Convergence Bounds for Sequential Monte Carlo on Multimodal Distributions using Soft Decomposition

Abstract: We prove bounds on the variance of a function $f$ under the empirical measure of the samples obtained by the Sequential Monte Carlo (SMC) algorithm, with time complexity depending on local rather than global Markov chain mixing dynamics. SMC is a Markov Chain Monte Carlo (MCMC) method, which starts by drawing $N$ particles from a known distribution, and then, through a sequence of distributions, re-weights and re-samples the particles, at each instance applying a Markov chain for smoothing. In principle, SMC tries to alleviate problems from multi-modality. However, most theoretical guarantees for SMC are obtained by assuming global mixing time bounds, which are only efficient in the uni-modal setting. We show that bounds can be obtained in the truly multi-modal setting, with mixing times that depend only on local MCMC dynamics.

• The paper derives variance bounds based on local mixing dynamics, improving SMC sampling in multimodal settings.
• It decomposes variance into inter-mode and intra-mode components using local Poincaré inequalities and mixture models.
• The results offer a framework for efficient SMC algorithms applicable to complex high-dimensional models in various fields.

An Analysis of Local-Mixing Dependent Variance Bounds in Sequential Monte Carlo Methods

This paper addresses an important limitation in the theoretical guarantees of Sequential Monte Carlo (SMC) algorithms by providing variance bounds based on local, rather than global mixing dynamics. The main contribution is the derivation of variance bounds that maintain efficiency in multi-modal settings, where traditional SMC methods often falter due to poor global mixing times.

Introduction to Sequential Monte Carlo Methods

SMC methods effectively sample from a target distribution $p(x) \propto e^{-V(x)}$, particularly in high-dimensional settings. These methods begin by sampling $N$ particles from an initial distribution. Subsequently, they re-weight and re-sample these particles through a sequence of intermediate distributions, applying a Markov chain at each step to ensure appropriate mixing. Despite their robustness in dealing with uni-modal distributions, traditional SMC methods suffer in multi-modal contexts due to their reliance on global mixing time bounds.

Local Mixing Dynamics Framework

The paper departs from the conventional approach by focusing on local mixing time dynamics rather than global ones. The authors consider SMC algorithms that utilize Markov processes with generators exhibiting a specific decomposition: $$\langle f, \mathscr{L}_kf \rangle_{\mu_k} \le \sum_{i=1}^m w_i\langle f, \mathscr{L}_{ki}f \rangle_{\mu_k^{(i)}}.$$ Here, $\mathscr{L}_{ki}$ represents the generator over the $i$-th mixture component and $w_i$ are weights in the mixture model. This decomposition allows capturing the dynamics within each mode separately and also between modes for mixtures such as Langevin dynamics and Metropolis random walks.

Main Results and Methodology

Variance Bound on Empirical Measures

The main theoretical result is a non-asymptotic bound on the variance of a function $f$ under the empirical measure $\mu_n^N$: $Var_{\xi_n}(\eta_n^N(f)) \leq \epsilon,$ for a number of samples $N$ and steps $t$ that are polynomial in the relevant parameters. This bound is significant as it only depends on local mixing properties instead of global ones, thus making it feasible for multi-modal distributions.

Inter-mode and Intra-mode Variance Decomposition

The authors further enhance the analysis by decomposing the variance into inter-mode and intra-mode components. Using local Poincaré inequalities, the intra-mode variance is bounded, which depends on the local mixing conditions. The inter-mode variance is treated separately by leveraging the mixture structure and employing minimum mode weight terms.

Hypercontractivity and Log-Sobolev Inequalities

A key component of the paper is the hypercontractivity analysis based on log-Sobolev inequalities. By connecting local log-Sobolev constants with hypercontractive properties, the authors further demonstrate how the smoothing effect of Markov kernels can be rigorously bounded: $$\|P_k f\|_{L^p(\mu_k)} \le \theta(p, p/2) \|f\|_{L^{p/2}(\mu_k)},$$ where $\theta(p, p/2)$ is a constant conducive to holding the inequality for decomposition over modes.

Practical Implications

Sampling and Telescoping Ratio Estimation

The implications of these results are substantial in practical sampling scenarios, especially when dealing with complex distributions such as those encountered in Noise Contrastive Estimation and Telescoping Ratio Estimation (TRE). The authors illustrate applications in both simulated and annealed importance sampling methods, demonstrating significant improvements in the number of samples required and the resultant variance bounds.

Efficient SMC with Local Properties

By leveraging local Poincaré and log-Sobolev inequalities, the results suggest alternative designs for SMC algorithms that are substantially more efficient for multi-modal distributions. This adaptation is particularly relevant for high-dimensional models widely used in machine learning and statistical physics.

Conclusion and Future Directions

The paper provides a significant theoretical framework for understanding and improving SMC methods using local mixing dynamics. It challenges the traditionally global approach, opening new avenues for efficient multi-modal sampling. Future work could expand on these results by exploring higher-order mixture decompositions or extending the methodology to other types of Markov processes.

In conclusion, this work establishes a pivotal foundation for SMC algorithms that are robust in multi-modal contexts, thereby contributing significantly to the field of Monte Carlo sampling and its applications in computational disciplines.

Reference

[Schweizer12] Schweizer, N. (2012), Non-asymptotic analysis of Sequential Monte Carlo methods.