Non-normal spectral signatures of instability in neural network training dynamics
arXiv:2605.23476v1 Announce Type: new Abstract: Training instabilities in deep networks - loss spikes, oscillatory convergence, and gradient pathologies - are empirically prevalent but lack a rigorous operator-theoretic explanation. We show that the linearized update operators for practically used optimizers are generically non-normal: for Adam, non-normality is controlled by the commutator [H, M] between the Hessian and the diagonal adaptive preconditioner, while for SGD with momentum it arises from the augmented state-space structure of the update map. Applying non-normal stability theory to these operators, we derive a conservative pseudospectral precursor bound in which \kappa(V) serves as an early-warning indicator of transient amplification even when the spectral radius remains below one, and we establish that exceptional points of the update operator appear as the \kappa(V) -> \infty limiting case of this framework. Numerical experiments on two-layer networks confirm that the sp