Near-tight closure bounds for Littlestone and threshold dimensions

07/07/2020

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by   Badih Ghazi, et al.

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We study closure properties for the Littlestone and threshold dimensions of binary hypothesis classes. Given classes ℋ_1, ..., ℋ_k of Boolean functions with bounded Littlestone (respectively, threshold) dimension, we establish an upper bound on the Littlestone (respectively, threshold) dimension of the class defined by applying an arbitrary binary aggregation rule to ℋ_1, ..., ℋ_k. We also show that our upper bounds are nearly tight. Our upper bounds give an exponential (in k) improvement upon analogous bounds shown by Alon et al. (COLT 2020), thus answering a question posed by their work.