arXiv subjects

Julien Hendrickx

Publications and source records attributed to Julien Hendrickx.

5 recordsLinked to original sources

arXiv2024

PEPit: computer-assisted worst-case analyses of first-order optimization methods in Python

PEPit is a Python package aiming at simplifying the access to worst-case analyses of a large family of first-order optimization methods possibly involving gradient, projection, proximal, or linear optimization oracles, along with their approximate, or Bregman variants. In short, PEPit is a package enabling computer-assisted worst-case analyses of first-order optimization methods. The key underlying idea is to cast the problem of performing a worst-case analysis, often referred to as a performance estimation problem (PEP), as a semidefinite program (SDP) which can be solved numerically. To do that, the package users are only required to write first-order methods nearly as they would have implemented them. The package then takes care of the SDP modeling parts, and the worst-case analysis is performed numerically via a standard solver.

math.OC↗

arXiv2024

Convex SGD: Generalization Without Early Stopping

We consider the generalization error associated with stochastic gradient descent on a smooth convex function over a compact set. We show the first bound on the generalization error that vanishes when the number of iterations $T$ and the dataset size $n$ go to zero at arbitrary rates; our bound scales as $\tilde{O}(1/\sqrt{T} + 1/\sqrt{n})$ with step-size $α_t = 1/\sqrt{t}$. In particular, strong convexity is not needed for stochastic gradient descent to generalize well.

cs.LG↗

arXiv2022

Rethinking the Micro-Foundation of Opinion Dynamics: Rich Consequences of the Weighted-Median Mechanism

To identify the main mechanisms underlying complex opinion formation processes in social systems, researchers have long been exploring simple mechanistic mathematical models. Most existing opinion dynamics models are built on a common micro-foundation, i.e., the weighted-averaging opinion update. However, we argue that this universally-adopted mechanism features a non-negligible unrealistic feature, which brings unnecessary difficulties in seeking a proper balance between model complexity and predictive power. In this paper, we propose the weighted-median mechanism as a new micro-foundation of opinion dynamics, which, with minimal assumptions, fundamentally resolves the inherent unrealistic feature of the weighted-averaging mechanism. Derived from the cognitive dissonance theory in psychology, the weighted-median mechanism is supported by online experiment data and broadens the applicability of opinion dynamics models to multiple-choice issues with ordered discrete options. Moreover, the weighted-median mechanism, despite being the simplest in form, captures various non-trivial real-world features of opinion evolution, while some widely-studied averaging-based models fail to.

cs.SI↗

arXiv2018

Improved mixing rates of directed cycles by added connection

We investigate the mixing rate of a Markov chain where a combination of long distance edges and non-reversibility is introduced: as a first step, we focus here on the following graphs: starting from the cycle graph, we select random nodes and add all edges connecting them. We prove a square factor improvement of the mixing rate compared to the reversible version of the Markov chain.

math.PR↗

arXiv2017

Spectral identification of networks with inputs

We consider a network of interconnected dynamical systems. Spectral network identification consists in recovering the eigenvalues of the network Laplacian from the measurements of a very limited number (possibly one) of signals. These eigenvalues allow to deduce some global properties of the network, such as bounds on the node degree. Having recently introduced this approach for autonomous networks of nonlinear systems, we extend it here to treat networked systems with external inputs on the nodes, in the case of linear dynamics. This is more natural in several applications, and removes the need to sometimes use several independent trajectories. We illustrate our framework with several examples, where we estimate the mean, minimum, and maximum node degree in the network. Inferring some information on the leading Laplacian eigenvectors, we also use our framework in the context of network clustering.

eess.SY↗