Convergence Rates for Non-Log-Concave Sampling and Log-Partition Estimation

2023-03-06 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences, Research Methodology & Innovation · Depth: Expert, short

Summary

A study by David Holzmüller and Francis Bach investigates convergence rates for sampling from Gibbs distributions and computing their log-partition functions, particularly in non-log-concave settings. While log-concave densities have efficient algorithms, non-log-concave scenarios typically face the curse of dimensionality. The research explores whether the faster convergence rates observed in smooth optimization problems can be replicated for non-log-concave sampling. The authors analyze the information-based complexity of these problems, demonstrating that optimal rates for sampling and log-partition computation can sometimes equal or even surpass those for optimization. They also evaluate several polynomial-time sampling algorithms, including an extension of a recent optimization approach, noting interesting behaviors but no near-optimal rates. This work, published in JMLR, offers insights into the relationships among sampling, log-partition, and optimization problems.

Key takeaway

For research scientists developing or applying sampling algorithms in complex, non-log-concave distributions, you should consider that the theoretical optimal convergence rates for sampling and log-partition estimation might be faster than for optimization. This suggests exploring novel sampling algorithms, rather than solely adapting optimization techniques, could yield significant performance improvements in high-dimensional problems.

Key insights

Optimal sampling and log-partition computation rates can sometimes exceed optimization rates in non-log-concave settings.

Principles

• Smoothness can alleviate the curse of dimensionality.
• Optimization is the low-temperature limit of sampling.

Topics

• Gibbs Distributions
• Log-Partition Function
• Non-Log-Concave Sampling
• Convergence Rates
• Information-Based Complexity

Code references

• dholzmueller/sampling_experiments

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.