A uniform lower error bound for half-space learning

Abstract. We give a lower bound for the error of any unitarily invari-ant algorithm learning half-spaces against the uniform or related distri-butions on the unit sphere. The bound is uniform in the choice of the target half-space and has an exponentially decaying deviation probability in the sample. The technique of proof is related to a proof of the John-son Lindenstrauss Lemma. We argue that, unlike previous lower bounds, our result is well suited to evaluate the bene\u85ts of multi-task or transfer learning, or other cases where an expense in the acquisition of domain knowledge has to be justi\u85ed.