Fast Gauss-Newton for Multiclass Cross-Entropy

In multiclass softmax cross-entropy, the full generalized Gauss-Newton (GGN) curvature couples all output logits through the softmax covariance, making second-order updates harder to scale as the number of classes grows. We show that the standard multiclass GGN can be decomposed exactly into a rank-one true-vs-rest term and a positive semidefinite within-competitor covariance term. Fast Gauss-Newton (FGN) retains the first term and drops the second, yielding a positive semidefinite under-approximation of the multiclass GGN that is exact for binary classification. The derivation uses an exact true-vs-rest scalar-margin representation of softmax cross-entropy: the loss and gradient are unchanged, and the approximation enters only at the curvature level. Exploiting the FGN curvature structure, the damped update can be written as an equivalent whitened row-space system with one row per mini-batch example. We solve this system matrix-free by conjugate gradient using Jacobian-vector and vector-Jacobian products of the scalar margin map. Targeted mechanism experiments and a fixed-feature head evaluation support the predictions from the decomposition: FGN stays closest to the full softmax GGN when competitor mass is concentrated or damping is large, and deviates as the dropped within-competitor covariance grows.