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
Oct 19

Scalable Levenberg-Marquardt Training Framework for Deep Learning

The paper introduces LeMA, a distributed Levenberg-Marquardt algorithm designed for large-scale neural network training on multi‑GPU systems. By deriving an exact dual formulation that eliminates memory dependence on parameter count and implementing a Jacobian sharding strategy, LeMA reduces the memory footprint from quadratic in parameters to quadratic in batch size. This enables exact LMA steps for billion‑parameter models, achieving a 7.97× speedup and 99.7% strong scaling efficiency on an 8‑GPU node for a convolutional network with over a billion parameters.

By Xianghao Wang, Johan Barth\'elemy, John Taylor
arXiv Machine Learning
Sep 17

Gradient Descent with Stochastic Subspaces via Persistence of Memory

The paper introduces a novel technique called "persistence of memory" to enhance stochastic subspace methods for large‑scale optimisation. By using a weakly correlated guidance vector that is refreshed only at wide intervals, the method provides a structured direction for random subspace descent. The authors demonstrate that this guidance can be efficiently computed in sparse or minibatch settings and present the first theoretical analysis of classical SSD methods for sparse functions, showing alignment with low‑lying Hessian eigenvectors near the optimum.

By Subhroshekhar Ghosh, Clement Z. Q. Ng, Pierre-Louis Poirion, Akiko Takeda