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 $\omega(\log \log N)$ 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 $N^{-\Omega(k)}$, or exponentially many queries -- in particular ruling out SQ algorithms with polynomial accuracy for $\omega(1)$ 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：我们在非适应设置下证明了弱学习半空间交的强下界（且简洁得出奇）。关于该问题已知结果极为有限：例如，甚至不知道是否存在多项式时间算法可学习仅两个半空间的交。另一方面，基于成熟假设（如最坏情形格问题的近似或Feige的3SAT假设变体）的下界仅针对超对数多个半空间的交被已知（或可由现有结果推导）[KS09,KS06,DSS16]，而更少半空间的交仅在非标准假设[DV21]（如存在大拉伸局部伪随机生成器）下被排除。我们显著缩小了这一差距：证明在标准最坏情形格问题假设（即SVP和SIVP难以在多项式因子内近似）下，学习N维空间中ω(log log N)个半空间仍需超多项式时间。此外，我们在统计查询框架中给出无条件难度结果：对任意k（甚至常数），学习N维空间中k个半空间需要精度N^{-Ω(k)}或指数级查询——特别排除了针对ω(1)个半空间的多项式精度SQ算法。据我们所知，这是关于学习超常数个半空间的首次无条件难度结果。我们通过建立半空间交与近年（鲁棒）高维统计中众多下界构造核心的“平行薄饼分布”[DKS17,BLPR19,BRST21]之间的新颖关联，以统一方式获得上述下界。