Agnostic Online Learning and Excellent Sets
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We use algorithmic methods from online learning to revisit a key idea from the interaction of model theory and combinatorics, the existence of large "indivisible" sets, called "ϵ-excellent," in k-edge stable graphs (equivalently, Littlestone classes). These sets arise in the Stable Regularity Lemma, a theorem characterizing the appearance of irregular pairs in Szemerédi's celebrated Regularity Lemma. Translating to the language of probability, we find a quite different existence proof for ϵ-excellent sets in Littlestone classes, using regret bounds in online learning. This proof applies to any ϵ < 1/2, compared to < 1/2^2^k or so in the original proof. We include a second proof using closure properties and the VC theorem, with other advantages but weaker bounds. As a simple corollary, the Littlestone dimension remains finite under some natural modifications to the definition. A theme in these proofs is the interaction of two abstract notions of majority, arising from measure, and from rank or dimension; we prove that these densely often coincide and that this is characteristic of Littlestone (stable) classes. The last section lists several open problems.
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