The Difficulty of Monte Carlo Approximation of Multivariate Monotone Functions

TL;DR

This paper investigates the complexity of approximating multivariate monotone functions using Monte Carlo methods, revealing inherent difficulties and proving the problem is not weakly tractable due to its exponential complexity in the square root of the dimension.

Contribution

It presents a Monte Carlo algorithm with complexity growing exponentially in 0dd, and establishes lower bounds showing the problem's inherent difficulty and non-weak tractability.

Findings

Monte Carlo complexity grows exponentially in 0dd

Deterministic methods are less effective for small 

The problem is proven to be not weakly tractable

Abstract

We study the $L_1$-approximation of $d$-variate monotone functions based on information from $n$ function evaluations. It is known that this problem suffers from the curse of dimensionality in the deterministic setting, that is, the number $n(\varepsilon,d)$ of function evaluations needed in order to approximate an unknown monotone function within a given error threshold $\varepsilon$ grows at least exponentially in $d$. This is not the case in the randomized setting (Monte Carlo setting) where the complexity $n(\varepsilon,d)$ grows exponentially in $\sqrt{d}$ (modulo logarithmic terms) only. An algorithm exhibiting this complexity is presented. Still, the problem remains difficult as best known methods are deterministic if $\varepsilon$ is comparably small, namely $\varepsilon \preceq 1/\sqrt{d}$. This inherent difficulty is confirmed by lower complexity bounds which reveal a joint $(\varepsilon,d)$-dependency and from which we deduce that the problem is not weakly tractable.