Agnostic Sample Compression Schemes for Regression

Abstract

We obtain the first positive results for bounded sample compression in the agnostic regression setting with the _p loss, where p [1,]. We construct a generic approximate sample compression scheme for real-valued function classes exhibiting exponential size in the fat-shattering dimension but independent of the sample size. Notably, for linear regression, an approximate compression of size linear in the dimension is constructed. Moreover, for _1 and _ losses, we can even exhibit an efficient exact sample compression scheme of size linear in the dimension. We further show that for every other _p loss, p (1,), there does not exist an exact agnostic compression scheme of bounded size. This refines and generalizes a negative result of David, Moran, and Yehudayoff for the _2 loss. We close by posing general open questions: for agnostic regression with _1 loss, does every function class admits an exact compression scheme of size equal to its pseudo-dimension? For the _2 loss, does every function class admit an approximate compression scheme of polynomial size in the fat-shattering dimension? These questions generalize Warmuth's classic sample compression conjecture for realizable-case classification.

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Reproductions