We consider the problem of distribution-free learning for Boolean function classes in the PAC and agnostic models. Generalizing a beautiful work of Malach and Shalev-Shwartz (2022) that gave tight correlational SQ (CSQ) lower bounds for learning DNF formulas, we give new proofs that lower bounds on the threshold or approximate degree of any function class directly imply CSQ lower bounds for PAC or agnostic learning respectively. While such bounds implicitly follow by combining prior results by Feldman (2008, 2012) and Sherstov (2008, 2011), to our knowledge the precise statements we give had not appeared in this form before. Moreover, our proofs are simple and largely self-contained. These lower bounds match corresponding positive results using upper bounds on the threshold or approximate degree in the SQ model for PAC or agnostic learning, and in this sense these results show that the polynomial method is a universal, best-possible approach for distribution-free CSQ learning.

翻译：我们研究PAC和不可知模型下布尔函数类的分布无关学习问题。通过推广Malach与Shalev-Shwartz（2022）关于DNF公式学习的紧致关联SQ下界优美工作，我们给出了新证明：任何函数类的阈值度或近似度下界直接分别蕴含PAC或不可知学习的关联SQ下界。虽然此类下界可隐含地通过综合Feldman（2008, 2012）与Sherstov（2008, 2011）的先前成果获得，据我们所知，本文给出的精确表述此前尚未以该形式出现。此外，我们的证明简洁且基本自包含。这些下界与SQ模型中基于阈值度或近似度上界得到的PAC或不可知学习正面对应结果相匹配，在该意义上表明：多项式方法是分布无关关联SQ学习中最优的普适性途径。