Abstract
arXiv:2408.05560v2 Announce Type: replace Abstract: Stochastic gradient updates are widely used for their efficiency and scalability, but their effective step sizes can depend strongly on feature scaling and local model sensitivity. Gauss-Newton methods address such scale effects through curvature information, but in their standard mini-batch form they require matrix-vector products, linear solves, or structured approximations. This paper studies the special case of scalar-output losses evaluated one sample at a time. In this setting, the generalized Gauss-Newton matrix has rank at most one, and its only possible nonzero curvature direction is aligned with the stochastic gradient. As a result, the damped Gauss-Newton direction reduces to a closed-form scalar normalization of the sample gradient. The resulting update, Incremental Gauss-Newton Descent (IGND), requires no curvature matrix storage, factorization, or iterative linear solve. We derive the update, characterize its behavior, and relate it to normalized gradient descent, adaptive first-order methods, stochastic Polyak step sizes, and mini-batch Gauss-Newton updates. Under explicit smoothness, alignment, and stochastic approximation assumptions, we prove a stationarity result for the IGND update. Experiments on supervised learning, a controlled test of scale robustness, and a linear-quadratic control case study show that IGND improves robustness to sensitivity scaling and can be competitive with, or complementary to, common stochastic optimizers while retaining a simple incremental update.