New Lower Bounds for Statistical Query Learning

Abstract

We prove two lower bounds on the Statistical Query (SQ) learning model. The first lower bound is on weak-learning. We prove that for a concept class of SQ-dimension d , a running time of Ω ( d /log d ) is needed. The SQ-dimension of a concept class is defined to be the maximum number of concepts that are “uniformly correlated”, in that each pair of them have nearly the same correlation. This lower bound matches the upper bound in [ BFJ+94 ], up to a logarithmic factor. We prove this lower bound against an “honest SQ-oracle”, which gives a stronger result than the ones against the more frequently used “adversarial SQ-oracles”. The second lower bound is more general. It gives a continuous trade-off between the “advantage” of an algorithm in learning the target function and the number of queries it needs to make, where the advantage of an algorithm is the probability it succeeds in predicting a label minus the probability it doesn’t. Both lower bounds extend and/or strengthen previous results, and solved an open problem left in [ Y01 ].