Fast Monte Carlo integration with repulsive processes

ongoing

Expensive computer simulations have become routine in the experimental sciences. Astrophysicists design complex models of the evolution of galaxies, biologists develop intricate models of cells, ecologists model the dynamics of ecosystems at a world scale. A single evaluation of such complex models takes minutes or hours on today's hardware. On the other hand, fitting these models to data can require millions of serial evaluations. Monte Carlo methods, for example, are ubiquitous in statistical inference for scientific data, but they scale poorly with the number of model evaluations. Meanwhile, the use of parallel computing architectures for Monte Carlo is often limited to running independent copies of the same algorithm. Blackjack will provide Monte Carlo methods that unlock inference for expensive models in biology by directly addressing the slow rate of convergence and the parallelization of Monte Carlo methods. The key to take down the Monte Carlo rate is to introduce repulsiveness between the quadrature nodes. For instance, we recently proved that determinantal point processes, a prototypal repulsive distribution introduced in physics, improve the Monte Carlo convergence rate, just like electrons lead to low-variance estimation of volumes by efficiently filling a box. Such results lead to open computational and statistical challenges. We propose to solve these challenges, and make repulsive processes a novel tool for applied statisticians, signal processers, and machine learners. Still with repulsiveness as a hammer, we will design the first parallel Markov chain Monte Carlo algorithms that are qualitatively different from running independent copies of known algorithms, i.e., that explicitly improve the order of convergence of the single-machine algorithm. To this end, we will turn mathematical tools such as repulsive particle systems and non-colliding processes into computationally cheap, communication-efficient Monte Carlo schemes with fast convergence.