A Rate Separation for Agnostic Direct Sums

Hanneke, Moran, and Waknine \cite{HannekeMoranWaknine2024} asked how the agnostic PAC learning curve of the direct sum C^r depends on the single-instance learning curve \epsagn(n\mid C) and on r. We show that the single-instance learning rate does not determine the direct-sum rate. Let \F be the class of the two constant binary functions and let \G consist of the zero function and the identity function. Both classes have agnostic learning curve of order n^{-1/2}.