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Sampling and Integration of Logconcave Functions by Algorithmic Diffusion

Yunbum Kook, Santosh S. Vempala

Abstract

We study the complexity of sampling, rounding, and integrating arbitrary logconcave functions. Our new approach provides the first complexity improvements in nearly two decades for general logconcave functions for all three problems, and matches the best-known complexities for the special case of uniform distributions on convex bodies. For the sampling problem, our output guarantees are significantly stronger than previously known, and lead to a streamlined analysis of statistical estimation based on dependent random samples.

Sampling and Integration of Logconcave Functions by Algorithmic Diffusion

Abstract

We study the complexity of sampling, rounding, and integrating arbitrary logconcave functions. Our new approach provides the first complexity improvements in nearly two decades for general logconcave functions for all three problems, and matches the best-known complexities for the special case of uniform distributions on convex bodies. For the sampling problem, our output guarantees are significantly stronger than previously known, and lead to a streamlined analysis of statistical estimation based on dependent random samples.

Paper Structure

This paper contains 57 sections, 33 theorems, 124 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Model and problems.
  3. Complexity of logconcave sampling.
  4. Classical sampling algorithms.
  5. Sampling and isoperimetry.
  6. Sampling, isoperimetry, and diffusion.
  7. Algorithmic diffusion.
  8. Results
  9. Result 1: Logconcave sampling from a warm start.
  10. Result 2: Warm-start generation (sampling without a warm start).
  11. Result 3: Isotropic rounding of logconcave distributions.
  12. Result 4: Integration of logconcave functions.
  13. Techniques and background
  14. Going beyond the uniform distribution
  15. Reduction to an exponential distribution.
  16. ...and 42 more sections

Key Result

Theorem 1.5

For any logconcave distribution $\pi^{X}$ specified by a well-defined function oracle $\mathsf{Eval}(V)$, for any given $\eta,\varepsilon\in(0,1)$, $q\geq2$, and $\pi_{0}^{X}$ with $\EuScript{R}_{\infty}(\pi_{0}^{X}\mathbin{\|}\pi^{X})=\log M$, we can use the $\mathsf{Proximal\ Sampler}$$\mathsf{PS}

Figures (1)

  • Figure 1.1: Reduction to an exponential distribution and sampling via the $\mathsf{Proximal\ Sampler}$$\mathsf{PS}_{\textup{exp}}$.

Theorems & Definitions (62)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 1.9
  • Theorem 2.2: Restatement of Theorem \ref{['thm:lc-warmstart-intro']}
  • ...and 52 more