# Subspace Levenberg Marquardt Algorithms in Training Neural Networks

arXiv: https://arxiv.org/abs/2609.00789 · HTML: https://mlsanity.com/paper/2609.00789/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 17 references that are in the map.
- First posted on arXiv 1 Sep 2026.

## Builds on (most cited first, 3 of 3)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [Adaptive Hybrid Subspace Levenberg Marquardt Algorithm with Adequacy Monitor for Large Scale Least Squares Problems](https://mlsanity.com/paper/2608.25524/) (2026)

## Abstract (arXiv)

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 computational and memory costs increase significantly as the number of parameters in an NN grows. To address this limitation, subspace methods have been proposed, such as the Krylov subspace LM (KSLM) and the hybrid subspace LM (HSLM), making second-order algorithms more efficient. In this work, we evaluate the subspace Levenberg-Marquardt algorithms for regression and classification tasks in neural networks. We compare the performance of subspace LM variants with the classical LM method, as well as other popular first-order algorithms, such as stochastic gradient descent (SGD) and Adam.