Optimal global rates of convergence for noiseless regression estimation problems with adaptively chosen design

Given the values of a measurable function m:Rd→R at n arbitrarily chosen points in Rd the problem of estimating m on whole Rd is considered. Here the estimate has to be defined such that the L1 error of the estimate (with integration with respect to a fixed but unknown probability measure) is small. Under the assumption that m is (p,C)-smooth (i.e., roughly speaking, m is p-times continuously differentiable) it is shown that the optimal minimax rate of convergence of the L1 error is n−p/d, where the upper bound is valid even if the support of the design measure is unbounded but the design measure satisfies some moment condition. Furthermore it is shown that this rate of convergence cannot be improved even if the function is not allowed to change with the size of the data.