Accelerated High-Accuracy Sampling from a Warm Start via the Proximal Bouncy Particle Sampler

Fan Chen; Sinho Chewi; Jianfeng Lu; Matthew S Zhang

math.STSep 7, 2026

Accelerated High-Accuracy Sampling from a Warm Start via the Proximal Bouncy Particle Sampler

Authors: Fan Chen, Sinho Chewi, Jianfeng Lu, Matthew S Zhang

Organizations: Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology. · Department of Statistics and Data Science, Yale University. · Department of Mathematics, Duke University. · Department of Mathematics, Massachusetts Institute of Technology.

Abstract

We study the problem of sampling from $μ(\mathrm{d}x)\propto e^{-V(x)}\,\mathrm{d}x$ on $\mathbb{R}^d$ , where $V$ is $α$ -strongly convex and $β$ -smooth, and write $κ:=β/α$ . We design and analyze the Proximal Bouncy Particle Sampler (Proximal BPS), a new sampler that combines ideas from the proximal sampler and the bouncy particle sampler. From a warm start initialization with $O(1)$ Rényi divergence w.r.t. $μ$ , Proximal BPS returns a sample whose law is $\varepsilon$ -close to $μ$ in total variation distance using $\widetilde O(\sqrtκ\,d^{1/4} \,\mathrm{polylog}(1/\varepsilon))$ gradient queries in expectation.

Figures & tables

Symbol	Meaning	Symbol	Meaning
Target and problem parameters
$d$	Ambient dimension.	$V$	Potential $V:\mathbb{R}^{d}\to\mathbb{R}$ .
$\mu$	Target law $\mu(\mathrm{d}x)\propto e^{-V(x)}\,\mathrm{d}x$ .	$\alpha,\beta$	Strong-convexity and smoothness constants of $V$ .
$\kappa$	Condition number $\kappa=\beta/\alpha$ .	$\varepsilon,\Delta$	Target total variation accuracy and order-2 Rényi warm start budget.
Proximal augmentation
$\eta$	Proximal scale.	$\pi_{\eta}$	Augmented target on $(X,Y)$ .

Table 1: Frequently used notation, grouped by theme.

Figure 1: Geometric overview of Proximal BPS. (a) The initial law is lifted to the augmented distribution. At each iteration, (b) the auxiliary point is reflected through the current position, and (c) a conditional reference point is drawn. With probability

\rho

, that point becomes the next position; with probability

1-\rho

, it determines the equilibrium position

c=Y^{+}-\eta\nabla V(\widetilde{X})

of the deterministic harmonic flow for the time-

\pi

BPS half-turn process in (d), which starts from the current position.

\dot{x}=\sqrt{\eta}\,p\,,\hskip 20.00003pt\dot{p}=-\frac{x-y}{\sqrt{\eta}}-\sqrt{\eta}\,\nabla V(\widetilde{X})\,,

(3.8)

Algorithm 1 Simulation of the ideal conditional half-turn process

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Sep 7, 2026J/K move · Enter open · S save

Smoothed Picard Hamiltonian Monte Carlo

Fan Chen, Sinho Chewi, Jianfeng Lu, Matthew S Zhang

Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology. · Department of Statistics and Data Science, Yale University. · Department of Mathematics, Duke University.+1

Markov Chain Monte Carlo Log-Concave Sampling

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We develop a new low-accuracy sampler, called smoothed Picard Hamiltonian Monte Carlo, which combines Gaussian smoothing, Picard iteration, and higher-order discretization. For a log-concave target

π\propto \exp(-V)

in dimension

d

satisfying

0 \prec αI \preceq \nabla^2 V \preceq βI

, with condition number

κ:= β/α

, smoothed Picard HMC returns a sample with

\sqrt α\,W_2(\cdot,π) \le \varepsilon

using

\widetilde O(κ^2 + κ^{7/6} d^{1/6}/\varepsilon^{1/3})

gradient queries. We also prove stronger

W_q

bounds, and then develop an algorithmic framework, the recursive warm start generator, to upgrade these

W_q

bounds to stronger divergence guarantees. This produces a warm start for the proximal bouncy particle sampler, introduced in a companion work, leading to a high-accuracy log-concave sampler with complexity

\widetilde O((κ^{7/6} d^{1/6} + κ^{1/2} d^{1/4})\mathrm{polylog}(1/\varepsilon))

Abstract

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