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Local Geometric Mixing via Dobrushin Contraction with Applications to Diffusion Path Monte Carlo and the Proximal Sampler

Stefan Oberdörster

arXiv:2609.28338v1math.PRcs.LGstat.COstat.ML

TL;DR

Uniform geometric mixing can overlook rapid local equilibration when convergence rates vary with the starting distribution. The paper combines local Dobrushin contraction with exit control and applies the resulting bounds to ideal and Metropolis-adjusted Diffusion Path Monte Carlo, establishing local mixing guarantees under stated conditions.

  • Problem

    Uniform geometric mixing can be limited by slow global convergence and fail to capture faster local equilibration from some initial distributions.

    Source: §1 · p. 3

  • Method

    The paper combines local Dobrushin contraction with exit control and applies this approach to ideal and Metropolis-adjusted Diffusion Path Monte Carlo.

    Source: §1 · p. 5

  • Results

    With sufficient noising, ideal Diffusion Path Monte Carlo has one-step local mixing for any target and start, while finite-variance targets have geometric local mixing for arbitrary initialization; its adjusted counterpart has local guarantees under local acceptance control and stable dynamics.

    Source: §1 · p. 5

  • Takeaways & Limitations

    The bounds capture rapid local equilibration even when global mixing is slower, and the ideal-method guarantees turn spectral-gap estimates into mixing-time bounds without a warm start.

    Source: §1 · pp. 3, 5

    Discretization and score-approximation errors enter implicitly through Metropolis rejection probabilities, and their explicit analysis is beyond the scope of this work.

    Source: §6 · p. 33

Abstract

from arXiv · show

Local geometric mixing localizes geometric mixing by requiring geometric convergence to equilibrium in total variation only over finitely many transitions. It accommodates local convergence rates and captures rapid local equilibration, even when global mixing is much slower. We establish and discuss local geometric mixing bounds through Dobrushin contraction. We then apply this approach to Diffusion Path Monte Carlo, a recently proposed Markov chain Monte Carlo method, aimed at leveraging advances in score-based modeling, whose ideal transitions coincide with those of the Proximal Sampler. Our analysis covers both the ideal method and its implementable Metropolis-adjusted counterpart, providing mixing guarantees under minimal assumptions. For the ideal method, these guarantees complement recent spectral gap estimates, which we develop into mixing time bounds.

1 Introduction

Uniform geometric mixing can miss rapid local equilibration when global convergence is slower. The paper defines local geometric mixing and develops Dobrushin-based bounds, then applies them to ideal and Metropolis-adjusted Diffusion Path Monte Carlo.

1 Introduction

  • Uniform geometric mixing misses fast equilibration from some starts when its rate must reflect slower global convergence.In the seldom-trapped chain, circle starts equilibrate faster than origin starts, but the uniform rate is limited by the slow rate ε/2.Source: §1 · pp. 2–3
  • Local Dobrushin contraction alone does not ensure convergence to a unique invariant distribution; at ε = 0, the seldom-trapped chain can retain initialization-dependent invariant mixtures.The circle’s local contraction rate improves from the global ε/2 rate to 1/2, but the chain may still converge to different invariant measures.Source: §1 · p. 3
  • Local geometric mixing requires geometric total-variation convergence only over finitely many transitions, allowing rates to depend on the initialization and transition horizon.The paper combines local Dobrushin contraction with exit control to obtain local mixing bounds.Source: §1 · pp. 4–5
  • For sufficiently noised ideal Diffusion Path Monte Carlo, the paper proves one-step local mixing for any target and start, and geometric local mixing for finite-variance targets with arbitrary initialization.A regularization estimate converts recent spectral-gap estimates into mixing-time bounds without a warm start.Source: §1 · p. 5
  • For Metropolis-adjusted Diffusion Path Monte Carlo, the paper gives quantitative local mixing guarantees when acceptance probabilities are locally controlled in a domain with suitably stable dynamics.The method is presented as an implementable counterpart of the ideal method.Source: §1 · p. 5

2 Local Geometric Mixing via Dobrushin Contraction

The section derives local geometric mixing bounds by combining Dobrushin contraction inside a domain with control of the probability of leaving it. The resulting bounds capture local equilibration over a finite horizon.

2 Local Geometric Mixing via Dobrushin Contraction

  • Local contraction inside a coupling domain yields geometric decay, while the probability that the coupled chain exits the domain bounds the loss of that guarantee.The bound is aimed at local geometric mixing; its right-hand side generally need not vanish as the transition count grows.Source: §2 · p. 6
  • Total-variation contraction extends earlier Wasserstein-style bounds and can be straightforward to establish for chains with Metropolis adjustment.The approach also extends to general coupling sets.Source: §2 · p. 7
  • For a state-space domain D, the probability of exit before n is bounded by the initial mass outside D plus (n − 1) times the largest one-step flow from D to its complement.The bound applies for all n ≥ 1.Source: §2 · p. 8
  • On the circle in the seldom-trapped example, local Dobrushin contraction has rate 1/2, and the resulting bound captures rapid local mixing.The exit-probability bounds are used to control departures from the circle.Source: §2 · p. 8

3 Diffusion Path Monte Carlo and the Proximal Sampler

The section presents Diffusion Path Monte Carlo and the Proximal Sampler as two formulations of the same Markov chain. It describes their transitions and shows how their parameters correspond despite different practical motivations.

3 Diffusion Path Monte Carlo and the Proximal Sampler

  • The section identifies the two samplers as equivalent formulations whose practical motivations target different step-size regimes.The Proximal Sampler’s discussed implementations target small step sizes, whereas Diffusion Path Monte Carlo is aimed at larger noising times or equivalent step sizes.Source: §3 · p. 9, §3.3 · pp. 11–12

3.1 Diffusion Path Monte Carlo

  • Diffusion Path Monte Carlo first applies Ornstein–Uhlenbeck noising, then the noising process’s time reversal in law; its ideal kernel uses the exact denoising transition.The exact denoising kernel is not known in practice, motivating a discretized, score-based approximation with Metropolis–Hastings correction.Source: §3.1 · pp. 9–10

3.2 The Proximal Sampler

  • The Proximal Sampler alternates a Gaussian forward draw with a conditional backward draw in an extended distribution.It is presented as Gibbs sampling with step size h.Source: §3.2 · pp. 10–11

3.3 Two Sides of the Same Coin: Small vs. Large Noising Time

  • Diffusion Path Monte Carlo and the Proximal Sampler have identical transition kernels when t = log(h + 1), despite using different transition constructions.Their kernels correspond up to rescaling by a dilation before a change of variables establishes equality.Source: §3.3 · p. 11

4 Mixing of Ideal Diffusion Path Monte Carlo and the Proximal Sampler

This section reviews convergence results for ideal Diffusion Path Monte Carlo and the Proximal Sampler, then develops mixing guarantees including local one-step and geometric bounds and spectral-gap-based guarantees without warm starts. For mixture targets, it discusses the consistency of spectral-gap and Dobrushin approaches and their differing noising-time scalings in accuracy dependence.

4.1.1 Small Noising Times or Step Sizes

  • Prior small-step results establish contraction for strongly log-concave targets and for targets satisfying log-Sobolev or Poincaré inequalities, assumptions suited to short-step regimes.The cited results include Wasserstein contraction [43] and relative-entropy or χ2-contraction [23].Source: §4.1.1 · pp. 13, 15

4.1.2 Large Noising Times or Step Sizes

  • For large noising times, [20] shows that multimodal mixtures of Poincaré components slow χ2-convergence by at most an absolute constant, while shorter times can have exponentially degenerating spectral-gap bounds.The mixture result allows strongly separated modes but assumes each component satisfies a Poincaré inequality.Source: §4.1.2 · p. 15

4.2 Local Mixing for Large Noising Times in General Target Distributions

  • For general target probability measures, large noising times yield mixing guarantees without assumptions on the target or initial distribution, via local geometric mixing and Dobrushin contraction.The bounds apply uniformly over starting points in a fixed ball.Source: §4.2 · p. 16

4.2.1 Local Mixing in One Step

  • For any target probability measure, sufficiently large noising times give one-step mixing uniformly from starts in a centered ball, with sufficient noising time growing logarithmically in its radius.The bound has mild initialization sensitivity compared with spectral-gap mixing bounds that require warm starts.Source: §4.2.1 · p. 17

4.2.2 Local Geometric Mixing

  • For sub-exponential tails, geometric mixing requires noising times scaling as log log ε^-1, versus log ε^-1 for one-step mixing; this scaling difference disappears for heavier polynomial tails.The geometric-mixing result assumes finite variance, whereas the one-step bound does not.Source: §4.2.2 · pp. 19–21
  • Theorem 7’s local-mixing proof controls the probability of exiting BR to establish finite-step mixing and a mixing-time bound.The proof uses Proposition 3 to bound one-step exit probabilities uniformly for starts in BR.Source: §4.2.2 · pp. 21–22

4.3 Mixing under a Spectral Gap

  • Theorem 9 shows that one ideal Diffusion Path Monte Carlo transition makes even a point-mass start warm, enabling spectral-gap mixing bounds without a warm-start restriction.The theorem applies equivalently to the Proximal Sampler with step size h = e^t − 1.Source: §4.3 · p. 24
  • Relative entropy from each transition π(t)(x, ·) to equilibrium is uniformly controlled for x ∈ BR, yielding mixing-time bounds for general initial distributions via Pinsker’s inequality.The bound relies on contraction to equilibrium in relative entropy.Source: §4.3 · p. 24
  • For mixture targets with Poincaré components, spectral-gap and Dobrushin approaches give consistent noising-time results, but the Dobrushin scaling has a mild extra double-log dependence on inverse accuracy.The extra dependence reflects that Dobrushin contraction guarantees only local geometric mixing, so accuracy enters the noising time to ensure sufficient stability.Source: §4.3 · p. 25

5 Metropolis-adjusted Diffusion Path Monte Carlo

Section 5 develops an implementable Metropolis-adjusted version of Diffusion Path Monte Carlo, correcting approximate noising–denoising transitions through a path-space construction. Under the stated absolute-continuity assumptions, the resulting projected kernel is reversible with respect to the target, though singular target components impose a scope limitation.

5.1 Implementing the Noising-Denoising Transition

  • Euler–Maruyama discretization and an approximate score produce a computational proposal because exact draws from the ideal denoising kernel are inaccessible.The theoretical analysis treats the approximate score as a black box.Source: §5.1 · p. 26

5.2 The Metropolis–Hastings Method

  • Discretization and score approximation can make the proposal kernel fail to preserve µ, while its endpoint-law acceptance probability is generally intractable to evaluate.The path-space adjustment provides the computationally feasible alternative described in the section.Source: §5.2 · pp. 27–28
  • General-state-space Metropolis–Hastings proposes a move, accepts it with its acceptance probability or stays at the current state, and satisfies detailed balance with µ.Detailed balance implies that µ is invariant.Source: §5.2 · pp. 27–28

5.3 Metropolis-adjusted Diffusion Path Monte Carlo

  • The feasible correction lifts the target to noising-denoising paths, adjusts deterministic path reversal, and combines it symmetrically with reversible path resampling.The path-space target projects back to µ.Source: §5.3 · pp. 28–29
  • Under absolutely continuous one-step transition laws, the path-space kernel and its projection are reversible with respect to the lifted target and µ, respectively.This result applies to any target probability distribution µ on Rd.Source: §5.3 · p. 30
  • Absolutely continuous transition kernels almost surely cannot end in a target’s singular component, so the method is not irreducible for targets with such components.For targets on lower-dimensional manifolds, the text points to Wasserstein-distance guarantees instead of total-variation convergence.Source: §5.3 · p. 30

6 Mixing of Metropolis-adjusted Diffusion Path Monte Carlo

The section applies local Dobrushin-contraction analysis to Metropolis-adjusted Diffusion Path Monte Carlo, giving local mixing guarantees when rejection probabilities are controlled in a stable domain. It also illustrates how acceptance can be high in typical Gaussian draws yet degenerate near the origin.

6 Mixing of Metropolis-adjusted Diffusion Path Monte Carlo

  • The section provides, to the authors’ knowledge, the first convergence analysis of Metropolis-adjusted Diffusion Path Monte Carlo.It transfers the local Dobrushin-contraction framework used for the ideal method to the Metropolis-adjusted method.Source: §6 · p. 32
  • Under controlled rejection and exit probabilities, sufficiently large noising times yield local geometric mixing for any target and initial probability distributions.The bounds apply once a domain with suitably stable dynamics is fixed, and the adjusted method mixes similarly to the ideal method when rejection probabilities are uniformly controlled there.Source: §6 · pp. 32–33
  • The theorem’s convenient rejection-probability assumption can be weakened using a bound on the probability of mutual acceptance, relaxing the requirement 2 Pr(D) < 1.The proof separates contraction of Q(t) from the Metropolis adjustment; uniform control of the counter probability of mutual acceptance suffices.Source: §6 · p. 33
  • Discretization and score-approximation errors enter implicitly through rejection probabilities, but their explicit control is outside this work’s scope.Suitable rejection-probability bounds can be inserted into Theorem 11 and Corollary 12 to make those error dependencies explicit.Source: §6 · p. 33
  • One or a couple of unadjusted Q(t) warm-up transitions may produce an informed initialization and prevent starting in tails where Metropolis adjustment is sensitive.This suggestion assumes Q(t) describes the target reasonably well.Source: §6 · p. 33
  • Unlike the ideal method, the adjusted method cannot generally mix to accuracy ε in one transition unless its rejection probability is of order ε.The text notes that score-approximation error may preclude such small rejection probabilities, regardless of discretization step size.Source: §6 · p. 34
  • For suitable h = Θ(d^-1/2), Gaussian-target acceptance stays bounded away from zero with high probability in typical stationary draws but degenerates exponentially at the origin.The section uses this contrast to illustrate rapid local mixing in a high-probability region alongside slower global mixing due to traps.Source: §6 · p. 36
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