Bayesian learning of expensive models, with applications to cell biology – Baccarat

Monte Carlo methods are randomized numerical quadratures, i. e. random sets of weighted points, where each point (or node) corresponds to a given value of the parameters of a biological model. Our workhorse is to design numerical quadratures with repulsive nodes. For instance, 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.

Here are some of our results:
* We have proved that determinantal point processes (DPPs) give numerical integration algorithms with an optimal convergence rate in reproducing kernel Hilbert spaces.
* We have given an asymptotic statistical test of hyperuniformity, the property of a point process that makes it a fast Monte Carlo algorithm.
* We have shown that the combination of a classical and a quantum computer allows to sample some DPPs faster than using only classical computers.

Perpectives are numerous, e.g.:
* there is still a gap between the processes that efficiently solve a nuemrical integration task on paper and the processes that we can efficiently sample on a (classical or quantum) computer.
* sampling a DPP on a quantum computer is a task that can be optimized in a variety of ways. This task is exciting, since it is possible that, if and once (large) quantum computers become easily accessible, DPPs shall become a standard distribution for every data scientist.

Our production is available as papers in scientific journals, as well as software packages. See
rbardenet.github.io

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. Baccarat 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.