Cône convexe

Résumé

En algèbre linéaire, un cône convexe est une partie d'un espace vectoriel sur un corps ordonné qui est stable par combinaisons linéaires à coefficients strictement positifs. droite|vignette|Exemple de cône convexe (en bleu clair). À l'intérieur de celui-ci se trouve le cône convexe rouge clair qui est composé des points \alpha x + \beta y avec\alpha, \beta > 0, x et y étant les points représentés sur la figure. Les courbes en haut à droite indiquent que les régions se prolongent à l'infini. Définition Soit K un corps ordonné, comme le corps des rationnels \mathbb{Q}, celui des réels algébriques ou (plus couramment) celui des réels \mathbb{R}. Un sous-ensemble C d'un K-espace vectoriel V est un cône convexe si \alpha x + \beta y appartient à C, pour tous scalaires strictement positifs \alpha, \beta

Source officielle

https://fr.wikipedia.org/wiki/C%C3%B4ne_convexe

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Invariant integrals on topological groups

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We generalize the fixed-point property for discrete groups acting on convex cones given by Monod in [23] to topological groups. At first, we focus on describing this fixed-point property from a functional point of view, and then we look at the class of groups that have it. Finally, we go through some applications of this fixed-point property. To accomplish these tasks, we introduce a new class of normed Riesz spaces that depend on group representation. (c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

ACADEMIC PRESS INC ELSEVIER SCIENCE2022

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Stochastic distributed learning with gradient quantization and double-variance reduction

Konstantin Mishchenko, Sebastian Urban Stich

We consider distributed optimization over several devices, each sending incremental model updates to a central server. This setting is considered, for instance, in federated learning. Various schemes have been designed to compress the model updates in order to reduce the overall communication cost. However, existing methods suffer from a significant slowdown due to additional variance omega > 0 coming from the compression operator and as a result, only converge sublinearly. What is needed is a variance reduction technique for taming the variance introduced by compression. We propose the first methods that achieve linear convergence for arbitrary compression operators. For strongly convex functions with condition number kappa, distributed among n machines with a finite-sum structure, each worker having less than in components, we also (i) give analysis for the weakly convex and the non-convex cases and (ii) verify in experiments that our novel variance reduced schemes are more efficient than the baselines. Moreover, we show theoretically that as the number of devices increases, higher compression levels are possible without this affecting the overall number of communications in comparison with methods that do not perform any compression. This leads to a significant reduction in communication cost. Our general analysis allows to pick the most suitable compression for each problem, finding the rig ht balance between additional variance and communication savings. Finally, we also (iii) give analysis for arbitrary quantized updates.

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MATHICSE Technical Report : A fast gradient method for nonnegative sparse regression with self dictionary

Robert Gerhard Jérôme Luce

Nonnegative matrix factorization (NMF) can be computed efficiently under the separability assumption, which asserts that all the columns of the input data matrix belong to the convex cone generated by only a few of its columns. The provably most robust methods to identify these basis columns are based on nonnegative sparse regression and self dictionary, and require the solution of large-scale convex optimization problems. In this paper we study a particular nonnegative sparse regression model with self dictionary. As opposed to previously proposed models, it is a smooth optimization problem where sparsity is enforced through appropriate linear constraints. We show that the Euclidean projection on the set defined by these constraints can be computed efficiently, and propose a fast gradient method to solve our model. We show the effectiveness of the approach compared to state-of-the-art methods on several synthetic data sets and real-world hyperspectral images.

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