Lénaïc Chizat
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Lénaïc Chizat has a total of 3 co-authors. Here are the top co-authors by number of shared publications:
- Alexis Thibault
Inria Bordeaux, 33400 Talence, France
1 shared publication - Charles Dossal
Institut de Mathématiques de Toulouse, Univ. Toulouse, INSA Toulouse, Toulouse, France
1 shared publication - Nicolas Papadakis
Université de Bordeaux
1 shared publication
Latest Publications
- Open Access
Convergence Rates of Gradient Methods for Convex Optimization in the Space of Measures
We study the convergence rate of Bregman gradient methods for convex optimization in the space of measures on a $d$-dimensional manifold. Under basic regularity assumptions, we show that the suboptimality gap at iteration $k$ is in $O(log(k)k^{--1})$ for multiplicative updates, while it is in $O(k^{--q/(d+q)})$ for additive updates for some $q \in {1, 2, 4}$ determined by the structure of the objective function. Our flexible proof strategy, based on approximation arguments, allows us to painlessly cover all Bregman Proximal Gradient Methods (PGM) and their acceleration (APGM) under various geometries such as the hyperbolic entropy and $L^p$ divergences. We also prove the tightness of our analysis with matching lower bounds and confirm the theoretical results with numerical experiments on low dimensional problems. Note that all these optimization methods must additionally pay the computational cost of discretization, which can be exponential in $d$.Open AccessConvergence Rates of Gradient Methods for Convex Optimization in the Space of Measures
We study the convergence rate of Bregman gradient methods for convex optimization in the space of measures on a $d$-dimensional manifold. Under basic regularity assumptions, we show that the suboptimality gap at iteration $k$ is in $O(log(k)k^{--1})$ for multiplicative updates, while it is in $O(k^{--q/(d+q)})$ for additive updates for some $q \in {1, 2, 4}$ determined by the structure of the objective function. Our flexible proof strategy, based on approximation arguments, allows us to painlessly cover all Bregman Proximal Gradient Methods (PGM) and their acceleration (APGM) under various geometries such as the hyperbolic entropy and $L^p$ divergences. We also prove the tightness of our analysis with matching lower bounds and confirm the theoretical results with numerical experiments on low dimensional problems. Note that all these optimization methods must additionally pay the computational cost of discretization, which can be exponential in $d$. - Open Access
- 30 April 2021
- Published by MDPI AG in Algorithms
- Vol. 14 (5) , 143
Overrelaxed Sinkhorn–Knopp Algorithm for Regularized Optimal Transport
This article describes a set of methods for quickly computing the solution to the regularized optimal transport problem. It generalizes and improves upon the widely used iterative Bregman projections algorithm (or Sinkhorn–Knopp algorithm). We first proposed to rely on regularized nonlinear acceleration schemes. In practice, such approaches lead to fast algorithms, but their global convergence is not ensured. Hence, we next proposed a new algorithm with convergence guarantees. The idea is to overrelax the Bregman projection operators, allowing for faster convergence. We proposed a simple method for establishing global convergence by ensuring the decrease of a Lyapunov function at each step. An adaptive choice of the overrelaxation parameter based on the Lyapunov function was constructed. We also suggested a heuristic to choose a suitable asymptotic overrelaxation parameter, based on a local convergence analysis. Our numerical experiments showed a gain in convergence speed by an order of magnitude in certain regimes.
- 30 April 2021
- Published by MDPI AG in Algorithms
- Vol. 14 (5) , 143
Open AccessOverrelaxed Sinkhorn–Knopp Algorithm for Regularized Optimal Transport
This article describes a set of methods for quickly computing the solution to the regularized optimal transport problem. It generalizes and improves upon the widely used iterative Bregman projections algorithm (or Sinkhorn–Knopp algorithm). We first proposed to rely on regularized nonlinear acceleration schemes. In practice, such approaches lead to fast algorithms, but their global convergence is not ensured. Hence, we next proposed a new algorithm with convergence guarantees. The idea is to overrelax the Bregman projection operators, allowing for faster convergence. We proposed a simple method for establishing global convergence by ensuring the decrease of a Lyapunov function at each step. An adaptive choice of the overrelaxation parameter based on the Lyapunov function was constructed. We also suggested a heuristic to choose a suitable asymptotic overrelaxation parameter, based on a local convergence analysis. Our numerical experiments showed a gain in convergence speed by an order of magnitude in certain regimes.