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Investigate convergence of jitter standard deviation in AutoStep MCMC tuning

Investigate the convergence behavior of the jitter standard deviation σ in the round-based tuning scheme for AutoStep MCMC and determine why σ tends to converge to approximately 0.1 across a wide range of target distributions. Provide a deeper analysis explaining this phenomenon within the AutoStep MCMC framework.

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Background

AutoStep MCMC introduces a round-based procedure to tune key parameters, including the step size jitter standard deviation σ. In empirical studies across diverse targets, the authors observed that σ varies in early rounds due to limited sample sizes but stabilizes as rounds progress.

A consistent empirical pattern emerged: σ often converges to a value near 0.1 across many different target distributions. The authors explicitly flag understanding this pattern as an open question, seeking a principled explanation for the observed convergence behavior in the AutoStep MCMC tuning process.

References

Figure \ref{fig:convergence_of_jitter} shows that in early tuning rounds the jitter standard deviation varies due to the use of relatively small sample sizes, but eventually stabilizes and converges as the rounds progress. An interesting observation is that across a wide range of targets, the jitter standard deviation tends to converge to $\approx 0.1$; we leave a deeper investigation of this observation as an open question.

— AutoStep: Locally adaptive involutive MCMC (2410.18929 - Liu et al., 24 Oct 2024) in Section 5.3 (Stability of round-based jitter tuning)