Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The adaptive complexity of parallelized log-concave sampling

Huanjian Zhou, Baoxiang Wang, Masashi Sugiyama

TL;DR

This work establishes the first adaptive lower bounds on the parallel runtime of sampling from high-dimensional log-concave targets. By constructing hardness potentials via a random partition chain and applying smoothing, the authors show that almost-linear adaptive schemes cannot achieve exponentially small total-variation error for unconstrained or box-constrained log-concave distributions in high dimensions. The results emphasize the essential $\ obreak\log(1/\\varepsilon)$ dependence in accuracy and reveal fundamental limits for parallel sampling, with implications for privacy-preserving and rare-event contexts. The methodology blends concentration of conditional Bernoulli variables with chain-structured hardness arguments, and suggests directions for exploring optimal algorithms and diffusion-model sampling regimes.

Abstract

In large-data applications, such as the inference process of diffusion models, it is desirable to design sampling algorithms with a high degree of parallelization. In this work, we study the adaptive complexity of sampling, which is the minimum number of sequential rounds required to achieve sampling given polynomially many queries executed in parallel at each round. For unconstrained sampling, we examine distributions that are log-smooth or log-Lipschitz and log strongly or non-strongly concave. We show that an almost linear iteration algorithm cannot return a sample with a specific exponentially small error under total variation distance. For box-constrained sampling, we show that an almost linear iteration algorithm cannot return a sample with sup-polynomially small error under total variation distance for log-concave distributions. Our proof relies upon novel analysis with the characterization of the output for the hardness potentials based on the chain-like structure with random partition and classical smoothing techniques.

The adaptive complexity of parallelized log-concave sampling

TL;DR

This work establishes the first adaptive lower bounds on the parallel runtime of sampling from high-dimensional log-concave targets. By constructing hardness potentials via a random partition chain and applying smoothing, the authors show that almost-linear adaptive schemes cannot achieve exponentially small total-variation error for unconstrained or box-constrained log-concave distributions in high dimensions. The results emphasize the essential dependence in accuracy and reveal fundamental limits for parallel sampling, with implications for privacy-preserving and rare-event contexts. The methodology blends concentration of conditional Bernoulli variables with chain-structured hardness arguments, and suggests directions for exploring optimal algorithms and diffusion-model sampling regimes.

Abstract

In large-data applications, such as the inference process of diffusion models, it is desirable to design sampling algorithms with a high degree of parallelization. In this work, we study the adaptive complexity of sampling, which is the minimum number of sequential rounds required to achieve sampling given polynomially many queries executed in parallel at each round. For unconstrained sampling, we examine distributions that are log-smooth or log-Lipschitz and log strongly or non-strongly concave. We show that an almost linear iteration algorithm cannot return a sample with a specific exponentially small error under total variation distance. For box-constrained sampling, we show that an almost linear iteration algorithm cannot return a sample with sup-polynomially small error under total variation distance for log-concave distributions. Our proof relies upon novel analysis with the characterization of the output for the hardness potentials based on the chain-like structure with random partition and classical smoothing techniques.
Paper Structure (55 sections, 21 theorems, 103 equations, 1 figure, 1 table)

This paper contains 55 sections, 21 theorems, 103 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Lower bound for unconstrained samplers.
  3. Lower bound for box-constrained samplers.
  4. Comparision to query complexity.
  5. Exponential accuracy requirements with privacy as an example.
  6. Preliminaries
  7. Sampling task
  8. Distribution and function class
  9. Oracle
  10. Adaptive algorithm class
  11. Measure of the output
  12. Initialization
  13. Notion of complexity
  14. Technical overview
  15. Existing techniques
  16. ...and 40 more sections

Key Result

Theorem 4.1

Let $d$ be sufficiently large, and $c\in (0,1)$ be a fixed uniform constant. Consider the function class $\mathcal{F}$, consisting of $1$-strongly convex and $2$-smooth functions. For any $\alpha = \omega(1)$ and $\varepsilon= {\mathcal{O}}(c^{d})$, there exists $\gamma = {\mathcal{O}}(d^{-\alpha})$

Figures (1)

  • Figure 1: Comparison with existing parallel methods for strongly log-concave and log-smooth distributions

Theorems & Definitions (28)

  • Theorem 4.1: Lower bound for strongly log-concave and log-smooth samplers
  • Theorem 4.2: Smoothing operators
  • Lemma 4.3
  • proof
  • Theorem 4.4: Lower bound for weakly log-concave samplers
  • Theorem 4.5: Lower bound for composite potentials
  • Theorem 5.1: Lower bound for log-smooth and log-concave samplers
  • Lemma A.1: Concentration of Lipshictz function of conditioned Bernoullis
  • Lemma A.2: Irwin-Hall tail bound (Corollary 5 in zhang2020non)
  • Theorem A.3: Concentration of linear functions over the Boolean slice (Theorem 4.2.5 in polaczyk2023concentration)
  • ...and 18 more