We revisit the PAC learnability of the class of intersections of polynomially many halfspaces over boolean cubes in the membership query model. The previous works (Klivans-Sherstov 2009, Angluin-Kharitonov 1995) imply the unlearnability of this class based on cryptographic assumptions, which is established in the case that the learner cannot generate queries of positive labels. We investigate this issue, focusing on the case that the learner is given representations of input distributions (so it may generate queries of positive and negative labels).

Our result is that assuming some new cryptographic primitives, the class is still unlearnable in the query model even if the learner has representations of the distributions. The result is established via a new argument. We show that if the class is learnable, we can come up with a differing-inputs obfuscator, which, however, does not exist given the cryptographic primitives. Thus the hardness result follows from the contradiction.

我们重新审视了成员查询模型中布尔立方体上多项式多个半空间的交集类的 PAC 可学习性。之前的工作（Klivans-Sherstov 2009，Angluin-Kharitonov 1995）暗示了该类基于密码学假设的不可学习性，这是在学习器无法生成正标签查询的情况下建立的。我们调查了这个问题，重点是学习器被赋予输入分布的表示的情况（因此它可能会生成正标签和负标签的查询）。

我们的结果是，假设一些新的密码原语，即使学习器具有分布的表示，该类在查询模型中仍然无法学习。结果是通过新的参数确定的。我们表明，如果该类是可学习的，我们可以想出一个不同的输入混淆器，但是，鉴于加密原语，它不存在。因此，硬度结果是从矛盾中得出的。