The Levenberg-Marquardt (LM) algorithm is a well-known second-order method for rapid convergence and strong robustness when training small- to medium-sized neural networks (NNs). However, its computat...
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.
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.
arXiv:2609.37308v1 Announce Type: cross Abstract: This paper introduces a hybrid joint-selective optimization (HJSO) framework for large-scale numerical problems in which a small subset of trainable...
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.
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.
arXiv:2608. 05892v1 Announce Type: new Abstract: Physics-informed neural networks (PINNs) often fail to accurately resolve partial differential equations (PDEs) with high-frequency or multi-scale solutions, as well as strongly nonlinear problems.
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.
arXiv:2608. 15665v1 Announce Type: new Abstract: Zeroth-order (ZO) optimization enables backpropagation-free fine-tuning of large language models, but existing ZO methods suffer from high-variance gradient estimators, making convergence unstable and highly sensitive to learning rates.
Physics-informed neural networks (PINNs) often fail to accurately resolve partial differential equations (PDEs) with high-frequency or multi-scale solutions, as well as strongly nonlinear problems. Two factors underlie this difficulty: spectral bias, the tendency of neural networks to underfit high-frequency features; and representation-coefficient coupling, the entanglement of representation learning and coefficient fitting within a single nonconvex optimization objective.
arXiv:2608. 16760v1 Announce Type: new Abstract: Reliable optimization is central to neural network (NN) training, yet Adam, the default optimizer for modern LLMs, rests on a fragile foundation.
arXiv:2606. 02328v1 Announce Type: new Abstract: We explore Riemannian optimization techniques for rank-factored matrix parameters, targeting contemporary deep learning applications.