arXiv subjects

Mikalai Korbit

Publications and source records attributed to Mikalai Korbit.

4 recordsLinked to original sources

arXiv2026

Incremental Gauss-Newton Descent for Machine Learning

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.

cs.LG↗

arXiv2026

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 curvature-vector products harder to scale as the number of classes grows. We show that the standard multiclass GGN can be decomposed exactly into a 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 an evaluation on a fixed-feature multiclass head 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.

cs.LG↗

arXiv2026

Second-Order, First-Class: A Composable Stack for Curvature-Aware Training

Second-order methods promise improved stability and faster convergence, yet they remain underused due to implementation overhead, tuning brittleness, and the lack of composable APIs. We introduce Somax, a composable Optax-native stack that treats curvature-aware training as a single JIT-compiled step governed by a static plan. Somax exposes first-class modules -- curvature operators, estimators, linear solvers, preconditioners, and damping policies -- behind a single step interface and composes with Optax by applying standard gradient transformations (e.g., momentum, weight decay, schedules) to the computed direction. This design makes typically hidden choices explicit and swappable. Somax separates planning from execution: it derives a static plan (including cadences) from module requirements, then runs the step through a specialized execution path that reuses intermediate results across modules. We report system-oriented ablations showing that (i) composition choices materially affect scaling behavior and time-to-accuracy, and (ii) planning reduces per-step overhead relative to unplanned composition with redundant recomputation.

cs.LG↗

arXiv2025

Exact Gauss-Newton Optimization for Training Deep Neural Networks

We present Exact Gauss-Newton (EGN), a stochastic second-order optimization algorithm that combines the generalized Gauss-Newton (GN) Hessian approximation with low-rank linear algebra to compute the descent direction. Leveraging the Duncan-Guttman matrix identity, the parameter update is obtained by factorizing a matrix which has the size of the mini-batch. This is particularly advantageous for large-scale machine learning problems where the dimension of the neural network parameter vector is several orders of magnitude larger than the batch size. Additionally, we show how improvements such as line search, adaptive regularization, and momentum can be seamlessly added to EGN to further accelerate the algorithm. Moreover, under mild assumptions, we prove that our algorithm converges in expectation to a stationary point of the objective. Finally, our numerical experiments demonstrate that EGN consistently exceeds, or at most matches the generalization performance of well-tuned SGD, Adam, GAF, SQN, and SGN optimizers across various supervised and reinforcement learning tasks.

cs.LG↗