Agnostic Sample Compression Schemes for Regression

We obtain the first positive results for bounded sample compression in the agnostic regression setting with the ℓ_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 ℓ₁ and ℓ_∞ losses, we can even exhibit an efficient exact sample compression scheme of size linear in the dimension. We further show that for every other ℓ_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 (2016) for the ℓ₂ loss. We close by posing general open questions: for agnostic regression with ℓ₁ loss, does every function class admit an exact compression scheme of polynomial size in the pseudo-dimension? For the ℓ₂ 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 (Warmuth, 2003).