Acknowledgement7.1 Introduction7.2 Formal setting7.2.1 Problem formulation7.2.2 An illustrative example7.2.3 Approximation with inputs lying in a ball7.3 Bounding the norm of the input function based on a pilot sample7.3.1 The cone and the optimal algorithm7.3.2 Bounds on the information cost for ρ=τ, ρ′=∞7.3.3 Tractability7.4 Tracking the decay rate of the series coefficients of the input function7.4.1 The adaptive algorithm and its information cost7.4.2 Bounds on the information cost for ρ=τ, ρ′=∞7.4.3 Essential optimality of the algorithm7.4.4 Tractability7.5 Inferring coordinate and smoothness importance7.5.1 Product-, order- and smoothness-dependent (POSD) weights7.5.2 Inferring POSD weights from an initial sample7.5.3 Numerical Examples