# A Method for Enhancing Generalization of Adam by Multiple Integrations

arXiv: https://arxiv.org/abs/2412.12473 · HTML: https://mlsanity.com/paper/2412.12473/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 34 references that are in the map.
- First posted on arXiv 17 Dec 2024.

## Builds on (most cited first, 20 of 20)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Fantastic Generalization Measures and Where to Find Them](https://mlsanity.com/paper/1912.02178/) (2019)
- [Sharp Minima Can Generalize For Deep Nets](https://mlsanity.com/paper/1703.04933/) (2017)
- [PyHessian: Neural Networks Through the Lens of the Hessian](https://mlsanity.com/paper/1912.07145/) (2019)
- [The Marginal Value of Adaptive Gradient Methods in Machine Learning](https://mlsanity.com/paper/1705.08292/) (2017)
- [ADAHESSIAN: An Adaptive Second Order Optimizer for Machine Learning](https://mlsanity.com/paper/2006.00719/) (2020)
- [Adaptive Gradient Methods with Dynamic Bound of Learning Rate](https://mlsanity.com/paper/1902.09843/) (2019)
- [Deep Pyramidal Residual Networks](https://mlsanity.com/paper/1610.02915/) (2016)
- [Improving Generalization Performance by Switching from Adam to SGD](https://mlsanity.com/paper/1712.07628/) (2017)
- [Relative Flatness and Generalization](https://mlsanity.com/paper/2001.00939/) (2020)
- [BertGCN: Transductive Text Classification by Combining GCN and BERT](https://mlsanity.com/paper/2105.05727/) (2021)
- [Understanding the Generalization of Adam in Learning Neural Networks with Proper Regularization](https://mlsanity.com/paper/2108.11371/) (2021)
- [Dynamic Regularized Sharpness Aware Minimization in Federated Learning: Approaching Global Consistency and Smooth Landscape](https://mlsanity.com/paper/2305.11584/) (2023)
- [Adaptive Inertia: Disentangling the Effects of Adaptive Learning Rate and Momentum](https://mlsanity.com/paper/2006.15815/) (2020)
- [AdaScale SGD: A User-Friendly Algorithm for Distributed Training](https://mlsanity.com/paper/2007.05105/) (2020)
- [Identifying Generalization Properties in Neural Networks](https://mlsanity.com/paper/1809.07402/) (2018)
- [Stochastic Gradient Variance Reduction by Solving a Filtering Problem](https://mlsanity.com/paper/2012.12418/) (2020)

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

- [HomeAdam: Adam and AdamW Algorithms Sometimes Go Home to Obtain Better Provable Generalization](https://mlsanity.com/paper/2603.02649/) (2026)
- [Combining Adam and its Inverse Counterpart to Enhance Generalization of Deep Learning Optimizers](https://mlsanity.com/paper/2603.07122/) (2026)

## Abstract (arXiv)

The insufficient generalization of adaptive moment estimation (Adam) has hindered its broader application. Recent studies have shown that flat minima in loss landscapes are highly associated with improved generalization. Inspired by the filtering effect of integration operations on high-frequency signals, we propose multiple integral Adam (MIAdam), a novel optimizer that integrates a multiple integral term into Adam. This multiple integral term effectively filters out sharp minima encountered during optimization, guiding the optimizer towards flatter regions and thereby enhancing generalization capability. We provide a theoretical explanation for the improvement in generalization through the diffusion theory framework and analyze the impact of the multiple integral term on the optimizer's convergence. Experimental results demonstrate that MIAdam not only enhances generalization and robustness against label noise but also maintains the rapid convergence characteristic of Adam, outperforming Adam and its variants in state-of-the-art benchmarks.