VC基于样本的无量纲和无分布测试
VC Dimension and Distribution-Free Sample-Based Testing
摘要：
我们考虑的问题，确定哪些类的功能可以更有效地测试比他们可以学习，在分布免费的基于样本的模型，对应于标准的PAC学习设置。我们的主要结果表明，虽然VC维本身并不总是提供严格的界限所需的样本数量来测试在这个模型中的一类功能，它可以结合一个密切相关的变体，我们称之为“低VC”（或LVC）维，以获得强有力的下界这个样本的复杂性。我们使用这个结果，以获得强大的，在许多情况下，接近最佳的样本复杂性的下界测试工会的间隔，半空间，半空间的交叉点，多项式阈值函数，和决策树。相反，我们表明，两个自然类的功能，juntas和单调功能，可以测试的样本数是多项式小于PAC学习所需的样本数。最后，我们还利用VC维数和性质测试之间的联系，建立了新的下界来测试半径可聚性和线性约束系统的可行性。
英文摘要：
We consider the problem of determining which classes of functions can be tested more efficiently than they can be learned, in the distribution-free sample-based model that corresponds to the standard PAC learning setting. Our main result shows that while VC dimension by itself does not always provide tight bounds on the number of samples required to test a class of functions in this model, it can be combined with a closely-related variant that we call "lower VC" (or LVC) dimension to obtain strong lower bounds on this sample complexity. We use this result to obtain strong and in many cases nearly optimal lower bounds on the sample complexity for testing unions of intervals, halfspaces, intersections of halfspaces, polynomial threshold functions, and decision trees. Conversely, we show that two natural classes of functions, juntas and monotone functions, can be tested with a number of samples that is polynomially smaller than the number of samples required for PAC learning. Finally, we also use the connection between VC dimension and property testing to establish new lower bounds for testing radius clusterability and testing feasibility of linear constraint systems.
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