Path coding penalties for directed acyclic graphs

We consider supervised learning problems where the features are embedded in a graph, such as gene expressions in a gene network. In this context, it is of much interest to automatically select a subgraph which has a small number of connected components, either to improve the prediction performance, or to obtain better interpretable results. Existing regularization or penalty functions for this purpose typically require solving among all connected subgraphs a selection problem which is combinatorially hard. In this paper, we address this issue for directed acyclic graphs (DAGs) and propose structured sparsity penalties over paths on a DAG (called “path coding” penalties). We design minimum cost flow formulations to compute the penalties and their proximal operator in polynomial time, allowing us in practice to efficiently select a subgraph with a small number of connected components. We present experiments on image and genomic data to illustrate the sparsity and connectivity benefits of path coding penalties over some existing ones as well as the scalability of our approach for prediction tasks.