arXiv subjects

Jérôme Bolte

Publications and source records attributed to Jérôme Bolte.

At least 19 recordsLinked to original sources

arXiv2026

When majority rules, minority loses: bias amplification of gradient descent

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.

cs.LG↗

arXiv2026

SSA: Improving Performance With a Better Scoring Function

While transformer models exhibit strong in-context learning (ICL) abilities, they often fail to generalize under simple distribution shifts. We analyze these failures and identify Softmax, the scoring function in the attention mechanism, as a contributing factor. We propose \textbf{Scaled Signed Averaging (SSA)}, a novel attention scoring function that mitigates these failures. SSA significantly improves performance on our ICL tasks and outperforms transformer models with Softmax on several NLP benchmarks and linguistic probing tasks, in both decoder-only and encoder-only architectures.

cs.CL↗

arXiv2026

The adjoint state method for parametric definable optimization without smoothness or uniqueness

We establish that nonconvex definable parametric optimization problems with possibly nonsmooth objectives, inequality constraints, conic constraint systems, and non-unique primal and dual solutions admit an adjoint state formula under a mere qualification condition. The adjoint construction yields a selection of a conservative field for the value function, providing a computable first-order object without requiring differentiation of the solution mapping. Through examples, we show that even in smooth problems, the formal adjoint construction fails without conservativity or definability, illustrating the relevance of these concepts to grasp theoretical aspects of the method. This work provides a tool which can be directly combined with existing primal-dual solvers for a wide range of parametric optimization problems.

math.OC↗

arXiv2026

Gradient extremals, talwegs, valleys, and directional alignment for generic gradient descent

Gradient extremals are loci along which the gradient is an eigenvector of the Hessian. These objects provide a natural geometric framework connecting several notions, notably valleys and talwegs, which we analyze from a variational viewpoint in the generic case. We then show that trajectories of the gradient flow and of its discrete counterpart exhibit directional alignment with the tangent spaces to gradient extremals, and generically to the talweg. Under non-resonance assumptions, and in contrast with the quadratic case, alignment rates are governed either by the first spectral gap or by the smallest eigenvalue of the Hessian at the limit point. Nonlinearities and the step length may both distort these rates in a complex manner. We further prove a volume concentration phenomenon emphasizing the structuring role of gradient extremals: for large times, the images of sets of initial conditions concentrate inside valleys and asymptotically around talwegs.

math.OC↗

arXiv2026

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

Inexact subgradient methods for semialgebraic functions

Motivated by the extensive application of approximate gradients in machine learning and optimization, we investigate inexact subgradient methods subject to persistent additive errors. Within a nonconvex semialgebraic framework, assuming boundedness or coercivity, we establish that the method yields iterates that eventually fluctuate near the critical set at a proximity characterized by an $O(ε^ρ)$ distance, where $ε$ denotes the magnitude of subgradient evaluation errors, and $ρ$ encapsulates geometric characteristics of the underlying problem. Our analysis comprehensively addresses both vanishing and constant step-size regimes. Notably, the latter regime inherently enlarges the fluctuation region, yet this enlargement remains on the order of $ε^ρ$. In the convex scenario, employing a universal error bound applicable to coercive semialgebraic functions, we derive novel complexity results concerning averaged iterates. Additionally, our study produces auxiliary results of independent interest, including descent-type lemmas for nonsmooth nonconvex functions and an invariance principle governing the behavior of algorithmic sequences under small-step limits.

math.OC↗

arXiv2025

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 ($Σ^p_2$-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

A second-order-like optimizer with adaptive gradient scaling for deep learning

In this empirical article, we introduce INNAprop, an optimization algorithm that combines the INNA method with the RMSprop adaptive gradient scaling. It leverages second-order information and rescaling while keeping the memory requirements of standard DL methods as AdamW or SGD with momentum. After giving geometrical insights, we evaluate INNAprop on CIFAR-10, Food101, and ImageNet with ResNets, VGG, DenseNet, and ViT, and on GPT-2 (OpenWebText) train from scratch and with LoRA fine-tuning (E2E). INNAprop consistently matches or outperforms AdamW both in training speed and accuracy, with minimal hyperparameter tuning in large-scale settings. Our code is publicly available at \url{https://github.com/innaprop/innaprop}.

cs.LG↗

arXiv2024

Subgradient sampling for nonsmooth nonconvex minimization

Risk minimization for nonsmooth nonconvex problems naturally leads to first-order sampling or, by an abuse of terminology, to stochastic subgradient descent. We establish the convergence of this method in the path-differentiable case and describe more precise results under additional geometric assumptions. We recover and improve results from Ermoliev and Norkin [Cybern. Syst. Anal., 34 (1998), pp. 196--215] by using a different approach: conservative calculus and the ODE method. In the definable case, we show that first-order subgradient sampling avoids artificial critical points with probability one and applies moreover to a large range of risk minimization problems in deep learning, based on the backpropagation oracle. As byproducts of our approach, we obtain several results on integration of independent interest, such as an interchange result for conservative derivatives and integrals or the definability of set-valued parameterized integrals.

math.OC↗

arXiv2023

Numerical influence of ReLU'(0) on backpropagation

In theory, the choice of ReLU(0) in [0, 1] for a neural network has a negligible influence both on backpropagation and training. Yet, in the real world, 32 bits default precision combined with the size of deep learning problems makes it a hyperparameter of training methods. We investigate the importance of the value of ReLU'(0) for several precision levels (16, 32, 64 bits), on various networks (fully connected, VGG, ResNet) and datasets (MNIST, CIFAR10, SVHN, ImageNet). We observe considerable variations of backpropagation outputs which occur around half of the time in 32 bits precision. The effect disappears with double precision, while it is systematic at 16 bits. For vanilla SGD training, the choice ReLU'(0) = 0 seems to be the most efficient. For our experiments on ImageNet the gain in test accuracy over ReLU'(0) = 1 was more than 10 points (two runs). We also evidence that reconditioning approaches as batch-norm or ADAM tend to buffer the influence of ReLU'(0)'s value. Overall, the message we convey is that algorithmic differentiation of nonsmooth problems potentially hides parameters that could be tuned advantageously.

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↗

arXiv2023

On the complexity of nonsmooth automatic differentiation

Using the notion of conservative gradient, we provide a simple model to estimate the computational costs of the backward and forward modes of algorithmic differentiation for a wide class of nonsmooth programs. The overhead complexity of the backward mode turns out to be independent of the dimension when using programs with locally Lipschitz semi-algebraic or definable elementary functions. This considerably extends Baur-Strassen's smooth cheap gradient principle. We illustrate our results by establishing fast backpropagation results of conservative gradients through feedforward neural networks with standard activation and loss functions. Nonsmooth backpropagation's cheapness contrasts with concurrent forward approaches, which have, to this day, dimensional-dependent worst-case overhead estimates. We provide further results suggesting the superiority of backward propagation of conservative gradients. Indeed, we relate the complexity of computing a large number of directional derivatives to that of matrix multiplication, and we show that finding two subgradients in the Clarke subdifferential of a function is an NP-hard problem.

math.NA↗

arXiv2022

Differentiating Nonsmooth Solutions to Parametric Monotone Inclusion Problems

We leverage path differentiability and a recent result on nonsmooth implicit differentiation calculus to give sufficient conditions ensuring that the solution to a monotone inclusion problem will be path differentiable, with formulas for computing its generalized gradient. A direct consequence of our result is that these solutions happen to be differentiable almost everywhere. Our approach is fully compatible with automatic differentiation and comes with assumptions which are easy to check, roughly speaking: semialgebraicity and strong monotonicity. We illustrate the scope of our results by considering three fundamental composite problem settings: strongly convex problems, dual solutions to convex minimization problems and primal-dual solutions to min-max problems.

cs.LG↗

arXiv2022

Automatic differentiation of nonsmooth iterative algorithms

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 nonexpansivity 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.

math.OC↗

arXiv2022

Nonsmooth Implicit Differentiation for Machine Learning and Optimization

In view of training increasingly complex learning architectures, we establish a nonsmooth implicit function theorem with an operational calculus. Our result applies to most practical problems (i.e., definable problems) provided that a nonsmooth form of the classical invertibility condition is fulfilled. This approach allows for formal subdifferentiation: for instance, replacing derivatives by Clarke Jacobians in the usual differentiation formulas is fully justified for a wide class of nonsmooth problems. Moreover this calculus is entirely compatible with algorithmic differentiation (e.g., backpropagation). We provide several applications such as training deep equilibrium networks, training neural nets with conic optimization layers, or hyperparameter-tuning for nonsmooth Lasso-type models. To show the sharpness of our assumptions, we present numerical experiments showcasing the extremely pathological gradient dynamics one can encounter when applying implicit algorithmic differentiation without any hypothesis.

cs.LG↗

arXiv2022

Swarm gradient dynamics for global optimization: the density case

Using jointly geometric and stochastic reformulations of nonconvex problems and exploiting a Monge-Kantorovich gradient system formulation with vanishing forces, we formally extend the simulated annealing method to a wide class of global optimization methods. Due to an inbuilt combination of a gradient-like strategy and particles interactions, we call them swarm gradient dynamics. As in the original paper of Holley-Kusuoka-Stroock, the key to the existence of a schedule ensuring convergence to a global minimizer is a functional inequality. One of our central theoretical contributions is the proof of such an inequality for one-dimensional compact manifolds. We conjecture the inequality to be true in a much wider setting. We also describe a general method allowing for global optimization and evidencing the crucial role of functional inequalities {à} la Łojasiewicz.

math.AP↗

arXiv2022

The Iterates of the Frank-Wolfe Algorithm May Not Converge

The Frank-Wolfe algorithm is a popular method for minimizing a smooth convex function $f$ over a compact convex set $\mathcal{C}$. While many convergence results have been derived in terms of function values, hardly nothing is known about the convergence behavior of the sequence of iterates $(x_t)_{t\in\mathbb{N}}$. Under the usual assumptions, we design several counterexamples to the convergence of $(x_t)_{t\in\mathbb{N}}$, where $f$ is $d$-time continuously differentiable, $d\geq2$, and $f(x_t)\to\min_\mathcal{C}f$. Our counterexamples cover the cases of open-loop, closed-loop, and line-search step-size strategies. We do not assume \emph{misspecification} of the linear minimization oracle and our results thus hold regardless of the points it returns, demonstrating the fundamental pathologies in the convergence behavior of $(x_t)_{t\in\mathbb{N}}$.

math.OC↗

arXiv2021

Second-order step-size tuning of SGD for non-convex optimization

In view of a direct and simple improvement of vanilla SGD, this paper presents a fine-tuning of its step-sizes in the mini-batch case. For doing so, one estimates curvature, based on a local quadratic model and using only noisy gradient approximations. One obtains a new stochastic first-order method (Step-Tuned SGD), enhanced by second-order information, which can be seen as a stochastic version of the classical Barzilai-Borwein method. Our theoretical results ensure almost sure convergence to the critical set and we provide convergence rates. Experiments on deep residual network training illustrate the favorable properties of our approach. For such networks we observe, during training, both a sudden drop of the loss and an improvement of test accuracy at medium stages, yielding better results than SGD, RMSprop, or ADAM.

cs.LG↗