A Direct Sum Result for the Information Complexity of Learning

TL;DR

This paper establishes a lower bound on the mutual information needed for PAC learning classes with VC dimension d, showing that the information complexity scales with d log log(|X|/d), and proves a direct sum property for this complexity.

Contribution

It introduces a lower bound on the information complexity for classes with VC dimension d and proves a direct sum theorem for the information complexity of combined classes.

Findings

Lower bound of Ω(d log log(|X|/d)) bits for information complexity

Information complexity sums when combining multiple classes

Generalization of previous results for VC dimension d

Abstract

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 $\mathrm{I}(S;A(S))$ 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 $\mathcal{X}$ with information complexity at least $\Omega\left(d\log \log \frac{|\mathcal{X}|}{d}\right)$ 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 $\mathcal{H}$. Intuitively, it is the minimum amount of information that an algorithm for $\mathcal{H}$ 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.