Title: Gumbel-Matrix Routing for Flexible Multi-task Learning. Abstract: This paper proposes a novel per-task routing method for multi-task applications. Multi-task neural networks can learn to transfer knowledge across different tasks by using parameter sharing. However, sharing parameters between unrelated tasks can hurt performance. To address this issue, routing networks can be applied to learn to share each group of parameters with a different subset of tasks to better leverage tasks relatedness. However, this use of routing methods requires to address the challenge of learning the routing jointly with the parameters of a modular multi-task neural network. We propose the Gumbel-Matrix routing, a novel multi-task routing method based on the Gumbel-Softmax, that is designed to learn fine-grained parameter sharing. When applied to the Omniglot benchmark, the proposed method improves the state-of-the-art error rate by 17%.