论文标题
关于高维的蒙特卡洛整合的注释
A Note on Monte Carlo Integration in High Dimensions
论文作者
论文摘要
蒙特卡洛整合是一种通常用于计算顽固性积分的技术，通常被认为对非常高的积分的性能很差。为了证明情况并非总是如此，我们通过允许积分的维度增加来研究蒙特卡洛的整合。在此过程中，我们通过浓度不等式的某些一般函数类别的近似相对和绝对误差得出了非反应界限。我们提供了具体的示例，其中所需的点数的大小以确保多项式之间的一致估计与指数级别不同，并证明在理论上任意快速或缓慢的速率是可能的。这表明蒙特卡洛整合在高维度上的行为并不统一。通过我们的方法，我们还获得了蒙特卡洛估计值的非反应置信区间，无论采样的点数多少，它们都是有效的。
Monte Carlo integration is a commonly used technique to compute intractable integrals and is typically thought to perform poorly for very high-dimensional integrals. To show that this is not always the case, we examine Monte Carlo integration using techniques from the high-dimensional statistics literature by allowing the dimension of the integral to increase. In doing so, we derive non-asymptotic bounds for the relative and absolute error of the approximation for some general classes of functions through concentration inequalities. We provide concrete examples in which the magnitude of the number of points sampled needed to guarantee a consistent estimate vary between polynomial to exponential, and show that in theory arbitrarily fast or slow rates are possible. This demonstrates that the behaviour of Monte Carlo integration in high dimensions is not uniform. Through our methods we also obtain non-asymptotic confidence intervals for Monte Carlo estimate which are valid regardless of the number of points sampled.
