Speeding up the Zig-Zag process
Zig-Zag is a Piecewise Deterministic Markov Process, efficiently used for simulation in a Markov Chain Monte Carlo setting. However, it fails to be exponentially ergodic on heavy tailed target distributions. In this talk we introduce an extension of the Zig-Zag process by allowing the process to move with a non-constant speed function $s$, depending on the current state of the process. We call this process Speed Up Zig-Zag (SUZZ). We provide conditions that guarantee stability properties for the SUZZ process, including non-explosivity, exponential ergodicity on heavy tailed targets and central limit theorem. Interestingly, we find that using speed functions that induce explosive deterministic dynamics may lead to stable algorithms that can even mix faster. We support our findings with simulation results and time permitting, we will discuss the choice of an efficient speed function by providing an efficiency criterion for the one-dimensional process.
This is joint work with Gareth Roberts.