Trading information complexity for error II: the case of a large error and external information complexity

TL;DR

This paper investigates the external and internal information costs required to compute Boolean functions with small errors, establishing bounds and tight results, including the first exact example of information complexity with error.

Contribution

It provides tight bounds on information costs for computing Boolean functions with small errors and presents the first exact example of information complexity with error.

Findings

Information cost of order ε² is necessary, order ε is sufficient.

Significant external information cost savings are possible with small errors.

Exact information complexity for XOR with error is determined as 1-2ε.

Abstract

Two problems are studied in this paper. (1) How much external or internal information cost is required to compute a Boolean-valued function with an error at most $1/2-\epsilon$ for a small $\epsilon$? It is shown that information cost of order $\epsilon^2$ is necessary and of order $\epsilon$ is sufficient. (2) How much external information cost can be saved to compute a function with a small error $\epsilon>0$ comparing to the case when no error is allowed? It is shown that information cost of order at least $\epsilon$ and at most $h(\sqrt{\epsilon})$ can be saved. Except the $O(h(\sqrt{\epsilon}))$ upper bound, the other three bounds are tight. For distribution $\mu$ that is equally distributed on $(0,0)$ and $(1,1)$, it is shown that $IC^{ext}_\mu(XOR, \epsilon)=1-2\epsilon$ where XOR is the two-bit xor function. This equality seems to be the first example of exact information complexity when an error is allowed.