提出一种新采样算法，高效生成复合对数凹分布，适用于高维数据。
A proximal gradient algorithm for composite log-concave sampling
- 结合梯度与限制高斯采样器，设计近端梯度采样框架
- 在强凸和光滑条件下，误差达ε仅需约κ√d log⁴(1/ε)步
- 可推广至非对数凹或非光滑情形，适用性广
我们提出一种在ℝᵈ上采样复合对数凹分布π ∝ exp(−f−g)的算法，假设可获得f的梯度信息，并具备g的受限高斯采样器（RGO）访问权限。RGO对应于g的近端算子的采样类比。当f+g为α-强凸且f为β-光滑时，该采样器在总变差距离下达到ε误差，仅需$ ilde{ m O}(κ oot d floor ext{log}^4(1/ ext{ε}))$次迭代，其中κ=β/α，与g=0时的最先进结果一致。进一步将结果扩展至：(1) π虽非对数凹但满足Poincaré或对数Sobolev不等式；(2) f非光滑但Lipschitz连续。
原文摘要 · Abstract (English)
We propose an algorithm to sample from composite log-concave distributions over $\mathbb{R}^d$, i.e., densities of the form $π\propto e^{-f-g}$, assuming access to gradient evaluations of $f$ and a restricted Gaussian oracle (RGO) for $g$. The latter requirement means that we can easily sample from the density $\text{RGO}_{g,h,y}(x) \propto \exp(-g(x) -\frac{1}{2h}||y-x||^2)$, which is the sampling analogue of the proximal operator for $g$. If $f + g$ is $α$-strongly convex and $f$ is $β$-smooth, our sampler achieves $\varepsilon$ error in total variation distance in $\widetilde{\mathcal O}(κ\sqrt d \log^4(1/\varepsilon))$ iterations where $κ:= β/α$, which matches prior state-of-the-art results for the case $g=0$. We further extend our results to cases where (1) $π$ is non-log-concave but satisfies a Poincaré or log-Sobolev inequality, and (2) $f$ is non-smooth but Lipschitz.
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