Transductive Learning is Compact
Keywords: Sample Complexity, Compactness, One-Inclusion Graphs, Metric Space, Transductive Learning, PAC Learning
TL;DR: We demonstrate that the transductive sample complexity of learning a hypothesis class is exactly equal to the sample complexity of learning its most difficult "finite projection," for a wide range of losses.
Abstract: We demonstrate a compactness result holding broadly across supervised learning with a general class of loss functions: Any hypothesis class $\mathcal{H}$ is learnable with transductive sample complexity $m$ precisely when all of its finite projections are learnable with sample complexity $m$. We prove that this exact form of compactness holds for realizable and agnostic learning with respect to all proper metric loss functions (e.g., any norm on $\mathbb{R}^d$) and any continuous loss on a compact space (e.g., cross-entropy, squared loss). For realizable learning with improper metric losses, we show that exact compactness of sample complexity can fail, and provide matching upper and lower bounds of a factor of 2 on the extent to which such sample complexities can differ. We conjecture that larger gaps are possible for the agnostic case. Furthermore, invoking the equivalence between sample complexities in the PAC and transductive models (up to lower order factors, in the realizable case) permits us to directly port our results to the PAC model, revealing an almost-exact form of compactness holding broadly in PAC learning.
Primary Area: Learning theory
Submission Number: 8015
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