We study Boolean classification problems over relational background structures in the logical framework introduced by Grohe and Tur\'an (TOCS 2004). It is known (Grohe and Ritzert, LICS 2017) that classifiers definable in first-order logic over structures of polylogarithmic degree can be learned in sublinear time, where the degree of the structure and the running time are measured in terms of the size of the structure. We generalise the results to the first-order logic with counting FOCN, which was introduced by Kuske and Schweikardt (LICS 2017) as an expressive logic generalising various other counting logics. Specifically, we prove that classifiers definable in FOCN over classes of structures of polylogarithmic degree can be consistently learned in sublinear time. This can be seen as a first step towards extending the learning framework to include numerical aspects of machine learning. We extend the result to agnostic probably approximately correct (PAC) learning for classes of structures of degree at most $(\log \log n)^c$ for some constant $c$. Moreover, we show that bounding the degree is crucial to obtain sublinear-time learning algorithms. That is, we prove that, for structures of unbounded degree, learning is not possible in sublinear time, even for classifiers definable in plain first-order logic.

翻译：摘要：我们在Grohe和Turán（TOCS 2004）引入的逻辑框架中，研究关系背景结构上的布尔分类问题。已知（Grohe和Ritzert，LICS 2017）在多项式对数度数结构上可用一阶逻辑定义的分类器可以在亚线性时间内学习，其中结构的度数和运行时间均以结构大小衡量。我们将该结果推广至由Kuske和Schweikardt（LICS 2017）引入的带计数的一阶逻辑FOCN，这是一种具有表达力的逻辑，可泛化多种其他计数逻辑。具体而言，我们证明：在多项式对数度数结构类上，FOCN可定义的分类器可以在亚线性时间内一致学习。这可视作将学习框架扩展至包含机器学习数值方面的第一步。我们将结果扩展至对某些常数c下度数最多为$(\log \log n)^c$的结构类的不可知可能近似正确（PAC）学习。此外，我们证明限制度数是获得亚线性时间学习算法的关键。即，对于无界度数结构，即使是对纯一阶逻辑可定义的分类器，也无法在亚线性时间内学习。