arXiv subjects

Samuel Vaiter

Publications and source records attributed to Samuel Vaiter.

At least 19 recordsLinked to original sources

arXiv2026

Proximal basin hopping: global optimization with guarantees

Global optimization is a challenging problem, with plenty of algorithms displaying empirical success, but scarce theoretical backing. In this work, we propose a new theoretical framework called Proximal Basin Hopping (PBH), carefully tailored to combine proximal optimization and local minimization. We use it to construct a practical algorithm that converges to the global minimizer with high probability, when using a finite amount of samples. Proximal Basin Hopping outperforms well known algorithms with theoretical backing on standard synthetic hard functions, and real problems such as fitting scaling laws for deep learning. Furthermore, the higher the dimension, the better the performance gap.

cs.LG↗

arXiv2026

On the Hardness of Junking LLMs

Large language models (LLMs) are known to be vulnerable to jailbreak attacks, which typically rely on carefully designed prompts containing explicit semantic structure. These attacks generally operate by fixing an adversarial instruction and optimizing small adversarial components (e.g., suffixes or prefixes). In this setting, prompt structure is fundamental for performance, and recent results show that even simple random search can achieve strong performance when combined with sophisticated prompt design. Recently, it has been observed that harmful behaviors can be elicited even without the adversarial prompt, relying solely on optimized token sequences. This suggests the existence of natural backdoors, i.e., token sequences naturally emerged during LLMs training that trigger unsafe outputs without any meaningful instruction. However, despite these observations, this setting remains largely unexplored, and in particular the hardness of finding natural backdoors has not been assessed yet. In this work, we provide a first proof-of-concept study investigating the hardness of this task, which we refer to as the junking problem. We formalize it as the problem of finding token sequences that maximize the probability of generating a target prefix of harmful responses, propose a greedy random-search method to assess is such sequences can be discovered easily. Our results show that this problem is harder than standard jailbreak attacks, confirming the importance of semantic information in prompt design. At the same time, we find that our simple strategy is sufficient to solve it with a high success rate, suggesting that natural backdoors are present and easily recoverable. Finally, through perplexity analysis, we observe that the discovered token sequences lie in low-probability regions of the model distribution, supporting the hypothesis that they emerged implicitly from the training process.

cs.LG↗

arXiv2026

Characterizations of inexact proximal operators

Proximal operators are now ubiquitous in non-smooth optimization. Since their introduction in the seminal work of Moreau, many papers have shown their effectiveness on a wide variety of problems, culminating in their use to construct convergent deep learning methods. The characterization of these operators for non-convex penalties was completed recently in [Gribonval et al, A characterization of proximity operators, 2020]. In this paper, we propose to follow this line of work by characterizing inexact proximal operators, thus providing an answer to what constitutes a good approximation of these operators. We propose several definitions of approximations and discuss their regularity, approximation power, and their fixed points. Equipped with these characterizations, we investigate the convergence of proximal algorithms in the presence of errors that may be non-summable and/or non-vanishing. In particular, we look at the proximal point algorithm, and at the forward-backward, Peaceman-Rachford and Douglas-Rachford algorithms when we minimize the sum of a weakly convex function (whose proximal operator is approximated) and a strongly convex function.

math.OC↗

arXiv2026

Towards Understanding Steering Strength

A popular approach to post-training control of large language models (LLMs) is the steering of intermediate latent representations. Namely, identify a well-chosen direction depending on the task at hand and perturbs representations along this direction at inference time. While many propositions exist to pick this direction, considerably less is understood about how to choose the magnitude of the move, whereas its importance is clear: too little and the intended behavior does not emerge, too much and the model's performance degrades beyond repair. In this work, we propose the first theoretical analysis of steering strength. We characterize its effect on next token probability, presence of a concept, and cross-entropy, deriving precise qualitative laws governing these quantities. Our analysis reveals surprising behaviors, including non-monotonic effects of steering strength. We validate our theoretical predictions empirically on eleven language models, ranging from a small GPT architecture to modern models.

cs.LG↗

arXiv2026

ZOBA: An Efficient Single-loop Zeroth-order Bilevel Optimization Algorithm

Bilevel optimization problems consist of minimizing a value function whose evaluation depends on the solution of an inner optimization problem. These problems are typically tackled using first-order methods that require computing the gradient of the value function ({\it the hypergradient}). In several practical settings, however, first-order information is unavailable ({\it zeroth-order setting}), rendering these methods inapplicable. Finite-difference methods provide an alternative by approximating hypergradients using function evaluations along a set of directions. Nevertheless, such surrogates are notoriously expensive, and existing finite-difference bilevel methods rely on two-loop algorithms that are poorly parallelizable. In this work, we propose ZOBA, the first finite-difference single-loop algorithm for bilevel optimization. Our method leverages finite-difference hypergradient approximations based on delayed information to eliminate the need for nested loops. We analyze the proposed algorithm and establish convergence rates in the non-convex setting, achieving a complexity of $\mathcal{O}(p(d + p)^2\varepsilon^{-2})$, where $p$ and $d$ denote the dimension of inner and outer spaces respectively, which is better than prior approaches based on Hessian approximation. We further introduce and analyze HF-ZOBA, a Hessian-free variant that yields additional complexity improvements. Finally, we corroborate our findings with numerical experiments on synthetic functions and a real-world black-box task in adversarial machine learning. Our results show that our methods achieve accuracy comparable to state-of-the-art techniques while requiring less computation time.

math.OC↗

arXiv2026

Fairness-informed Pareto Optimization : An Efficient Bilevel Framework

Despite their promise, fair machine learning methods often yield Pareto-inefficient models, in which the performance of certain groups can be improved without degrading that of others. This issue arises frequently in traditional in-processing approaches such as fairness-through-regularization. In contrast, existing Pareto-efficient approaches are biased towards a certain perspective on fairness and fail to adapt to the broad range of fairness metrics studied in the literature. In this paper, we present BADR, a simple framework to recover the optimal Pareto-efficient model for any fairness metric. Our framework recovers its models through a Bilevel Adaptive Rescalarisation procedure. The lower level is a weighted empirical risk minimization task where the weights are a convex combination of the groups, while the upper level optimizes the chosen fairness objective. We equip our framework with two novel large-scale, single-loop algorithms, BADR-GD and BADR-SGD, and establish their convergence guarantees. We release badr, an open-source Python toolbox implementing our framework for a variety of learning tasks and fairness metrics. Finally, we conduct extensive numerical experiments demonstrating the advantages of BADR over existing Pareto-efficient approaches to fairness.

cs.LG↗

arXiv2025

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

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.

cs.LG↗

arXiv2025

Deep Equilibrium models for Poisson Imaging Inverse problems via Mirror Descent

Deep Equilibrium Models (DEQs) are implicit neural networks with fixed points, which have recently gained attention for learning image regularization functionals, particularly in settings involving Gaussian fidelities, where assumptions on the forward operator ensure contractiveness of standard (proximal) Gradient Descent operators. In this work, we extend the application of DEQs to Poisson inverse problems, where the data fidelity term is more appropriately modeled by the Kullback--Leibler divergence. To this end, we introduce a novel DEQ formulation based on Mirror Descent defined in terms of a tailored non-Euclidean geometry that naturally adapts with the structure of the data term. This enables the learning of neural regularizers within a principled training framework. We derive sufficient conditions and establish refined convergence results based on the Kurdyka--Lojasiewicz framework for subanalytic functions with non-closed domains to guarantee the convergence of the learned reconstruction scheme and propose computational strategies that enable both efficient training and parameter-free inference. Numerical experiments show that our method outperforms traditional model-based approaches and it is comparable to the performance of Bregman Plug-and-Play methods, while mitigating their typical drawbacks, such as time-consuming tuning of hyper-parameters. The code is publicly available at https://github.com/christiandaniele/DEQ-MD.

math.OC↗

arXiv2025

Differentiable Generalized Sliced Wasserstein Plans

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 to 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.

cs.LG↗

arXiv2025

Faster Computation of Entropic Optimal Transport via Stable Low Frequency Modes

In this paper, we propose an accelerated version for the Sinkhorn algorithm, which is the reference method for computing the solution to Entropic Optimal Transport. Its main draw-back is the exponential slow-down of convergence as the regularization weakens $\varepsilon \rightarrow 0$. Thanks to spectral insights on the behavior of the Hessian, we propose to mitigate the problem via an original spectral warm-start strategy. This leads to faster convergence compared to the reference method, as also demonstrated in our numerical experiments.

math.NA↗

arXiv2025

Bilevel gradient methods and the Morse parametric qualification condition

We introduce the Morse parametric qualification condition for bilevel programming. Generic semi-algebraic functions are Morse parametric in a piecewise sense. Thus, bilevel programs with a Morse parametric lower level constitute a relevant intermediate class between strongly convex and fully generic lower levels. In this framework, we study bilevel gradient algorithms with two strategies: the single-step multi-step strategy, which involves a sequence of steps on the lower-level problems followed by one step on the upper-level problem, and a differentiable programming strategy that optimizes a smooth approximation of the bilevel problem. While the first is shown to be a biased gradient method on the problem with rich properties, the second, inspired by meta-learning applications, is less stable but offers simplicity and ease of implementation.

math.OC↗

arXiv2025

Learning Theory for Kernel Bilevel Optimization

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.

cs.LG↗

arXiv2024

Risk Estimate under a Time-Varying Autoregressive Model for Data-Driven Reproduction Number Estimation

COVID-19 pandemic has brought to the fore epidemiological models which, though describing a wealth of behaviors, have previously received little attention in signal processing literature. In this work, a generalized time-varying autoregressive model is considered, encompassing, but not reducing to, a state-of-the-art model of viral epidemics propagation. The time-varying parameter of this model is estimated via the minimization of a penalized likelihood estimator. A major challenge is that the estimation accuracy strongly depends on hyperparameters fine-tuning. Without available ground truth, hyperparameters are selected by minimizing specifically designed data-driven oracles, used as proxy for the estimation error. Focusing on the time-varying autoregressive Poisson model, Stein's Unbiased Risk Estimate formalism is generalized to construct asymptotically unbiased risk estimators based on the derivation of an original autoregressive counterpart of Stein's lemma. The accuracy of these oracles and of the resulting estimates are assessed through intensive Monte Carlo simulations on synthetic data. Then, elaborating on recent epidemiological models, a novel weekly scaled Poisson model is proposed, better accounting for intrinsic variability of the contaminations while being robust to reporting errors. Finally, the data-driven procedure is particularized to the estimation of COVID-19 reproduction number from weekly infection counts demonstrating its ability to tackle real-world applications.

stat.ME↗

arXiv2024

Geometric and computational hardness of bilevel programming

We first show a simple but striking result in bilevel optimization: unconstrained $C^\infty$ smooth bilevel programming is as hard as general extended-real-valued lower semicontinuous minimization. We then proceed to a worst-case analysis of box-constrained bilevel polynomial optimization. We show in particular that any extended-real-valued semi-algebraic function, possibly non-continuous, can be expressed as the value function of a polynomial bilevel program. Secondly, from a computational complexity perspective, the decision version of polynomial bilevel programming is one level above NP in the polynomial hierarchy ($\Sigma^p_2$-hard). Both types of difficulties are uncommon in non-linear programs for which objective functions are typically continuous and belong to the class NP. These results highlight the irremediable hardness attached to general bilevel optimization and the necessity of imposing some form of regularity on the lower level.

cs.CC↗

arXiv2024

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

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 dataset. 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.

math.ST↗

arXiv2024

Derivatives of Stochastic Gradient Descent in parametric optimization

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.

math.OC↗

arXiv2023

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

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 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 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.

cs.LG↗

arXiv2023

One-step differentiation of iterative algorithms

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.

math.OC↗