January
Combining Normalizing Flows and Quasi-Monte Carlo
Charly Andral Ceremade, Université Paris Dauphine-PSL
Recent advances in machine learning have led to the development of new methods for enhancing Monte Carlo methods such as Markov chain Monte Carlo (MCMC) and importance sampling (IS). One such method is normalizing flows, which use a neural network to approximate a distribution by evaluating it pointwise. Normalizing flows have been shown to improve the performance of MCMC and IS. On the other side, (randomized) quasi-Monte Carlo methods are used to perform numerical integration. They replace the random sampling of Monte Carlo by a sequence which cover the hypercube more uniformly, resulting in better convergence rates for the error that plain Monte Carlo. In this work, we combine these two methods by using quasi-Monte Carlo to sample the initial distribution that is transported by the flow. We demonstrate through numerical experiments that this combination can lead to an estimator with significantly lower variance than if the flow was sampled with a classic Monte Carlo.
Optimizing the diffusion of overdamped Langevin dynamics
Régis Santet, CERMICS, École des Ponts & MATHERIALS, INRIA Paris
Overdamped Langevin dynamics are reversible stochastic differential equations which are commonly used to sample probability measures in high dimensional spaces, such as the ones appearing in computational statistical physics and Bayesian inference. By varying the diffusion coefficient, there are in fact infinitely many reversible overdamped Langevin dynamics which preserve the target probability measure at hand. This suggests to optimize the diffusion coefficient in order to increase the convergence rate of the dynamics, as measured by the spectral gap of the generator associated with the stochastic differential equation. We analytically study this problem here, obtaining in particular necessary conditions on the optimal diffusion coefficient. We also derive an explicit expression of the optimal diffusion in some homogenized limit. Numerical results, both relying on discretizations of the spectral gap problem and Monte Carlo simulations of the stochastic dynamics, demonstrate the increased quality of the sampling arising from an appropriate choice of the diffusion coefficient.
This is joint work with Tony Lelièvre, Grigorios Pavliotis, Geneviève Robin and Gabriel Stoltz.