Strong minimax lower bounds for learning

Minimax lower bounds for concept learning state, for example, that for each sample size n and learning rule g(n), there exists a distribution of the observation X and a concept C to be learnt such that the expected error of g(n) 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 {g(n)}, there exists a fixed distribution of X and a fixed 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.