[Paper Review] Scan Order in Gibbs Sampling: Models in Which it Matters and Bounds on How Much

Bryan He, Christopher De+2 more|arXiv (Cornell University)|Jun 10, 2016

Markov Chains and Monte Carlo Methods24 references19 citations

TL;DR

This paper investigates the impact of scan order in Gibbs sampling, demonstrating that systematic scan can mix significantly faster or slower than random scan by polynomial factors, contrary to the long-standing conjecture that differences are at most logarithmic. The authors introduce an augmented state space method using conductance to prove that, under mild conditions, scan order affects mixing time only up to a polynomial factor, resolving a key theoretical question in MCMC sampling.

ABSTRACT

Gibbs sampling is a Markov Chain Monte Carlo sampling technique that iteratively samples variables from their conditional distributions. There are two common scan orders for the variables: random scan and systematic scan. Due to the benefits of locality in hardware, systematic scan is commonly used, even though most statistical guarantees are only for random scan. While it has been conjectured that the mixing times of random scan and systematic scan do not differ by more than a logarithmic factor, we show by counterexample that this is not the case, and we prove that that the mixing times do not differ by more than a polynomial factor under mild conditions. To prove these relative bounds, we introduce a method of augmenting the state space to study systematic scan using conductance.

Motivation & Objective

• To challenge the longstanding conjecture that systematic and random scan Gibbs sampling differ in mixing time by at most a logarithmic factor.
• To investigate whether systematic scan can mix significantly faster or slower than random scan in specific models.
• To establish relative bounds on mixing times across different scan orders under mild conditions.
• To develop a novel method for analyzing systematic scan using conductance via state space augmentation.

Proposed method

• Introduce an augmented state space that includes both the original variables and a scan index to model systematic scan as a Markov chain.
• Define a transition kernel for the augmented chain that preserves the dynamics of systematic scan while enabling conductance analysis.
• Use conductance as a tool to bound mixing times, leveraging known inequalities from Markov chain theory.
• Prove that the conductance of the augmented systematic scan chain is lower bounded by a fraction of the random scan conductance.
• Establish relative bounds on mixing times by relating the conductance of the augmented systematic scan to that of random scan.
• Apply these bounds to show that scan order can only affect mixing time by a polynomial factor under mild conditions.

Experimental results

Research questions

• RQ1 Can systematic scan mix faster than random scan by more than a logarithmic factor in any model?
• RQ2 Can systematic scan with a poor permutation mix slower than random scan by more than a constant factor?
• RQ3 What is the tightest possible relative bound on mixing times between random scan and systematic scan across all models?
• RQ4 How does the choice of permutation in systematic scan affect mixing time in practice?
• RQ5 Can conductance be effectively applied to analyze systematic scan by augmenting the state space?

Key findings

• A counterexample is constructed where systematic scan with a specific permutation mixes more than a logarithmic factor faster than random scan, disproving direction (2) of the conjecture.
• Another model is presented where the worst-case systematic scan permutation mixes slower than random scan by a polynomial factor, disproving direction (1) of the conjecture.
• The mixing time of systematic scan is bounded above by a polynomial factor of the random scan mixing time under mild conditions, establishing a relative bound.
• The conductance of the augmented systematic scan chain is shown to be at least a fraction of the conductance of the random scan chain, enabling theoretical bounds.
• Empirical validation confirms that mixing times vary significantly with scan permutation, supporting the theoretical findings.
• The proposed state space augmentation method enables the application of conductance techniques to systematic scan, which was previously difficult due to lack of reversibility.