arXiv Machine Learning
Aug 27

Adaptive Hybrid Subspace Levenberg Marquardt Algorithm with Adequacy Monitor for Large Scale Least Squares Problems

The paper introduces an Adaptive Hybrid Subspace Levenberg–Marquardt (HSLM) algorithm that tackles large‑scale nonlinear least‑squares problems by building a low‑dimensional subspace from gradient, memory, Krylov‑subspace, and randomized curvature data. It employs a deterministic adequacy monitor to adaptively enrich the subspace and decouples step acceptance from damping adjustment, using Armijo backtracking for step length and a ratio of actual to predicted reduction for damping updates. The authors prove global convergence to stationarity and local linear and superlinear convergence, and demonstrate that HSLM matches the convergence of classical and Krylov‑subspace LM while significantly reducing per‑iteration cost, especially as the parameter dimension increases.

By M. Duc Hoang, Timothy J. Lewis
arXiv Machine Learning
Sep 4

Linearized subspace refinement framework to expose hidden accuracy in trained neural networks

The paper introduces Linearized Subspace Refinement (LSR), a post‑training framework that uses the local linearized model of a trained neural network to compute a low‑dimensional correction via a reduced least‑squares problem. LSR is architecture‑agnostic and improves accuracy across tasks such as function approximation, operator learning, physics‑informed fine‑tuning, and noisy inverse problems, often achieving order‑of‑magnitude error reductions. The method reveals that standard training can leave significant accuracy plateaus due to numerical ill‑conditioning, and it offers a subspace rank that balances correction strength, stability, and noise sensitivity.

By Wenbo Cao, Weiwei Zhang
arXiv Machine Learning
Aug 26

A Data-dependent Early Stopping Rule using Rademacher Complexity with L1-norm

The paper proposes an analytic method for determining the optimal early‑stopping time in training neural networks, avoiding the need for gradient‑descent training. It uses Rademacher complexity with an L1‑norm to estimate generalization error, offering a more general approach than previous random‑matrix‑theory based methods. The framework is demonstrated on linear regression and extended to nonlinear neural networks via linear probing, as shown in a MNIST classification example.

By Duy Hoang, Bastien Berret, Olivier Bruneau, Laurent Fribourg