Simple online learning with consistent oracle
Alexander Kozachinskiy, Tomasz Steifer
TL;DR
The paper investigates online learning when the learner has access to a consistent oracle for a hypothesis class with Littlestone dimension $d$, addressing a computational bottleneck of traditional SOA-based methods. It introduces a simpler algorithm that maintains an evolving set of consistent functions and uses majority voting with pruning to achieve an upper bound of $O(256^d)$ mistakes, along with a modular recursive construction to guarantee progress in Littlestone dimension. A complementary lower bound shows that no algorithm in this model can beat $3^d$ mistakes, establishing a tight exponential gap and clarifying the landscape of possible exponents. The work also discusses implications for finite truth-table hypothesis classes and for recursively enumerable (RER) classes, highlighting the practicality of consistent-oracle-based online learning when ERM/SOA are infeasible.
Abstract
We consider online learning in the model where a learning algorithm can access the class only via the \emph{consistent oracle} -- an oracle, that, at any moment, can give a function from the class that agrees with all examples seen so far. This model was recently considered by Assos et al.~(COLT'23). It is motivated by the fact that standard methods of online learning rely on computing the Littlestone dimension of subclasses, a computationally intractable problem. Assos et al.~gave an online learning algorithm in this model that makes at most $C^d$ mistakes on classes of Littlestone dimension $d$, for some absolute unspecified constant $C > 0$. We give a novel algorithm that makes at most $O(256^d)$ mistakes. Our proof is significantly simpler and uses only very basic properties of the Littlestone dimension. We also show that there exists no algorithm in this model that makes less than $3^d$ mistakes.