Recent work due to Goel et al. gave the first efficient algorithms for learning with distribution shift in the challenging PQ framework. In this setting, a learner receives labeled training examples, unlabeled test examples, and must make correct predictions on the test set but is allowed to abstain from predicting on out-of-distribution points. Their results rely on ${\cal L}_2$ sandwiching approximations, a strong requirement that leads to poor bounds for several basic function classes such as DNF formulas. Here, we show that the weaker notion of ${\cal L}_1$ sandwiching suffices for efficient PQ learning. As a consequence, we obtain the first quasipolynomial-time PQ learning algorithm for DNFs under the uniform distribution and essentially match the guarantees known for ordinary PAC learning. More broadly, our bounds provide exponential improvements for several classes including constant depth circuits and constant degree polynomial threshold functions. Our main technical ingredient is Iterative Chow Filtering, a new procedure that uses low-degree Chow parameters to identify and remove test points incompatible with the training distribution.

翻译：Goel等人的近期研究首次在具有挑战性的PQ学习框架中提出了处理分布转移的高效算法。在该设定下，学习器接收带标签的训练样本和无标签的测试样本，必须对测试集做出正确预测，但允许对分布外数据点放弃预测。他们的结果依赖于${\cal L}_2$夹逼近似，这一强约束导致对诸如DNF公式等若干基础函数类产生较差的边界。本文证明，采用较弱的${\cal L}_1$夹逼近似足以实现高效PQ学习。基于此，我们首次提出均匀分布下针对DNF函数类的拟多项式时间PQ学习算法，且其保障基本匹配普通PAC学习的已知结果。更广泛而言，我们的边界为包括常数深度电路和常数度多项式阈值函数在内的多个函数类提供了指数级改进。本文的核心技术贡献是迭代乔滤波——一种利用低阶乔参数识别并移除与训练分布不兼容的测试点的新方法。