通过凸面几何学进行独立于边际的在线多类学习

我们考虑多类分类的问题，在这个问题上，有一串对抗性选择的查询到达，必须在线分配一个标签。与寻求最小化错误分类率的传统界限不同，我们最小化每个查询到与其正确标签对应的区域的总距离。当真正的标签是通过最近的邻居分区来确定的--即一个点的标签是由它在欧氏距离中最接近的k个中心来决定的--我们表明，我们可以实现一个与总的查询次数无关的损失。我们对这一结果进行了补充，表明学习一般的凸集需要每次查询的几乎线性损失。我们的结果建立在对上下文搜索的几何问题的遗憾保证之上。此外，我们开发了一种新的还原技术，从多类分类到二类分类，这可能是独立的兴趣。

原文题目：Margin-Independent Online Multiclass Learning via Convex Geometry

原文：We consider the problem of multi-class classification, where a stream of adversarially chosen queries arrive and must be assigned a label online. Unlike traditional bounds which seek to minimize the misclassification rate, we minimize the total distance from each query to the region corresponding to its correct label. When the true labels are determined via a nearest neighbor partition -- i.e. the label of a point is given by which of k centers it is closest to in Euclidean distance -- we show that one can achieve a loss that is independent of the total number of queries. We complement this result by showing that learning general convex sets requires an almost linear loss per query. Our results build off of regret guarantees for the geometric problem of contextual search. In addition, we develop a novel reduction technique from multiclass classification to binary classification which may be of independent interest.