Matthew S Zhang | Searcharxiv

arXiv2026

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

We study the problem of sampling from $\mu(\mathrm{d}x)\propto e^{-V(x)}\,\mathrm{d}x$ on $\mathbb{R}^d$, where $V$ is $\alpha$-strongly convex and $\beta$-smooth, and write $\kappa:=\beta/\alpha$. 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) $ R\'enyi divergence w.r.t. $\mu$, Proximal BPS returns a sample whose law is $\varepsilon$-close to $\mu$ in total variation distance using $\widetilde O(\sqrt\kappa\,d^{1/4} \,\mathrm{polylog}(1/\varepsilon))$ gradient queries in expectation.

math.ST↗

arXiv2026

Smoothed Picard Hamiltonian Monte Carlo

We develop a new low-accuracy sampler, called \emph{smoothed Picard Hamiltonian Monte Carlo}, which combines Gaussian smoothing, Picard iteration, and higher-order discretization. For a log-concave target $\pi \propto \exp(-V)$ in dimension $d$ satisfying $0 \prec \alpha I \preceq \nabla^2 V \preceq \beta I$, with condition number $\kappa := \beta/\alpha$, smoothed Picard HMC returns a sample with $\sqrt \alpha\,W_2(\cdot,\pi) \le \varepsilon$ using $\widetilde O(\kappa^2 + \kappa^{7/6} d^{1/6}/\varepsilon^{1/3})$ 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 $\widetilde O((\kappa^{7/6} d^{1/6} + \kappa^{1/2} d^{1/4})\mathrm{polylog}(1/\varepsilon))$.

math.ST↗