Optimality of zero refresh rate in the Zig-Zag sampler

Determine whether, for the Zig-Zag sampler targeting probability densities on Euclidean spaces and employing a refreshment mechanism with rate parameter λ≥0, setting the refresh rate to λ=0 minimizes the asymptotic variance of ergodic averages in comparison to choices with λ>0.

Background

The Zig-Zag sampler introduces velocity flips at rates depending on the gradient of the log-density and may also include refreshment events occurring at a user-specified rate λ≥0.

In the numerical illustration, the authors set λ=0 for the Zig-Zag sampler and note that this setting is conjectured to be optimal for minimizing asymptotic variance, referencing prior work. Establishing when and why λ=0 is optimal would clarify best-practice choices of refreshment in Zig-Zag implementations.

References

For the BPS we used refresh rate λ=0.66 (using methodological guidance from [bouchard2018bouncy]), while for ZZS we used λ=0, which is conjectured to be optimal in terms of minimising the asymptotic variance (e.g. [bierkens.duncan:17]).

— Some aspects of robustness in modern Markov Chain Monte Carlo  (2511.21563 - Power et al., 26 Nov 2025) in Piecewise-Deterministic Markov Processes: Bouncy Particle Sampler and the Zig-Zag Process (Section 2.2.2)