Comparison Theorems for the Mixing Times of Systematic and Random Scan Dynamics (2410.11136v1)

Abstract: A popular method for sampling from high-dimensional distributions is the Gibbs sampler, which iteratively resamples sites from the conditional distribution of the desired measure given the values of the other coordinates. But to what extent does the order of site updates matter in the mixing time? Two natural choices are (i) standard, or random scan, Glauber dynamics where the updated variable is chosen uniformly at random, and (ii) the systematic scan dynamics where variables are updated in a fixed, cyclic order. We first show that for systems of dimension $n$, one round of the systematic scan dynamics has spectral gap at most a factor of order $n$ worse than the corresponding spectral gap of a single step of Glauber dynamics, tightening existing bounds in the literature by He, et al. [NeurIPS '16] and Chlebicka, {\L}atuszy\'nski, and Miasodejow [Ann. Appl. Probab. '24]. This result is sharp even for simple spin systems by an example of Roberts and Rosenthal [Int. J. Statist. Prob. '15]. We complement this with a converse statement: if all, or even just one scan order rapidly mixes, the Glauber dynamics has a polynomially related mixing time, resolving a question of Chlebicka, {\L}atuszy\'nski, and Miasodejow. Our arguments are simple and only use elementary linear algebra and probability.

• The paper shows that one round of systematic scan dynamics has a spectral gap at most a factor of order n worse than random scan dynamics.
• It establishes that the mixing time for systematic scans can be at most n² times worse than for random scan dynamics, even in worst-case scenarios.
• The analysis introduces a versatile framework using linear algebra and probability to compare deterministic and randomized update strategies.

An Examination of Mixing Times in Systematic and Random Scan Dynamics

The paper "Comparison Theorems for the Mixing Times of Systematic and Random Scan Dynamics" presents a rigorous treatment of the mixing times associated with two Gibbs sampling procedures: the systematic scan and the random scan Glauber dynamics. This research is pivotal in understanding the efficiency of these algorithms in the context of sampling from high-dimensional distributions.

Highlights of the Research

• Gibbs Sampler Dynamics: The paper focuses on the Gibbs sampler, a widely-used method for sampling from high-dimensional distributions. The two update mechanisms explored are the systematic scan, where variables are updated in a fixed cyclic order, and the random scan Glauber dynamics, where updates are chosen randomly.
• Spectral Gap Comparisons: The authors improve upon existing bounds for the spectral gap. They show that one round of systematic scan dynamics has a spectral gap at most a factor of order $n$ worse than the random scan. This result tightens previous bounds and holds even for simple spin systems.
• Reversibility and Non-Reversibility: The systematic scan is not typically reversible, unlike the random scan, making its analysis more complex. However, its practical benefits include increased computational efficiency by avoiding additional randomness.
• Implications for Mixing Times: The research establishes that the mixing time of systematic scan dynamics can be at most $n^2$ times worse than that of Glauber dynamics, modulo logarithmic factors. This result is sharp even for worst-case scenarios, highlighting the intrinsic complexity introduced by the order of updates.

Theoretical and Practical Significance

• Algorithm Efficiency: Understanding the relationship between systematic and random updates aids in selecting the most computationally efficient method for specific applications. The systematic scan's deterministic update pattern can improve runtime efficiency in real-world implementations.
• Analytical Framework: The paper provides a framework to bound mixing times across different dynamics, leveraging elementary linear algebra and probability. These bounds extend to general settings, offering insights even when deterministic update sequences are employed.
• Future Directions: The results point to further exploration of how other functional inequalities may play a role in mixing dynamics and the potential for system-specific analyses to yield polynomial time savings.

Conclusions

This paper addresses open questions regarding the comparative performance of systematic scan and random scan dynamics, providing a more nuanced understanding of their relative efficiencies. The findings have substantial implications in fields such as statistical physics, economics, and beyond, where efficient sampling from complex distributions is crucial. Future work could focus on deriving tighter bounds or exploring alternative sequences of updates that maintain the desired spectral properties.