We obtain the first positive results for bounded sample compression in the agnostic regression setting with the $\ell_p$ loss, where $p\in [1,\infty]$. 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 $\ell_1$ and $\ell_\infty$ losses, we can even exhibit an efficient exact sample compression scheme of size linear in the dimension. We further show that for every other $\ell_p$ loss, $p\in (1,\infty)$, 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 $\ell_2$ loss. We close by posing general open questions: for agnostic regression with $\ell_1$ loss, does every function class admits an exact compression scheme of size equal to its pseudo-dimension? For the $\ell_2$ 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.

翻译：我们首次在$\ell_p$损失（其中$p\in [1,\infty]$）的不可知回归场景中获得了有界样本压缩的正面结果。我们为实值函数类构造了一个通用的近似样本压缩方案，其大小在脂肪破碎维度上呈指数级增长，但与样本量无关。值得注意的是，在线性回归中，我们构造了一个大小与维度线性相关的近似压缩方案。此外，对于$\ell_1$和$\ell_\infty$损失，我们甚至可以展示一个大小与维度线性相关的高效精确样本压缩方案。我们进一步证明，对于所有其他$\ell_p$损失（$p\in (1,\infty)$），不存在有界大小的精确不可知压缩方案。这细化和推广了David、Moran和Yehudayoff关于$\ell_2$损失的负面结论。最后，我们提出了一般性的开放问题：对于具有$\ell_1$损失的不可知回归，是否每个函数类都存在一个大小等于其伪维度的精确压缩方案？对于$\ell_2$损失，是否每个函数类都存在一个大小在脂肪破碎维度上呈多项式增长的近似压缩方案？这些问题推广了Warmuth关于可实现情况分类的经典样本压缩猜想。