arXiv subjects

Nicolas Brisebarre

Publications and source records attributed to Nicolas Brisebarre.

9 recordsLinked to original sources

arXiv2026

Mixed precision Newton's method for optimization

Second-order optimization methods, such as Newton's algorithm, achieve fast local convergence and high accuracy, but their practical use is often limited by high computational costs. To mitigate this issue, variants such as inexact and quasi-Newton methods are widely used. A complementary and promising approach to improve the efficiency of the method is to employ mixed precision arithmetic, using different floating-point precisions for different operations, based on their impact on the convergence and accuracy of the method. In this work, we perform an error analysis of Newton's method accounting for different sources of inexactness, including approximations and rounding errors. We present a convergence analysis for the generated sequence, establishing bounds on the convergence rate and attainable accuracy. This theoretical framework covers quasi-Newton and inexact Newton methods, and is leveraged to propose mixed precision algorithms. We present a wide set of numerical experiments to illustrate our theoretical results and the behavior of Newton's method and its approximate variants in mixed precision floating-point arithmetic.

math.OC↗

arXiv2026

Integer points close to a transcendental curve: an algorithmic approach

In this article, we propose an algorithmic approach to determine the integer points located near a transcendental curve. This approach is closely related to a celebrated work by Bombieri and Pila and to the so-called Coppersmith's method. We establish the underlying theoretical foundations, prove the algorithms, study their complexity and present practical experiments; we also compare our approach with previously existing ones. From a practical point of view, we focus on an instance of our general problem, called the Table Maker's Dilemma, whose solving makes it possible to evaluate a given function with correct rounding. Our experiments show a significant speedup. In particular, our results show that the development of a correctly rounded mathematical library for the binary128 format is now possible at a much smaller cost than with previously existing approaches.

math.NT↗

arXiv2026

Fractions of Recurrence Operators for Generalized Fourier Series in Classical Orthogonal Polynomials

We consider series expansions in bases of classical orthogonal polynomials. When such a series solves a linear differential equation with polynomial coefficients, its coefficients satisfy a linear recurrence equation. We interpret this equation as the numerator of a fraction of linear recurrence operators. This interpretation lets us give a simple and unified view of previous algorithms computing these recurrences, with a noncommutative Euclidean algorithm as the algorithmic engine. Finally, we demonstrate the effectiveness of our approach on various examples.

math.CA↗

arXiv2025

A Rescaling-Invariant Lipschitz Bound Based on Path-Metrics for Modern ReLU Network Parameterizations

Robustness with respect to weight perturbations underpins guarantees for generalization, pruning and quantization. Existing guarantees rely on Lipschitz bounds in parameter space, cover only plain feed-forward MLPs, and break under the ubiquitous neuron-wise rescaling symmetry of ReLU networks. We prove a new Lipschitz inequality expressed through the $\ell^1$-path-metric of the weights. The bound is (i) rescaling-invariant by construction and (ii) applies to any ReLU-DAG architecture with any combination of convolutions, skip connections, pooling, and frozen (inference-time) batch-normalization -- thus encompassing ResNets, U-Nets, VGG-style CNNs, and more. By respecting the network's natural symmetries, the new bound strictly sharpens prior parameter-space bounds and can be computed in two forward passes. To illustrate its utility, we derive from it a symmetry-aware pruning criterion and show -- through a proof-of-concept experiment on a ResNet-18 trained on ImageNet -- that its pruning performance matches that of classical magnitude pruning, while becoming totally immune to arbitrary neuron-wise rescalings.

cs.LG↗

arXiv2024

A path-norm toolkit for modern networks: consequences, promises and challenges

This work introduces the first toolkit around path-norms that fully encompasses general DAG ReLU networks with biases, skip connections and any operation based on the extraction of order statistics: max pooling, GroupSort etc. This toolkit notably allows us to establish generalization bounds for modern neural networks that are not only the most widely applicable path-norm based ones, but also recover or beat the sharpest known bounds of this type. These extended path-norms further enjoy the usual benefits of path-norms: ease of computation, invariance under the symmetries of the network, and improved sharpness on layered fully-connected networks compared to the product of operator norms, another complexity measure most commonly used. The versatility of the toolkit and its ease of implementation allow us to challenge the concrete promises of path-norm-based generalization bounds, by numerically evaluating the sharpest known bounds for ResNets on ImageNet.

stat.ML↗

arXiv2022

Differential-Difference Properties of Hypergeometric Series

Six families of generalized hypergeometric series in a variable $x$ and an arbitrary number of parameters are considered. Each of them is indexed by an integer $n$. Linear recurrence relations in $n$ relate these functions and their product by the variable $x$. We give explicit factorizations of these equations as products of first order recurrence operators. Related recurrences are also derived for the derivative with respect to $x$. These formulas generalize well-known properties of the classical orthogonal polynomials.

math.CA↗

arXiv2022

Approximation speed of quantized vs. unquantized ReLU neural networks and beyond

We deal with two complementary questions about approximation properties of ReLU networks. First, we study how the uniform quantization of ReLU networks with real-valued weights impacts their approximation properties. We establish an upper-bound on the minimal number of bits per coordinate needed for uniformly quantized ReLU networks to keep the same polynomial asymptotic approximation speeds as unquantized ones. We also characterize the error of nearest-neighbour uniform quantization of ReLU networks. This is achieved using a new lower-bound on the Lipschitz constant of the map that associates the parameters of ReLU networks to their realization, and an upper-bound generalizing classical results. Second, we investigate when ReLU networks can be expected, or not, to have better approximation properties than other classical approximation families. Indeed, several approximation families share the following common limitation: their polynomial asymptotic approximation speed of any set is bounded from above by the encoding speed of this set. We introduce a new abstract property of approximation families, called infinite-encodability, which implies this upper-bound. Many classical approximation families, defined with dictionaries or ReLU networks, are shown to be infinite-encodable. This unifies and generalizes several situations where this upper-bound is known.

cs.IT↗

arXiv2017

On a two-valued sequence and related continued fractions in power series fields

We explicitly describe a noteworthy transcendental continued fraction in the field of power series over Q, having irrationality measure equal to 3. This continued fraction is a generating function of a particular sequence in the set {1, 2}. The origin of this sequence, whose study was initiated in a recent paper, is to be found in another continued fraction, in the field of power series over $\mathbb{F}\_3$, which satisfies a simple algebraic equation of degree 4, introduced thirty years ago by D. Robbins.

math.NT↗

arXiv2003

Finding the "truncated" polynomial that is closest to a function

When implementing regular enough functions (e.g., elementary or special functions) on a computing system, we frequently use polynomial approximations. In most cases, the polynomial that best approximates (for a given distance and in a given interval) a function has coefficients that are not exactly representable with a finite number of bits. And yet, the polynomial approximations that are actually implemented do have coefficients that are represented with a finite - and sometimes small - number of bits: this is due to the finiteness of the floating-point representations (for software implementations), and to the need to have small, hence fast and/or inexpensive, multipliers (for hardware implementations). We then have to consider polynomial approximations for which the degree-$i$ coefficient has at most $m_i$ fractional bits (in other words, it is a rational number with denominator $2^{m_i}$). We provide a general method for finding the best polynomial approximation under this constraint. Then, we suggest refinements than can be used to accelerate our method.

cs.MS↗