The Curse of Dimensionality for Monotone and Convex Functions of Many Variables

TL;DR

This paper demonstrates that integrating and approximating monotone and convex functions in high dimensions require exponentially many samples, confirming the presence of the curse of dimensionality in these problems.

Contribution

It proves that the integration and approximation of monotone and convex functions in high dimensions suffer from the curse of dimensionality, requiring exponentially many function evaluations.

Findings

Exponential growth in sample complexity with dimension

Curse of dimensionality applies to monotone and convex functions

High-dimensional problems are inherently hard to approximate

Abstract

We study the integration and approximation problems for monotone and convex bounded functions that depend on $d$ variables, where $d$ can be arbitrarily large. We consider the worst case error for algorithms that use finitely many function values. We prove that these problems suffer from the curse of dimensionality. That is, one needs exponentially many (in $d$) function values to achieve an error $\epsilon$.