A Direct Sum Result for the Information Complexity of Learning

How many bits of information are required to PAC learn a class of hypotheses of VC dimension d? The mathematical setting we follow is that of Bassily et al. (2018), where the value of interest is the mutual information I(S;A(S)) between the input sample S and the hypothesis outputted by the learning algorithm A. We introduce a class of functions of VC dimension d over the domain X with information complexity at least Omegaleft(dlog log frac|X|dright) bits for any consistent and proper algorithm (deterministic or random). Bassily et al. proved a similar (but quantitatively weaker) result for the case d=1. The above result is in fact a special case of a more general phenomenon we explore. We define the notion of information complexity of a given class of functions H. Intuitively, it is the minimum amount of information that an algorithm for H must retain about its input to ensure consistency and properness. We prove a direct sum result for information complexity in this context; roughly speaking, the information complexity sums when combining several classes.