We introduce a notion of distance between supervised learning problems, which
we call the Risk distance. This optimal-transport-inspired distance facilitates
stability results; one can quantify how seriously issues like sampling bias,
noise, limited data, and approximations might change a given problem by
bounding how much these modifications can move the problem under the Risk
distance. With the distance established, we explore the geometry of the
resulting space of supervised learning problems, providing explicit geodesics
and proving that the set of classification problems is dense in a larger class
of problems. We also provide two variants of the Risk distance: one that
incorporates specified weights on a problem's predictors, and one that is more
sensitive to the contours of a problem's risk landscape.

我们引入了一种称为风险距离（Risk distance）的监督学习问题之间的距离概念，该概念能够促进稳定性结果；通过限定这些修改可以如何改变给定问题，我们可以量化采样偏差、噪声、有限数据和近似等问题的严重性。通过建立距离，我们探索了产生的监督学习问题空间的几何特征，提供了明确的测地线，并证明了分类问题集合在更大类别的问题中是稠密的。我们还提供了两个 Risk 距离的变种：一个在问题的预测者上加入指定权重的变种，另一个对问题的风险景观更为敏感。

监督学习问题的几何性质和稳定性

Geometry and Stability of Supervised Learning Problems

This article is a continuation of a previous work where we studied infinite
horizon control problems for which the dynamic, running cost and control space
may be different in two half-spaces of some euclidian space $\R^N$. In this
article we extend our results in several directions: $(i)$ to more general
domains; $(ii)$ by considering finite horizon control problems; $(iii)$ by
weaken the controlability assumptions. We use a Bellman approach and our main
results are to identify the right Hamilton-Jacobi-Bellman Equation (and in
particular the right conditions to be put on the interfaces separating the
regions where the dynamic and running cost are different) and to provide the
maximal and minimal solutions, as well as conditions for uniqueness. We also
provide stability results for such equations.

用 Bellman 方法研究了不同欧几里得空间的动态、运行成本和控制空间不同的无限控制问题，在更广泛的区域和有限时间控制问题方面扩展了结果，并提供了稳定性结果和合理的控制条件和最大、最小解。