Connections between optimization and sampling

In this talk, I am going to give a brief summary of two of my recent papers that exploit connections between optimization and sampling. "Hamiltonian descent methods" greatly extends the class of convex functions that can be optimized with linear rates compared to existing methods. This new optimization method is based on conformal Hamiltonian dynamics, and it was inspired by the literature on Hamiltonian MCMC methods. Besides being more efficient, this method is also more reliable and requires less tuning than existing techniques such as gradient descent and heavy ball method. "Randomized Hamiltonian Monte Carlo as Scaling Limit of the Bouncy Particle Sampler and Dimension-Free Convergence Rates" studies the high dimensional behavior of the Bouncy Particle Sampler (BPS), a non-reversible piecewise deterministic MCMC method. Although the paths of this method are straight lines, we show that in high dimensions they converge to a Randomised Hamiltonian Monte Carlo (RHMC) process, whose paths are determined by the Hamiltonian dynamics. We also give a characterization of the mixing rate of the RHMC process for log-concave target distributions that can be used to tune the parameters of BPS.