The Difficulty of Monte Carlo Approximation of Multivariate Monotone Functions

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.