The challenge of simulating random variables is a central problem in Statistics and Machine Learning. Given a tractable proposal distribution $P$, from which we can draw exact samples, and a target distribution $Q$ which is absolutely continuous with respect to $P$, that is $P \gg Q$, the A* sampling algorithm can be used to draw exact samples from $Q$, so long as we can evaluate the Radon-Nikodym derivative of $Q$ with respect to $P$. Maddison et al. originally showed that for a target distribution $Q$ and proposal distribution $P$, the runtime of A* sampling is upper bounded by $\mathcal{O}(\exp(D_{\infty}[Q||P]))$ where $D_{\infty}[Q||P]$ is the Renyi divergence from $Q$ to $P$. This runtime can be prohibitively long for many cases of practical interest. Here, we show that with additional restrictive assumptions on $Q$ and $P$, we can achieve much faster runtimes. Specifically, we show that if $Q$ and $P$ are distributions on $\mathbb{R}$ and their Radon-Nikodym derivative is unimodal, the runtime of A* sampling is $\mathcal{O}(D_{\infty}[Q||P])$, which is exponentially faster than A* sampling without further assumptions.
翻译:模拟随机变量是统计和机器学习的一个中心问题。 模拟随机变量的挑战是统计和机器学习中的一个中心问题。 最初,鉴于一个可移植的投标书分发量(P)美元(我们可以从中提取精确的样本)和一个目标分配额(Q)美元(绝对连续的美元)(P美元,即P=gg Q美元),A* 抽样算法可以用来从美元中提取精确的样本,只要我们可以评估Radon-Nikodym衍生物(Q)美元相对于美元。 Maddison等人(Addison et al.) 显示,对于一个目标分发量(Q) 美元和建议书分发量(P$),A* 抽样的运行时间比美元高得多。 具体地说,如果A* (DQ) 美元和RA* 美元(MDA) 的销售量(美元) 和RA* 美元(美元),我们显示,如果RQ=美元(美元) 美元和Rama* 美元(美元) 。