[Paper Review] Tree pyramidal adaptive importance sampling

Javier Felip, Nilesh Ahuja+1 more|arXiv (Cornell University)|Dec 18, 2019

Bayesian Methods and Mixture Models26 references4 citations

TL;DR

This paper proposes Tree-Pyramidal Adaptive Importance Sampling (TP-AIS), a parameter-free, anytime importance sampling method that uses a hierarchical tree pyramid structure to adaptively refine proposal distributions across subspaces. By iteratively subdividing high-density regions and retaining all weighted samples, TP-AIS achieves superior sampling efficiency and higher Normalized Effective Sample Size (N-ESS) than state-of-the-art methods like DM-PMC, M-PMC, and LAIS, especially in low-to-mid dimensional problems.

ABSTRACT

This paper introduces Tree-Pyramidal Adaptive Importance Sampling (TP-AIS), a novel iterated sampling method that outperforms state-of-the-art approaches like deterministic mixture population Monte Carlo (DM-PMC), mixture population Monte Carlo (M-PMC) and layered adaptive importance sampling (LAIS). TP-AIS iteratively builds a proposal distribution parameterized by a tree pyramid, where each tree leaf spans a subspace that represents its importance density. After each new sample operation, a set of tree leaves are subdivided for improving the approximation of the proposal distribution to the target density. Unlike the rest of the methods in the literature, TP-AIS is parameter free and requires no tuning to achieve its best performance. We evaluate TP-AIS with different complexity randomized target probability density functions (PDF) and also analyze its application to different dimensions. The results are compared to state-of-the-art iterative importance sampling approaches and other baseline MCMC approaches using Normalized Effective Sample Size (N-ESS), Jensen-Shannon Divergence, and time complexity.

Motivation & Objective

• Address the challenge of poor sampling efficiency in importance sampling due to suboptimal proposal distributions.
• Overcome the need for manual tuning of parameters in existing adaptive IS methods, which limits performance and usability.
• Enable anytime sampling with trade-offs between computation time and approximation quality, without burn-in or mixing issues.
• Improve sample efficiency and approximation accuracy in low-to-mid dimensional posterior distributions compared to MCMC and IS baselines.
• Develop a scalable, parallelizable sampling framework that maintains high-quality proposal distributions through hierarchical space partitioning.

Proposed method

• Parameterize the proposal distribution using a tree pyramid, where each leaf represents a K-dimensional subspace with an associated importance density.
• Iteratively subdivide tree leaves based on sample density and importance weights to refine the proposal distribution over time.
• Use a hierarchical, adaptive refinement strategy that allocates more samples to high-probability regions and fewer to low-probability ones.
• Maintain all samples with their weights, enabling parallel sampling and anytime stopping without loss of information.
• Leverage the structure of the tree pyramid to efficiently compute and update proposal densities across nested subspaces.
• Ensure the method is parameter-free by avoiding hyperparameters for adaptation, tuning, or kernel selection.

Experimental results

Research questions

• RQ1 Can a parameter-free, adaptive importance sampling method achieve higher sampling efficiency than state-of-the-art IS and MCMC methods?
• RQ2 How does TP-AIS perform in terms of Normalized Effective Sample Size (N-ESS) and Jensen-Shannon Divergence (JSD) across different target distributions and dimensions?
• RQ3 To what extent does the anytime property of TP-AIS enable practical trade-offs between computation time and approximation quality?
• RQ4 How does TP-AIS scale with increasing dimensionality compared to MCMC and other IS methods?
• RQ5 What are the limitations of TP-AIS in high-dimensional settings, and how does it compare to MCMC in terms of time complexity under high-dimensional target densities?

Key findings

• TP-AIS achieves significantly higher Normalized Effective Sample Size (N-ESS) than DM-PMC, M-PMC, and LAIS, with 100 samples outperforming other methods at 1000 samples in 1D Gaussian Mixture Models.
• In higher-dimensional problems (e.g., 9D), TP-AIS maintains high sampling efficiency, while MCMC-Metropolis-Hastings suffers from low acceptance rates and exponential time growth due to high-dimensional concentration.
• The method exhibits comparable time complexity to baseline methods for a reasonable number of samples, despite its more complex proposal structure.
• TP-AIS achieves lower Jensen-Shannon Divergence (JSD) than competing methods, indicating better approximation of the target posterior distribution.
• The parameter-free design eliminates the need for tuning, enabling consistent high performance without domain-specific configuration.
• The method’s anytime property allows halting the sampling process at any time, making it suitable for real-time or resource-constrained inference applications.