arXiv subjects

Armand Gissler

Publications and source records attributed to Armand Gissler.

6 recordsLinked to original sources

arXiv2026

Adjusted Scores for Discrete Langevin Algorithms

Sampling from discrete distributions is a ubiquitous task in machine learning, recently revisited by the emergence of discrete diffusion models. While Langevin algorithms constitute the state of the art for continuous spaces, discrete versions lack similar theoretical guarantees when the step-size becomes small. In this paper, we address this limitation by interpreting discrete sampling algorithms as discretizations of continuous-time dynamics on the hypercube. In particular, we describe several score functions for discrete algorithms which result in approximations of Glauber dynamics for the correct target distribution. We also compute upper bounds for the contraction of these algorithms, with or without Metropolis adjustment.

math.ST↗

arXiv2025

On the irreducibility and convergence of a class of nonsmooth nonlinear state-space models on manifolds and their applications to zeroth-order optimization

In this paper, we analyze a large class of general nonlinear state-space models on a state-space X, defined by the recursion $ϕ_{k+1} = F(ϕ_k,α(ϕ_k,U_{k+1}))$, $k \in\mathbb N$, where $F,α$ are some functions and $\{U_{k+1}\}_{k\in\mathbb N}$ is a sequence of i.i.d. random variables. More precisely, we extend conditions under which this class of Markov chains is irreducible, aperiodic and satisfies important continuity properties, relaxing two key assumptions from prior works. First, the state-space X is supposed to be a smooth manifold instead of an open subset of a Euclidean space. Second, we only suppose that $F$ is locally Lipschitz continuous. We demonstrate the significance of our results through their application to Markov chains underlying optimization algorithms. These schemes belong to the class of evolution strategies with covariance matrix adaptation and step-size adaptation.

math.OC↗

arXiv2024

Irreducibility of nonsmooth state-space models with an application to CMA-ES

We analyze a stochastic process resulting from the normalization of states in the zeroth-order optimization method CMA-ES. On a specific class of minimization problems where the objective function is scaling-invariant, this process defines a time-homogeneous Markov chain whose convergence at a geometric rate can imply the linear convergence of CMA-ES. However, the analysis of the intricate updates for this process constitute a great mathematical challenge. We establish that this Markov chain is an irreducible and aperiodic T-chain. These contributions represent a first major step for the convergence analysis towards a stationary distribution. We rely for this analysis on conditions for the irreducibility of nonsmooth state-space models on manifolds. To obtain our results, we extend these conditions to address the irreducibility in different hyperparameter settings that define different Markov chains, and to include nonsmooth state spaces.

math.OC↗

arXiv2024

Asymptotic estimations of a perturbed symmetric eigenproblem

We study ill-conditioned positive definite matrices that are disturbed by the sum of $m$ rank-one matrices of a specific form. We provide estimates for the eigenvalues and eigenvectors. When the condition number of the initial matrix tends to infinity, we bound the values of the coordinates of the eigenvectors of the perturbed matrix. Equivalently, in the coordinate system where the initial matrix is diagonal, we bound the rate of convergence of coordinates that tend to zero.

math.NA↗

arXiv2021

Scaling-invariant functions versus positively homogeneous functions

Scaling-invariant functions preserve the order of points when the points are scaled by the same positive scalar (with respect to a unique reference point). Composites of strictly monotonic functions with positively homogeneous functions are scaling-invariant with respect to zero. We prove in this paper that the reverse is true for large classes of scaling-invariant functions. Specifically, we give necessary and sufficient conditions for scaling-invariant functions to be composites of a strictly monotonic function with a positively homogeneous function. We also study sublevel sets of scaling-invariant functions generalizing well-known properties of positively homogeneous functions.

math.OC↗

arXiv2021

A note on the K-epigraph

We study the question as to when the closed convex hull of a K-convex map equals its K-epigraph. In particular, we shed light onto the smallest cone K such that a given map has convex and closed K-epigraph, respectively. We apply our findings to several examples in matrix space as well as to convex composite functions.

math.OC↗