We show strong (and surprisingly simple) lower bounds for weakly learning intersections of halfspaces in the improper setting. Strikingly little is known about this problem. For instance, it is not even known if there is a polynomial-time algorithm for learning the intersection of only two halfspaces. On the other hand, lower bounds based on well-established assumptions (such as approximating worst-case lattice problems or variants of Feige's 3SAT hypothesis) are only known (or are implied by existing results) for the intersection of super-logarithmically many halfspaces [KS09,KS06,DSS16]. With intersections of fewer halfspaces being only ruled out under less standard assumptions [DV21] (such as the existence of local pseudo-random generators with large stretch). We significantly narrow this gap by showing that even learning $ω(\log \log N)$ halfspaces in dimension $N$ takes super-polynomial time under standard assumptions on worst-case lattice problems (namely that SVP and SIVP are hard to approximate within polynomial factors). Further, we give unconditional hardness results in the statistical query framework. Specifically, we show that for any $k$ (even constant), learning $k$ halfspaces in dimension $N$ requires accuracy $N^{-Ω(k)}$, or exponentially many queries -- in particular ruling out SQ algorithms with polynomial accuracy for $ω(1)$ halfspaces. To the best of our knowledge this is the first unconditional hardness result for learning a super-constant number of halfspaces. Our lower bounds are obtained in a unified way via a novel connection we make between intersections of halfspaces and the so-called parallel pancakes distribution [DKS17,BLPR19,BRST21] that has been at the heart of many lower bound constructions in (robust) high-dimensional statistics in the past few years.

翻译：我们展示了在非适当设置下弱学习半空间交集问题的强大（且令人惊讶地简单）下界。关于该问题，已知结果极少。例如，甚至未知是否存在多项式时间算法用于仅学习两个半空间的交集。另一方面，基于公认假设（如近似最坏情况格问题或Feige的3SAT假设变体）的下界仅已知（或由现有结果隐含）于超对数个半空间的交集[KS09, KS06, DSS16]。而对较少半空间交集的排除仅基于不太标准的假设[DV21]（如存在具有大拉伸的局部伪随机生成器）。我们显著缩小了这一差距：证明在标准最坏情况格问题假设（即SVP和SIVP在多项式因子内难以近似）下，学习维度$N$中的$ω(\log \log N)$个半空间需要超多项式时间。此外，我们在统计查询框架中给出无条件难度结果。具体而言，我们证明对任意$k$（甚至常数），学习维度$N$中的$k$个半空间需要精度$N^{-Ω(k)}$，或指数级查询次数——特别地，排除了对$ω(1)$个半空间具有多项式精度的SQ算法。据我们所知，这是学习超常数个半空间的首个无条件难度结果。我们的下界通过一种新颖联系统一获得：该联系将半空间交集与所谓的平行薄饼分布[DKS17, BLPR19, BRST21]关联起来，后者近年来一直是（鲁棒）高维统计中许多下界构造的核心。