arXiv：cs.AI· Bo Yuan, Wei Guo, Yongxin Chen·· 6 小时前
反射近端动力学实现高精度对数凹采样加速
Reflected Proximal Dynamics for Accelerated High-Accuracy Log-Concave Sampling
AI 导读
研究者提出"反射近端动力学"，将近端采样与辅助变量的精确反射、条件动能运动及受控刷新相结合，用于从对数凹分布 π(x)∝e^{-V(x)} 中采样。在热启动条件下，两种实现达到总变分误差 ε 均需 Õ(√κ d^{1/4}) 次期望 oracle 查询，误差依赖为 polylog(1/ε)；结合近端初始化阶段去除热启动假设后，查询量增至 Õ(κ^{6/7} d^{1/4})。
正文
Abstract:We study sampling from $\pi(x)\propto\mathrm{e}^{-V(x)}$ on $\mathbb{R}^d$, where $V$ is $\alpha$-strongly convex and $\beta$-smooth with condition number $\kappa=\beta/\alpha$. For this problem, the proximal sampler preserves the target exactly and recovers overdamped Langevin dynamics in the small-step limit, but its Gaussian resampling step discards directional information. Underdamped Langevin dynamics retains such information through momentum to accelerate exploration, motivating a proximal construction that combines target preservation with directional persistence. We introduce \textit{reflected proximal dynamics}, which combines proximal sampling with an exact reflection of the auxiliary variable, conditional kinetic motion, and controlled refreshment. We develop two realizations: a flux-resampled underdamped half-turn and a persistent-momentum quarter-turn. From a warm start with constant $\chi^2$ divergence, both methods achieve total variation error $\varepsilon$ using $\widetilde O(\sqrt{\kappa}d^{1/4})$ expected oracle queries. We further remove the supplied warm-start assumption by combining either sampler with existing posterior samplers in a proximal initialization stage with an optimized coarse step size. Accounting for both initialization and sampling yields $\widetilde O(\kappa^{6/7}d^{1/4})$ expected oracle queries. In both regimes, the dependence on the target accuracy $\varepsilon$ is $\mathrm{polylog}(1/\varepsilon)$, making reflected proximal dynamics a promising framework for accelerated high-accuracy log-concave sampling.
| Comments: | Under Review |
| Subjects: | Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Statistics Theory (math.ST) |
| Cite as: | arXiv:2610.16248 [cs.LG] |
| (or arXiv:2610.16248v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.16248 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Bo Yuan [view email]
[v1]
Mon, 12 Oct 2026 06:22:06 UTC (145 KB)
来源：arXiv：cs.AI · arxiv.org