  When learning graph neural networks (GNNs) in node-level prediction tasks,
most existing loss functions are applied for each node independently, even if
node embeddings and their labels are non-i.i.d. because of their graph
structures. To eliminate such inconsistency, in this study we propose a novel
Quasi-Wasserstein (QW) loss with the help of the optimal transport defined on
graphs, leading to new learning and prediction paradigms of GNNs. In
particular, we design a ``Quasi-Wasserstein'' distance between the observed
multi-dimensional node labels and their estimations, optimizing the label
transport defined on graph edges. The estimations are parameterized by a GNN in
which the optimal label transport may determine the graph edge weights
optionally. By reformulating the strict constraint of the label transport to a
Bregman divergence-based regularizer, we obtain the proposed Quasi-Wasserstein
loss associated with two efficient solvers learning the GNN together with
optimal label transport. When predicting node labels, our model combines the
output of the GNN with the residual component provided by the optimal label
transport, leading to a new transductive prediction paradigm. Experiments show
that the proposed QW loss applies to various GNNs and helps to improve their
performance in node-level classification and regression tasks. The code of this
work can be found at \url{https://github.com/SDS-Lab/QW_Loss}.
