We develop an extension of recently developed methods for obtaining time-space tradeoff lower bounds for problems of learning from random test samples to handle the situation where the space of tests is signficantly smaller than the space of inputs, a class of learning problems that is not handled by prior work. This extension is based on a measure of how matrices amplify the 2-norms of probability distributions that is more refined than the 2-norms of these matrices.

As applications that follow from our new technique, we show that any algorithm that learns m -variate homogeneous polynomial functions of degree at most d over F 2 from evaluations on randomly chosen inputs either requires space ( mn ) or 2 ( m ) time where n = m ( d ) is the dimension of the space of such functions. These bounds are asymptotically optimal since they match the tradeoffs achieved by natural learning algorithms for the problems.