图上凸标记的自导向学习
Self-Directed Learning of Convex Labelings on Graphs
AI总结：
针对图上自导向节点分类问题，聚焦（测地线）凸簇情况设计多项式时间算法，在双凸簇图上错误界为3(h(G)+1)^4 ln n，对轻微非凸簇鲁棒且为同质性簇提出简单高效算法。
摘要：
我们研究自导向学习设置下图中节点的分类问题。这种学习设置是在线学习的一种变体，不同于由对手决定节点呈现顺序，学习者自主且自适应地选择节点。虽然最近已考虑欧几里得半空间、线性函数和一般多类假设类的自导向学习，但此前不存在专门针对图上自导向节点分类的结果。本文针对该问题开发高效算法。具体而言，我们聚焦（测地线）凸簇的情况，即对于任意两个具有相同标记的节点，它们之间每条最短路径上的所有节点也具有相同标记。我们设计了一种运行时间为n多项式的算法，在具有两个凸簇的图上仅产生3(h(G)+1)^4 ln n次错误，其中n是节点总数，h(G)是Hadwiger数（即图G的最大团子式的大小）。我们还表明，该算法对簇略微非凸的情况具有鲁棒性，仍能实现对数级的错误界。最后，我们为同质性簇设计了一种简单高效的算法，其中强连接节点倾向于属于同一类。
英文摘要：
We study the problem of classifying the nodes of a given graph in the self-directed learning setup. This learning setting is a variant of online learning, where rather than an adversary determining the sequence in which nodes are presented, the learner autonomously and adaptively selects them. While self-directed learning of Euclidean halfspaces, linear functions, and general multiclass hypothesis classes was recently considered, no results previously existed specifically for self-directed node classification on graphs. In this paper, we address this problem developing efficient algorithms for it. More specifically, we focus on the case of (geodesically) convex clusters, i.e., for every two nodes sharing the same label, all nodes on every shortest path between them also share the same label. In particular, we devise an algorithm with runtime polynomial in $n$ that makes only $3(h(G)+1)^4 \ln n$ mistakes on graphs with two convex clusters, where $n$ is the total number of nodes and $h(G)$ is the Hadwiger number, i.e., the size of the largest clique minor of the graph $G$. We also show that our algorithm is robust to the case that clusters are slightly non-convex, still achieving a mistake bound logarithmic in $n$. Finally, we devise a simple and efficient algorithm for homophilic clusters, where strongly connected nodes tend to belong to the same class.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
