We pose a fundamental question in computational learning theory: can we efficiently test whether a training set satisfies the assumptions of a given noise model? This question has remained unaddressed despite decades of research on learning in the presence of noise. In this work, we show that this task is tractable and present the first efficient algorithm to test various noise assumptions on the training data. To model this question, we extend the recently proposed testable learning framework of Rubinfeld and Vasilyan (2023) and require a learner to run an associated test that satisfies the following two conditions: (1) whenever the test accepts, the learner outputs a classifier along with a certificate of optimality, and (2) the test must pass for any dataset drawn according to a specified modeling assumption on both the marginal distribution and the noise model. We then consider the problem of learning halfspaces over Gaussian marginals with Massart noise (where each label can be flipped with probability less than $1/2$ depending on the input features), and give a fully-polynomial time testable learning algorithm. We also show a separation between the classical setting of learning in the presence of structured noise and testable learning. In fact, for the simple case of random classification noise (where each label is flipped with fixed probability $η= 1/2$), we show that testable learning requires super-polynomial time while classical learning is trivial.

翻译：我们提出了计算学习理论中的一个基本问题：能否高效测试训练集是否满足给定噪声模型的假设？尽管对含噪学习的研究已有数十年，但这一问题始终未被解决。本文证明该任务是可解的，并首次提出高效算法来测试训练数据中的多种噪声假设。为建模这一问题，我们扩展了Rubinfeld与Vasilyan（2023）近期提出的可测试学习框架，要求学习器运行满足以下两个条件的关联测试：(1) 当测试通过时，学习器需输出一个分类器及其最优性证书；(2) 对于任何根据边际分布与噪声模型的特定建模假设生成的数据集，该测试必须通过。随后，我们考虑在Massart噪声（每个标签的翻转概率取决于输入特征且低于1/2）下关于高斯边际学习半空间的问题，并给出全多项式时间可测试学习算法。我们还揭示了带结构噪声的传统学习场景与可测试学习之间的分离。事实上，对于随机分类噪声（每个标签以固定概率η=1/2翻转）的简单情形，可测试学习需要超多项式时间，而传统学习则易于实现。