Module-Aware Optimization for Graph Neural Networks

Guy Hadad, Haggai Roitman, Moshe Eliasof, Bracha Shapira

Different optimizers impose different geometries on parameter updates. Adaptive methods such as Adam and AdamW use coordinate-wise updates, while matrix-aware optimizers such as Muon utilize weight-matrix structure through orthogonalized updates. For Graph Neural Networks (GNNs), however, the effective geometry of its weight matrices is shaped not only by its learned channel-mixing weights, but also by the graph-propagated representations it acts on. Thus, even if an update is well normalized as a matrix, its effect on the layer output can be amplified when applied to graph-propagated features. To address this gap, we introduce GraphMuon, a graph-aware optimizer that adapts Muon's matrix updates to the covariance structure of graph-propagated layer inputs. GraphMuon performs orthogonalization in a metric defined by these propagated representations and normalizes step sizes according to the update's induced effect on the layer representation. We show that theoretically, vanilla Muon can suffer from graph-induced update amplification, while our GraphMuon can mitigate this effect. As such, our GraphMuon yields a graph-metric analogue of Muon's polar update step. Empirically,…