We examine the task of locating a target region among those induced by intersections of $n$ halfspaces in $\mathbb{R}^d$. This generic task connects to fundamental machine learning problems, such as training a perceptron and learning a $\phi$-separable dichotomy. We investigate the average teaching complexity of the task, i.e., the minimal number of samples (halfspace queries) required by a teacher to help a version-space learner in locating a randomly selected target. As our main result, we show that the average-case teaching complexity is $\Theta(d)$, which is in sharp contrast to the worst-case teaching complexity of $\Theta(n)$. If instead, we consider the average-case learning complexity, the bounds have a dependency on $n$ as $\Theta(n)$ for \tt{i.i.d.} queries and $\Theta(d \log(n))$ for actively chosen queries by the learner. Our proof techniques are based on novel insights from computational geometry, which allow us to count the number of convex polytopes and faces in a Euclidean space depending on the arrangement of halfspaces. Our insights allow us to establish a tight bound on the average-case complexity for $\phi$-separable dichotomies, which generalizes the known $\mathcal{O}(d)$ bound on the average number of "extreme patterns" in the classical computational geometry literature (Cover, 1965).

中文翻译：

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通过半空间查询教授凸多面体的平均情况复杂度

我们研究了在 $\mathbb{R}^d$ 中由 $n$ 个半空间的交叉引起的区域中定位目标区域的任务。这个通用任务与基本的机器学习问题相关，例如训练感知器和学习 $\phi$ 可分离的二分法。我们调查任务的平均教学复杂度，即教师帮助版本空间学习者定位随机选择的目标所需的最小样本数（半空间查询）。作为我们的主要结果，我们表明平均情况教学复杂度为 $\Theta(d)$，这与最坏情况教学复杂度 $\Theta(n)$ 形成鲜明对比。相反，如果我们考虑平均情况学习复杂度，则边界依赖于 $n$ 作为 $\Theta(n)$ 用于 \tt{iid} 查询和 $\Theta(d \log(n))$ 用于学习者主动选择的查询。我们的证明技术基于计算几何的新见解，这使我们能够根据半空间的排列来计算欧几里得空间中凸多面体和面的数量。我们的见解使我们能够为 $\phi$-可分离二分法建立平均情况复杂度的严格界限，这概括了已知的 $\mathcal{O}(d)$ 界限在“极端模式”的平均数量上经典计算几何文献（封面，1965 年）。