论文标题
带有受限高斯甲骨文的复合logconcave采样
Composite Logconcave Sampling with a Restricted Gaussian Oracle
论文作者
论文摘要
我们考虑从$ \ mathbb {r}^d上的复合密度进行抽样，用于良好的条件$ f $ and convex（但可能是非平滑型）$ g $，通过convex and convex设置的一般性限制的限制，用于convex and convex或abs abs abs a vance and actractian的一般限制，用于良好的条件$ f $ and convex（但可能是非平滑的）。对于带有条件号$κ$的$ f $，我们的算法以$ o \ left（κ^2 d \ log^2 \ tfrac {κD}ε\ right）$迭代运行，每种都查询$ f $的梯度和受限的高斯甲骨文，以达到总变化距离$ε$。限制的高斯甲骨文（Gaussian Oracle）从先前研究过的分布中绘制了一个分布的样品，其负模样总和是二次和$ g $，并且是复合优化中使用的近端甲骨文的自然扩展。我们的算法在概念上很简单，并且比现有的复合采样方法获得了更强的可证明保证和更大的通用性。我们进行的实验显示了我们的算法在撞击算法上大大改善了对（非二元）高斯对正矫正的限制的限制。
We consider sampling from composite densities on $\mathbb{R}^d$ of the form $dπ(x) \propto \exp(-f(x) - g(x))dx$ for well-conditioned $f$ and convex (but possibly non-smooth) $g$, a family generalizing restrictions to a convex set, through the abstraction of a restricted Gaussian oracle. For $f$ with condition number $κ$, our algorithm runs in $O \left(κ^2 d \log^2\tfrac{κd}ε\right)$ iterations, each querying a gradient of $f$ and a restricted Gaussian oracle, to achieve total variation distance $ε$. The restricted Gaussian oracle, which draws samples from a distribution whose negative log-likelihood sums a quadratic and $g$, has been previously studied and is a natural extension of the proximal oracle used in composite optimization. Our algorithm is conceptually simple and obtains stronger provable guarantees and greater generality than existing methods for composite sampling. We conduct experiments showing our algorithm vastly improves upon the hit-and-run algorithm for sampling the restriction of a (non-diagonal) Gaussian to the positive orthant.
