs) on latent position random graphs. In the large graph limit, GNNs are known to converge to certain ``continuous'' models known as c-GNNs, which directly enables a study of their approximation power on random graph models. In the absence of input node features however, just as GNNs are limited by the Weisfeiler-Lehman isomorphism test, c-GNNs will be severely limited on simple random graph models. For instance, they will fail to distinguish the communities of a well-separated Stochastic Block Model (SBM) with constant degree function. Thus, we consider recently proposed architectures that augment GNNs with unique node identifiers, referred to as Structural GNNs here (SGNNs). We study the convergence of SGNNs to their continuous counterpart (c-SGNNs) in the large random graph limit, under new conditions on the node identifiers. We then show that c-SGNNs are strictly more powerful than c-GNNs in the continuous limit, and prove their universality on several random graph models of interest, including most SBMs and a large class of random geometric graphs. Our results cover both permutation-invariant and permutation-equivariant architectures.

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NeurIPS Conference 2020 Conference Paper

Convergence and Stability of Graph Convolutional Networks on Large Random Graphs

Nicolas Keriven
Alberto Bietti
Samuel Vaiter

We study properties of Graph Convolutional Networks (GCNs) by analyzing their behavior on standard models of random graphs, where nodes are represented by random latent variables and edges are drawn according to a similarity kernel. This allows us to overcome the difficulties of dealing with discrete notions such as isomorphisms on very large graphs, by considering instead more natural geometric aspects. We first study the convergence of GCNs to their continuous counterpart as the number of nodes grows. Our results are fully non-asymptotic and are valid for relatively sparse graphs with an average degree that grows logarithmically with the number of nodes. We then analyze the stability of GCNs to small deformations of the random graph model. In contrast to previous studies of stability in discrete settings, our continuous setup allows us to provide more intuitive deformation-based metrics for understanding stability, which have proven useful for explaining the success of convolutional representations on Euclidean domains.

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JMLR Journal 2020 Journal Article

Dual Extrapolation for Sparse GLMs

Mathurin Massias
Samuel Vaiter
Alexandre Gramfort
Joseph Salmon

Generalized Linear Models (GLM) form a wide class of regression and classification models, where prediction is a function of a linear combination of the input variables. For statistical inference in high dimension, sparsity inducing regularizations have proven to be useful while offering statistical guarantees. However, solving the resulting optimization problems can be challenging: even for popular iterative algorithms such as coordinate descent, one needs to loop over a large number of variables. To mitigate this, techniques known as screening rules and working sets diminish the size of the optimization problem at hand, either by progressively removing variables, or by solving a growing sequence of smaller problems. For both techniques, significant variables are identified thanks to convex duality arguments. In this paper, we show that the dual iterates of a GLM exhibit a Vector AutoRegressive (VAR) behavior after sign identification, when the primal problem is solved with proximal gradient descent or cyclic coordinate descent. Exploiting this regularity, one can construct dual points that offer tighter certificates of optimality, enhancing the performance of screening rules and working set algorithms. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2020. ( edit, beta )

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ICML Conference 2020 Conference Paper

Implicit differentiation of Lasso-type models for hyperparameter optimization

Quentin Bertrand
Quentin Klopfenstein
Mathieu Blondel
Samuel Vaiter
Alexandre Gramfort
Joseph Salmon

Setting regularization parameters for Lasso-type estimators is notoriously difficult, though crucial for obtaining the best accuracy. The most popular hyperparameter optimization approach is grid-search on a held-out dataset. However, grid-search requires to choose a predefined grid of parameters and scales exponentially in the number of parameters. Another class of approaches casts hyperparameter optimization as a bi-level optimization problem, typically solved by gradient descent. The key challenge for these approaches is the estimation of the gradient w. r. t. the hyperparameters. Computing that gradient via forward or backward automatic differentiation usually suffers from high memory consumption, while implicit differentiation typically involves solving a linear system which can be prohibitive and numerically unstable. In addition, implicit differentiation usually assumes smooth loss functions, which is not the case of Lasso-type problems. This work introduces an efficient implicit differentiation algorithm, without matrix inversion, tailored for Lasso-type problems. Our proposal scales to high-dimensional data by leveraging the sparsity of the solutions. Empirically, we demonstrate that the proposed method outperforms a large number of standard methods for hyperparameter optimization.

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