Finite Littlestone Dimension Implies Finite Information Complexity

TL;DR

This paper demonstrates that classes of functions with finite Littlestone dimension can be learned with finite information complexity, and provides bounds on this complexity for specific classes, advancing understanding of online learning limits.

Contribution

It establishes that finite Littlestone dimension implies finite information complexity and introduces bounds for specific classes, improving theoretical understanding of online learning.

Findings

Globally stable algorithms have finite, roughly exponential, information complexity in Littlestone dimension.

For indicator functions of affine subspaces, information complexity is logarithmic in the dimension.

Finite Littlestone dimension guarantees finite information complexity in online learning.

Abstract

We prove that every online learnable class of functions of Littlestone dimension $d$ admits a learning algorithm with finite information complexity. Towards this end, we use the notion of a globally stable algorithm. Generally, the information complexity of such a globally stable algorithm is large yet finite, roughly exponential in $d$. We also show there is room for improvement; for a canonical online learnable class, indicator functions of affine subspaces of dimension $d$, the information complexity can be upper bounded logarithmically in $d$.