[Paper Review] Optimization assisted MCMC

Ricky Fok, Aijun An+1 more|arXiv (Cornell University)|Sep 9, 2017

Markov Chains and Monte Carlo Methods13 references3 citations

TL;DR

This paper proposes Optimization-Assisted MCMC, a framework that integrates global optimization via conformal geodesics and local search (e.g., BFGS) to accelerate mode discovery in high-dimensional multimodal distributions. By first locating modes using a geodesic path on a conformally transformed manifold and then refining them with local optimization, the method enables faster convergence and significantly reduced bias in MCMC sampling, outperforming WHMC by up to an order of magnitude in relative error for mean estimation.

ABSTRACT

Markov Chain Monte Carlo (MCMC) sampling methods are widely used but often encounter either slow convergence or biased sampling when applied to multimodal high dimensional distributions. In this paper, we present a general framework of improving classical MCMC samplers by employing a global optimization method. The global optimization method first reduces a high dimensional search to an one dimensional geodesic to find a starting point close to a local mode. The search is accelerated and completed by using a local search method such as BFGS. We modify the target distribution by extracting a local Gaussian distribution aound the found mode. The process is repeated to find all the modes during sampling on the fly. We integrate the optimization algorithm into the Wormhole Hamiltonian Monte Carlo (WHMC) method. Experimental results show that, when applied to high dimensional, multimodal Gaussian mixture models and the network sensor localization problem, the proposed method achieves much faster convergence, with relative error from the mean improved by about an order of magnitude than WHMC in some cases.

Motivation & Objective

• To address slow convergence and mode-hunting bias in MCMC sampling for high-dimensional multimodal posteriors.
• To overcome the limitations of WHMC, such as reliance on random starting points and fictitious mode creation via residual potential subtraction.
• To improve the efficiency and accuracy of mode discovery before sampling begins, reducing on-the-fly bias in MCMC estimates.
• To develop a method that ensures better exploration of all major modes by guiding optimization toward undiscovered modes from the start.
• To integrate global optimization into MCMC in a way that maintains detailed balance and enables faster, more reliable convergence.

Proposed method

• The method constructs a conformal geodesic on a Riemannian manifold derived from the residual log-likelihood, enabling a one-dimensional path that can traverse multiple local maxima.
• It uses a global optimization strategy on this geodesic to locate major modes before sampling, reducing the risk of missing significant modes.
• Local optimization via BFGS is applied to refine the mode estimates found along the geodesic, improving accuracy and convergence speed.
• The target distribution is modified by extracting a local Gaussian approximation around each discovered mode, enabling efficient sampling and mode separation.
• The framework is integrated into Wormhole Hamiltonian Monte Carlo (WHMC), using the discovered modes to guide regeneration and proposal distributions.
• A modified leapfrog integrator is used that incorporates a vector field to guide transitions between modes via wormhole metrics, shortening intermodal distances.

Experimental results

Research questions

• RQ1 Can a geodesic-based global optimization path reliably locate multiple modes in high-dimensional multimodal distributions?
• RQ2 Does pre-discovering modes via optimization reduce sampling bias and improve convergence speed in MCMC?
• RQ3 How does the proposed method compare to WHMC in terms of relative error for mean and covariance estimation?
• RQ4 Can the use of conformal geometry and geodesic paths avoid the gradient vanishing issues common in residual potential methods?
• RQ5 Does the integration of optimization into MCMC reduce the need for random restarts and improve regeneration efficiency?

Key findings

• The proposed method achieves significantly faster convergence than WHMC in high-dimensional multimodal Gaussian mixture models.
• Relative error in mean estimation was improved by approximately one order of magnitude compared to WHMC in tested cases.
• The method successfully avoids fictitious modes that commonly arise in residual potential-based approaches.
• By finding all major modes before sampling, the method reduces on-the-fly bias and improves the accuracy of posterior estimates.
• The use of conformal geodesics enables efficient traversal of multiple basins of attraction, enhancing mode discovery efficiency.
• Integration with WHMC via wormhole metrics and modified leapfrog integrators maintains Hamiltonian dynamics while accelerating intermodal transitions.