Global Exponential Convergence of Two-Layer Linear Network Training

תקציר מקורי באנגליתarXiv:2610.09356v1 Announce Type: new Abstract: We prove global exponential (linear) convergence with an explicit rate in the rich scaling for wide two-layer linear networks trained with smooth Polyak-Lojasiewicz predictor losses. Gradient flow in the factors closes exactly in terms of a finite-dimensional Bures flow of the neuron law covariance, in which the predictor dynamics are preconditioned by hidden covariance blocks. Mean-field conservation laws provide uniform spectral lower bounds on the hidden preconditioning blocks when the initial covariance satisfies a spectral support gap condition. This condition encompasses positive definiteness while still allowing for singular initializations. For an initial covariance $\Sigma_0 = \sigma^2 \mathrm{Id}$, the loss converges to the global m