Online Learning and Resource-Bounded Dimension: Winnow Yields New Lower Bounds for Hard Sets

TL;DR

This paper links online mistake-bound learning with resource-bounded dimension, using Winnow to derive new lower bounds on the density of hard sets, advancing understanding in computational complexity.

Contribution

It introduces a novel connection between online learning models and resource-bounded measure, providing improved bounds and solving a longstanding open problem.

Findings

Winnow yields new lower bounds for hard sets

Establishes a relationship between online mistake-bound and resource-bounded dimension

Solves one of the twelve open problems in resource-bounded measure

Abstract

We establish a relationship between the online mistake-bound model of learning and resource-bounded dimension. This connection is combined with the Winnow algorithm to obtain new results about the density of hard sets under adaptive reductions. This improves previous work of Fu (1995) and Lutz and Zhao (2000), and solves one of Lutz and Mayordomo's "Twelve Problems in Resource-Bounded Measure" (1999).