# Single Loop Gaussian Homotopy Method for Non-convex Optimization

arXiv: https://arxiv.org/abs/2203.05717 · HTML: https://mlsanity.com/paper/2203.05717/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Mar 2022.

## Built on by (most cited first, 16 of 16)

- [Homotopy-based training of NeuralODEs for accurate dynamics discovery](https://mlsanity.com/paper/2210.01407/) (2022)
- [Continuation Path Learning for Homotopy Optimization](https://mlsanity.com/paper/2307.12551/) (2023)
- [Global Optimality in Bivariate Gradient-based DAG Learning](https://mlsanity.com/paper/2306.17378/) (2023)
- [Learning (Approximately) Equivariant Networks via Constrained Optimization](https://mlsanity.com/paper/2505.13631/) (2025)
- [One-Line-of-Code Data Mollification Improves Optimization of Likelihood-based Generative Models](https://mlsanity.com/paper/2305.18900/) (2023)
- [Global Optimization with A Power-Transformed Objective and Gaussian Smoothing](https://mlsanity.com/paper/2412.05204/) (2024)
- [Improving Classifier-Free Guidance of Flow Matching via Manifold Projection](https://mlsanity.com/paper/2601.21892/) (2026)
- [Using Stochastic Gradient Descent to Smooth Nonconvex Functions: Analysis of Implicit Graduated Optimization](https://mlsanity.com/paper/2311.08745/) (2023)
- [Neural Predictor-Corrector: Solving Homotopy Problems with Reinforcement Learning](https://mlsanity.com/paper/2602.03086/) (2026)
- [Power Homotopy for Zeroth-Order Non-Convex Optimizations](https://mlsanity.com/paper/2511.13592/) (2025)
- [Direct Soft-Policy Sampling via Langevin Dynamics](https://mlsanity.com/paper/2602.07873/) (2026)
- [Estimating Multi-chirp Parameters using Curvature-guided Langevin Monte Carlo](https://mlsanity.com/paper/2501.18178/) (2025)
- [Progressive Power Homotopy for Non-convex Optimization](https://mlsanity.com/paper/2601.15915/) (2026)
- [Probabilistic Gaussian Homotopy: A Probability-Space Continuation Framework for Nonconvex Optimization](https://mlsanity.com/paper/2603.13546/) (2026)
- [Probabilistic Smoothing with Ratio-Monotone Transforms for Global Optimization](https://mlsanity.com/paper/2605.27316/) (2026)
- [Taming Nonconvexity with Epigraph Lifting](https://mlsanity.com/paper/2610.16429/) (2026)

## Abstract (arXiv)

The Gaussian homotopy (GH) method is a popular approach to finding better stationary points for non-convex optimization problems by gradually reducing a parameter value t, which changes the problem to be solved from an almost convex one to the original target one. Existing GH-based methods repeatedly call an iterative optimization solver to find a stationary point every time t is updated, which incurs high computational costs. We propose a novel single loop framework for GH methods (SLGH) that updates the parameter t and the optimization decision variables at the same. Computational complexity analysis is performed on the SLGH algorithm under various situations: either a gradient or gradient-free oracle of a GH function can be obtained for both deterministic and stochastic settings. The convergence rate of SLGH with a tuned hyperparameter becomes consistent with the convergence rate of gradient descent, even though the problem to be solved is gradually changed due to t. In numerical experiments, our SLGH algorithms show faster convergence than an existing double loop GH method while outperforming gradient descent-based methods in terms of finding a better solution.