We give the first tester-learner for halfspaces that succeeds universally over a wide class of structured distributions. Our universal tester-learner runs in fully polynomial time and has the following guarantee: the learner achieves error $O(\mathrm{opt}) + \epsilon$ on any labeled distribution that the tester accepts, and moreover, the tester accepts whenever the marginal is any distribution that satisfies a Poincar\'e inequality. In contrast to prior work on testable learning, our tester is not tailored to any single target distribution but rather succeeds for an entire target class of distributions. The class of Poincar\'e distributions includes all strongly log-concave distributions, and, assuming the Kannan--L\'{o}vasz--Simonovits (KLS) conjecture, includes all log-concave distributions. In the special case where the label noise is known to be Massart, our tester-learner achieves error $\mathrm{opt} + \epsilon$ while accepting all log-concave distributions unconditionally (without assuming KLS). Our tests rely on checking hypercontractivity of the unknown distribution using a sum-of-squares (SOS) program, and crucially make use of the fact that Poincar\'e distributions are certifiably hypercontractive in the SOS framework.

翻译：我们给出了首个在半空间问题上通用的测试-学习器，能够适用于广泛的结构化分布类。该通用测试-学习器在完全多项式时间内运行，并具有如下保证：对于测试器接受的任何有标签分布，学习器可实现误差$O(\mathrm{opt}) + \epsilon$；此外，只要边际分布满足庞加莱不等式，测试器即予以接受。与以往可测试学习的工作不同，我们的测试器并非针对单一目标分布设计，而是面向整个目标分布类成功运行。庞加莱分布类包含所有强对数凹分布，且若假设坎南-洛瓦斯-西蒙诺维茨（KLS）猜想成立，则包含所有对数凹分布。在标签噪声已知为莫萨特噪声的特殊情形下，我们的测试-学习器可实现误差$\mathrm{opt} + \epsilon$，同时无条件地接受所有对数凹分布（无需假设KLS）。我们的测试依赖于通过平方和（SOS）规划检查未知分布的超收缩性，并关键利用了庞加莱分布在SOS框架下可被验证具有超收缩性这一事实。