  We study adaptive approximation algorithms for general multivariate linear
problems where the sets of input functions are non-convex cones. While it is
known that adaptive algorithms perform essentially no better than non-adaptive
algorithms for convex input sets, the situation may be different for non-convex
sets. A typical example considered here is function approximation based on
series expansions. Given an error tolerance, we use series coefficients of the
input to construct an approximate solution such that the error does not exceed
this tolerance. We study the situation where we can bound the norm of the input
based on a pilot sample, and the situation where we keep track of the decay
rate of the series coefficients of the input. Moreover, we consider situations
where it makes sense to infer coordinate and smoothness importance. Besides
performing an error analysis, we also study the information cost of our
algorithms and the computational complexity of our problems, and we identify
conditions under which we can avoid a curse of dimensionality.
