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A Method for Enhancing Generalization of Adam by Multiple Integrations

Long Jin, Han Nong, Liangming Chen, Zhenming Su

TL;DR

MIAdam addresses Adam's generalization gap by steering optimization toward flat minima using a multiple-integral term, which filters high-frequency components of the optimization trajectory. The authors develop a diffusion-theory based generalization analysis, showing the mean escape time $\phi$ decreases for MIAdam variants relative to Adam, and provide a regret-based convergence analysis. Empirically, MIAdam improves generalization and robustness on image and text tasks while preserving fast convergence, supported by Hessian-based flatness metrics and label-noise experiments. Overall, MIAdam offers a practical optimizer that improves generalization without sacrificing convergence speed, with minimal computational overhead.

Abstract

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.

A Method for Enhancing Generalization of Adam by Multiple Integrations

TL;DR

MIAdam addresses Adam's generalization gap by steering optimization toward flat minima using a multiple-integral term, which filters high-frequency components of the optimization trajectory. The authors develop a diffusion-theory based generalization analysis, showing the mean escape time decreases for MIAdam variants relative to Adam, and provide a regret-based convergence analysis. Empirically, MIAdam improves generalization and robustness on image and text tasks while preserving fast convergence, supported by Hessian-based flatness metrics and label-noise experiments. Overall, MIAdam offers a practical optimizer that improves generalization without sacrificing convergence speed, with minimal computational overhead.

Abstract

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.

Paper Structure

This paper contains 27 sections, 2 theorems, 38 equations, 5 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Overview of Adam
  4. The Relationship between Flat Minima and Generalization
  5. MIAdam
  6. Generalization and Convergence Analyses
  7. Generalization Analyses
  8. Convergence Analyses
  9. Simulations and Experiments
  10. Simulations
  11. Experiments
  12. Image Classification Experiments
  13. Text Classification Experiments
  14. Robustness Against Label Noises
  15. Conclusion
  16. ...and 12 more sections

Key Result

Theorem 1

Suppose that Assumption asu:1, Assumption asu:2, and Assumption asu:3 hold while saddle point $\bm{u}$ is the exit from sharp minimum $\bm{a}$. Then the mean escape time of MIAdam1 from sharp minimum $\bm{a}$ to flat minimum $\bm{b}$ through saddle point $\bm{u}$ before the switch is where subscript $_{\bm{e}}$ denotes the escape direction; $\varrho$ is the path-dependent parameter; $\mathcal{b}

Figures (5)

  • Figure 1: The idea of this work and the filtering effect of integrations on optimizer trajectories. The blue integrated trajectory represents an equivalent path that does not actually exist on the original loss landscape.
  • Figure 2: Simulations of trajectory of Adam and MIAdam on 2-parameter loss landscapes.
  • Figure 3: Comparisons of top Hessian eigenvalues $\lambda_{\text{top}}$, Hessian traces $\lambda_{\text{trace}}$, and full Hessian eigenvalue densities for loss landscapes on the CIFAR-10 dataset using ResNet18.
  • Figure 4: Comparisons of the sum of the absolute values of the eigenvalues of the Hessian matrix at the convergence points of Adam, MIAdam1, MIAdam2, and MIAdam3 for 2500 rounds of simulation in the loss landscape shown in Fig. \ref{['fig.2']}(e).
  • Figure 5: Training and testing comparations of Adam, MIAdam1, MIAdam2, and MIAdam3 on CIFAR-100 using DenseNet121.

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • proof
  • Definition 1
  • Definition 2
  • proof