Strong Minimax Lower Bounds for Learning

Minimax lower bounds for concept learning state, for example, that for each sample size n and learning rule gn , there exists a distribution of the observation X and a concept C to be learnt such that the expected error of gn is at least a constant times V=n, where V is the vc dimension of the concept class. However, these bounds do not tell anything about the rate of decrease of the error for a fixed distribution-concept pair. In this paper we investigate minimax lower bounds in such a "stronger" sense. We show that for several natural k-parameter concept classes, including the class of linear halfspaces, the class of balls, the class of polyhedra with a certain number of faces, and a class of neural networks, for any sequence of learning rules fgn g, there exists a fixed distribution of X and axed concept C such that the expected error is larger than a constant times k=n for infinitely many n. We also obtain such strong minimax lower bounds for the tail distribution of the probability of error, which extend the corresponding minimax lower bounds.