We investigate the computational efficiency of agnostic learning for several fundamental geometric concept classes in the plane. While the sample complexity of agnostic learning is well understood, its time complexity has received much less attention. We study the class of triangles and, more generally, the class of convex polygons with $k$ vertices for small $k$, as well as the class of convex sets in a square. We present a proper agnostic learner for the class of triangles that has optimal sample complexity and runs in time $\tilde O({\epsilon^{-6}})$, improving on the algorithm of Dobkin and Gunopulos (COLT `95) that runs in time $\tilde O({\epsilon^{-10}})$. For 4-gons and 5-gons, we improve the running time from $O({\epsilon^{-12}})$, achieved by Fischer and Kwek (eCOLT `96), to $\tilde O({\epsilon^{-8}})$ and $\tilde O({\epsilon^{-10}})$, respectively. We also design a proper agnostic learner for convex sets under the uniform distribution over a square with running time $\tilde O({\epsilon^{-5}})$, improving on the previous $\tilde O(\epsilon^{-8})$ bound at the cost of slightly higher sample complexity. Notably, agnostic learning of convex sets in $[0,1]^2$ under general distributions is impossible because this concept class has infinite VC-dimension. Our agnostic learners use data structures and algorithms from computational geometry and their analysis relies on tools from geometry and probabilistic combinatorics. Because our learners are proper, they yield tolerant property testers with matching running times. Our results raise a fundamental question of whether a gap between the sample and time complexity is inherent for agnostic learning of these and other natural concept classes.

翻译：本文研究了平面上若干基本几何概念类的不可知学习计算效率。尽管不可知学习的样本复杂度已得到充分理解，但其时间复杂度却较少受到关注。我们研究了三角形类，以及更一般地，针对较小$k$值的$k$顶点凸多边形类，还有正方形内凸集类。我们提出了一种针对三角形类的恰当不可知学习器，该学习器具有最优样本复杂度，且运行时间为$\tilde O({\epsilon^{-6}})$，改进了Dobkin和Gunopulos（COLT `95）提出的运行时间为$\tilde O({\epsilon^{-10}})$的算法。对于四边形和五边形，我们分别将运行时间从Fischer和Kwek（eCOLT `96）实现的$O({\epsilon^{-12}})$改进为$\tilde O({\epsilon^{-8}})$和$\tilde O({\epsilon^{-10}})$。我们还设计了一种针对正方形上均匀分布下凸集类的恰当不可知学习器，其运行时间为$\tilde O({\epsilon^{-5}})$，以略高的样本复杂度为代价，改进了先前$\tilde O(\epsilon^{-8})$的界限。值得注意的是，在一般分布下对$[0,1]^2$中凸集进行不可知学习是不可能的，因为该概念类具有无限VC维。我们的不可知学习器使用了计算几何中的数据结构和算法，其分析依赖于几何与概率组合学的工具。由于我们的学习器是恰当的，它们可产生具有匹配运行时间的容错性质检验器。我们的结果提出了一个根本性问题：对于这些及其他自然概念类的不可知学习，样本复杂度与时间复杂度之间的差距是否具有内在必然性。