One-shot channel simulation has recently emerged as a promising alternative to quantization and entropy coding in machine-learning-based lossy data compression schemes. However, while there are several potential applications of channel simulation - lossy compression with realism constraints or differential privacy, to name a few - little is known about its fundamental limitations. In this paper, we restrict our attention to a subclass of channel simulation protocols called causal rejection samplers (CRS), establish new, tighter lower bounds on their expected runtime and codelength, and demonstrate the bounds' achievability. Concretely, for an arbitrary CRS, let $Q$ and $P$ denote a target and proposal distribution supplied as input, and let $K$ be the number of samples examined by the algorithm. We show that the expected runtime $\mathbb{E}[K]$ of any CRS scales at least as $\exp_2(D_\infty[Q || P])$, where $D_\infty[Q || P]$ is the R\'enyi $\infty$-divergence. Regarding the codelength, we show that $D_{KL}[Q || P] \leq D_{CS}[Q || P] \leq \mathbb{H}[K]$, where $D_{CS}[Q || P]$ is a new quantity we call the channel simulation divergence. Furthermore, we prove that our new lower bound, unlike the $D_{KL}[Q || P]$ lower bound, is achievable tightly, i.e. there is a CRS such that $\mathbb{H}[K] \leq D_{CS}[Q || P] + \log_2 (e + 1)$. Finally, we conduct numerical studies of the asymptotic scaling of the codelength of Gaussian and Laplace channel simulation algorithms.

翻译：单次信道模拟近期在基于机器学习的无损数据压缩方案中，作为量化和熵编码的替代方法展现出潜力。然而，尽管信道模拟具有多种潜在应用（例如带真实性约束的无损压缩或差分隐私），其基本局限性却鲜为人知。本文聚焦于一类名为因果拒绝采样器（CRS）的信道模拟协议，建立了其预期运行时间与码长更紧的新下界，并证明了该下界的可达性。具体而言，对于任意CRS，设输入的目标分布与提议分布分别为$Q$和$P$，算法检查的样本数为$K$。我们证明任何CRS的预期运行时间$\mathbb{E}[K]$至少以$\exp_2(D_\infty[Q || P])$为尺度增长，其中$D_\infty[Q || P]$为Rényi $\infty$-散度。关于码长，我们证明$D_{KL}[Q || P] \leq D_{CS}[Q || P] \leq \mathbb{H}[K]$，其中$D_{CS}[Q || P]$是我们定义的新量——信道模拟散度。进一步，我们证明该新下界（不同于$D_{KL}[Q || P]$下界）是紧可达的，即存在一个CRS使得$\mathbb{H}[K] \leq D_{CS}[Q || P] + \log_2 (e + 1)$。最后，我们对高斯与拉普拉斯信道模拟算法的码长渐近标度进行了数值研究。