RT Monograph
SR 00
A1 Crespo, Marelys
A1 Gadat, Sébastien
A1 Gendre, Xavier
T1 Stochastic Langevin Monte Carlo for (weakly) log-concave posterior distributions
YR 2023
FD 2023-01
VO 23-1398
SP 28
AB In this paper, we investigate a continuous time version of the Stochastic Langevin Monte Carlo method, introduced in [39], that incorporates a stochastic sampling step inside the traditional overdamped Langevin diffusion. This method is popular in machine learning for sampling posterior distribution. We will pay specific attention in our work to the computational cost in terms  of n (the number of observations that produces the posterior distribution), and d (the dimension of the ambient space where the parameter of interest is living). We derive our analysis in the weakly convex framework, which is parameterized with the help of the Kurdyka- Lojasiewicz (KL) inequality, that permits to handle a vanishing curvature settings, which is far less restrictive when compared to the simple strongly convex case. We establish that the final horizon of simulation to obtain an ε approximation (in terms of entropy) is of the order (d log(n)²)(1+r)² [log²(ε−1) + n²d²(1+r) log4(1+r)(n)] with a Poissonian subsampling of parameter n(d log²(n))1+r)−1, where the parameter r is involved in the KL inequality and varies between 0 (strongly convex case) and 1 (limiting Laplace situation).
T2 TSE Working Paper
PB TSE Working Paper
PP Toulouse
AV Published
LK https://publications.ut-capitole.fr/id/eprint/46718/
UL http://tse-fr.eu/pub/127747