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SVD-Preconditioned Gradient Descent Method for Solving Nonlinear Least Squares Problems

Zhipeng Chang, Wenrui Hao, Nian Liu

TL;DR

This work introduces SPGD, a SVD-based preconditioned gradient method for nonlinear least-squares problems, and integrates it with Adam-style adaptivity to form SPGD-Adam. By leveraging the local spectral information of the Jacobian, SPGD achieves a more favorable convergence factor than classical gradient descent, and the modified Adam variant provides global convergence under AMSGrad-style stabilization and regularized preconditioning. The authors supply a rigorous convergence analysis establishing local linear convergence for SPGD and global convergence for the modified Adam framework, along with detailed bounds on error terms. Empirically, SPGD and SPGD-Adam demonstrate faster convergence and lower residuals across function-approximation tasks, PDE-like problems, and image-classification settings (CIFAR-10) compared with standard Adam, highlighting the practical impact of problem-structure–driven preconditioning.

Abstract

This paper introduces a novel optimization algorithm designed for nonlinear least-squares problems. The method is derived by preconditioning the gradient descent direction using the Singular Value Decomposition (SVD) of the Jacobian. This SVD-based preconditioner is then integrated with the first- and second-moment adaptive learning rate mechanism of the Adam optimizer. We establish the local linear convergence of the proposed method under standard regularity assumptions and prove global convergence for a modified version of the algorithm under suitable conditions. The effectiveness of the approach is demonstrated experimentally across a range of tasks, including function approximation, partial differential equation (PDE) solving, and image classification on the CIFAR-10 dataset. Results show that the proposed method consistently outperforms standard Adam, achieving faster convergence and lower error in both regression and classification settings.

SVD-Preconditioned Gradient Descent Method for Solving Nonlinear Least Squares Problems

TL;DR

This work introduces SPGD, a SVD-based preconditioned gradient method for nonlinear least-squares problems, and integrates it with Adam-style adaptivity to form SPGD-Adam. By leveraging the local spectral information of the Jacobian, SPGD achieves a more favorable convergence factor than classical gradient descent, and the modified Adam variant provides global convergence under AMSGrad-style stabilization and regularized preconditioning. The authors supply a rigorous convergence analysis establishing local linear convergence for SPGD and global convergence for the modified Adam framework, along with detailed bounds on error terms. Empirically, SPGD and SPGD-Adam demonstrate faster convergence and lower residuals across function-approximation tasks, PDE-like problems, and image-classification settings (CIFAR-10) compared with standard Adam, highlighting the practical impact of problem-structure–driven preconditioning.

Abstract

This paper introduces a novel optimization algorithm designed for nonlinear least-squares problems. The method is derived by preconditioning the gradient descent direction using the Singular Value Decomposition (SVD) of the Jacobian. This SVD-based preconditioner is then integrated with the first- and second-moment adaptive learning rate mechanism of the Adam optimizer. We establish the local linear convergence of the proposed method under standard regularity assumptions and prove global convergence for a modified version of the algorithm under suitable conditions. The effectiveness of the approach is demonstrated experimentally across a range of tasks, including function approximation, partial differential equation (PDE) solving, and image classification on the CIFAR-10 dataset. Results show that the proposed method consistently outperforms standard Adam, achieving faster convergence and lower error in both regression and classification settings.
Paper Structure (16 sections, 3 theorems, 72 equations, 1 figure, 1 algorithm)

This paper contains 16 sections, 3 theorems, 72 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. SVD-Preconditioned Gradient Descent (SPGD) method
  3. Method derivation
  4. Local convergence
  5. Adam framework with SPGD method
  6. Modified Adam algorithm
  7. Global convergence
  8. Assumptions
  9. Convergence theorem
  10. Bounding $I_{1,t}$
  11. Bounding $I_{2,t}$
  12. Decomposing $I_{3,t}$
  13. Bounding $I_{4,t}$
  14. Summarization
  15. Numerical Experiments
  16. ...and 1 more sections

Key Result

Theorem 1

Suppose that $F$ satisfies the conditions in Assumption assump:regular. Consider the GD iteration where $\alpha>0$ denotes the step size. Then, for any sufficiently small $\alpha$, there exists a neighborhood $U_{\alpha}\subset U$ of $\theta^*$ such that the gradient descent sequence $\{\theta_t\}_{t\geq 0}$ generated by equ:gd iteration in thm with initial point $\theta_0\in U_{\alpha}$ satisfi

Figures (1)

  • Figure 1: Scenario I (Varying Frequency): Test loss versus the number of epochs with fixed dimension $d$ and varying frequency parameters $n \in \{5, 7\}$. The shaded region in each subplot indicates the interquartile range over 10 independent runs with different random seeds.

Theorems & Definitions (11)

  • Theorem 1: Local linear convergence of GD near a regular equilibrium
  • proof
  • Remark 1
  • Theorem 2: Local linear convergence of the SPGD method near a regular equilibrium
  • proof
  • Remark 2
  • Remark 3: Extension to cross-entropy loss
  • Remark 4: Efficient computation for large-scale networks
  • Remark 5
  • Theorem 3
  • ...and 1 more