Rejection sampling is a common tool for low dimensional problems ($d \leq 2$), often touted as an "easy" way to obtain valid samples from a distribution $f(\cdot)$ of interest. In practice it is non-trivial to apply, often requiring considerable mathematical effort to devise a good proposal distribution $g(\cdot)$ and select a supremum $C$. More advanced samplers require additional mathematical derivations, limitations on $f(\cdot)$, or even cross-validation, making them difficult to apply. We devise a new approximate baseline approach to rejection sampling that works with less information, requiring only a differentiable $f(\cdot)$ be specified, making it easier to use. We propose a new approach to rejection sampling by refining a parameterized proposal distribution with a loss derived from the acceptance threshold. In this manner we obtain comparable or better acceptance rates on current benchmarks by up to $7.3\times$, while requiring no extra assumptions or any derivations to use: only a differentiable $f(\cdot)$ is required. While approximate, the results are correct with high probability, and in all tests pass a distributional check. This makes our approach easy to use, reproduce, and efficacious.

翻译：拒绝采样是低维问题（$d \leq 2$）中常用的工具，常被标榜为从目标分布$f(\cdot)$获取有效样本的"简易"途径。然而在实践中其应用颇具挑战性，往往需要大量数学工作来设计良好的提议分布$g(\cdot)$并选取上确界$C$。更高级的采样器则需要额外的数学推导、对$f(\cdot)$施加限制条件，甚至需要交叉验证，导致其难以应用。我们提出了一种新的近似基线式拒绝采样方法，该方法在信息需求较少的条件下即可运行——仅需指定可微分的$f(\cdot)$，从而显著降低使用门槛。我们的核心创新在于：通过接受阈值衍生的损失函数对参数化提议分布进行优化，以此实现拒绝采样。在现有基准测试中，本方法可获得高达7.3倍的接受率提升，且无需引入额外假设或任何数学推导：仅需可微分的$f(\cdot)$作为输入。尽管该方法具有近似性，但结果具有高概率正确性，且在所有测试中均通过了分布一致性检验。这使得我们的方法易于使用、复现，且成效显著。