2018/02/28 by Kunsch, Robert J.
#41A25 #41A46 #41A63 #65C05 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.1803.00099
We study the L1-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(ε,d) of function evaluations needed in order to approximate an unknown monotone function within a given error threshold ε grows at least exponentially in d. This is not the case in the randomized setting (Monte Carlo setting) where the complexity n(ε,d) grows exponentially in √(d) (modulo logarithmic terms) only. An algorithm exhibiting this complexity is presented. Still, the problem remains difficult as best known methods are deterministic if ε is comparably small, namely ε \preceq 1/√(d). This inherent difficulty is confirmed by lower complexity bounds which reveal a joint (ε,d)-dependency and from which we deduce that the problem is not weakly tractable.