We study the complexity of sampling, rounding, and integrating arbitrary logconcave functions. Our new approach provides the first complexity improvements in nearly two decades for general logconcave functions for all three problems, and matches the best-known complexities for the special case of uniform distributions on convex bodies. For the sampling problem, our output guarantees are significantly stronger than previously known, and lead to a streamlined analysis of statistical estimation based on dependent random samples.

翻译：本文研究了任意对数凹函数的采样、舍入及积分计算的复杂度问题。我们提出的新方法为这三类问题提供了近二十年来针对一般对数凹函数的首次复杂度改进，并达到了凸体上均匀分布这一特殊情形的最佳已知复杂度。在采样问题方面，我们的输出保证显著优于已有结果，并为基于相依随机样本的统计估计提供了简化的分析框架。