  We present a fast, scalable, data-driven approach for solving relaxations of
0-1 integer linear programs. We use a combination of graph neural networks
(GNN) and the Lagrange decomposition based algorithm FastDOG (Abbas and Swoboda
2022b). We make the latter differentiable for end-to-end training and use GNNs
to predict its algorithmic parameters. This allows to retain the algorithm's
theoretical properties including dual feasibility and guaranteed non-decrease
in the lower bound while improving it via training. We overcome suboptimal
fixed points of the basic solver by additional non-parametric GNN update steps
maintaining dual feasibility. For training we use an unsupervised loss. We
train on smaller problems and test on larger ones showing strong generalization
performance with a GNN comprising only around $10k$ parameters. Our solver
achieves significantly faster performance and better dual objectives than its
non-learned version, achieving close to optimal objective values of LP
relaxations of very large structured prediction problems and on selected
combinatorial ones. In particular, we achieve better objective values than
specialized approximate solvers for specific problem classes while retaining
their efficiency. Our solver has better any-time performance over a large time
period compared to a commercial solver. Code available at
https://github.com/LPMP/BDD
