Label spreading is a general technique for semi-supervised learning with point cloud or network data, which can be interpreted as a diffusion of labels on a graph. While there are many variants of label spreading, nearly all of them are linear models, where the incoming information to a node is a weighted sum of information from neighboring nodes. Here, we add nonlinearity to label spreading via nonlinear functions involving higher-order network structure, namely triangles in the graph. For a broad class of nonlinear functions, we prove convergence of our nonlinear higher-order label spreading algorithm to the global solution of an interpretable semi-supervised loss function. We demonstrate the efficiency and efficacy of our approach on a variety of point cloud and network datasets, where the nonlinear higher-order model outperforms classical label spreading, hypergraph clustering, and graph neural networks.