Learned Edge Targets for Graph-Coupled Neural Layers

Graph regularization commonly encourages neighboring representations to agree. Endpoint differences can also inform prediction. We introduce Learned Edge targets for Graph-cOupled neural layers (LEGO), which learn preferred differences and compute coupled node representations through a fixed number of primal-dual updates. Learning the targets improves fixed total variation (TV) blocks on five low-label transductive benchmarks and on deduplicated Chameleon and Squirrel, and also benefits GCN and GAT. We also learn the response shape with a radial potential (LP). To explain these effects, we decompose the targets into integrable and cycle components. Integrable targets translate node coordinates at every update; cycle components can affect the output through heterogeneous edge responses. Anchor injection and integrable-target controls recover much of the predictive gain. Interventions at frozen checkpoints also indicate that predictions depend on cycle components and edge correspondence. We derive a perturbation bound relating radial optimality residuals to error in the unrolled output. Code is available at https://anonymous.4open.science/r/LEGO-977D.