# Explicit and Implicit Graduated Optimization in Deep Neural Networks

arXiv: https://arxiv.org/abs/2412.11501 · HTML: https://mlsanity.com/paper/2412.11501/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 39 references that are in the map.
- First posted on arXiv 16 Dec 2024.

## Builds on (most cited first, 10 of 10)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Understanding the Role of Momentum in Stochastic Gradient Methods](https://mlsanity.com/paper/1910.13962/) (2019)
- [On Graduated Optimization for Stochastic Non-Convex Problems](https://mlsanity.com/paper/1503.03712/) (2015)
- [Stochastic Gradient Descent in the Viewpoint of Graduated Optimization](https://mlsanity.com/paper/2308.06775/) (2023)
- [Using Stochastic Gradient Descent to Smooth Nonconvex Functions: Analysis of Implicit Graduated Optimization](https://mlsanity.com/paper/2311.08745/) (2023)

## Built on by (most cited first, 2 of 2)

- [Both Asymptotic and Non-Asymptotic Convergence of Quasi-Hyperbolic Momentum using Increasing Batch Size](https://mlsanity.com/paper/2506.23544/) (2025)
- [Taming Nonconvexity with Epigraph Lifting](https://mlsanity.com/paper/2610.16429/) (2026)

## Abstract (arXiv)

Graduated optimization is a global optimization technique that is used to minimize a multimodal nonconvex function by smoothing the objective function with noise and gradually refining the solution. This paper experimentally evaluates the performance of the explicit graduated optimization algorithm with an optimal noise scheduling derived from a previous study and discusses its limitations. It uses traditional benchmark functions and empirical loss functions for modern neural network architectures for evaluating. In addition, this paper extends the implicit graduated optimization algorithm, which is based on the fact that stochastic noise in the optimization process of SGD implicitly smooths the objective function, to SGD with momentum, analyzes its convergence, and demonstrates its effectiveness through experiments on image classification tasks with ResNet architectures.