# AdaSAM: Boosting Sharpness-Aware Minimization with Adaptive Learning Rate and Momentum for Training Deep Neural Networks

## TL;DR

This paper provides the first theoretical convergence analysis of AdaSAM, an optimizer combining sharpness-aware minimization with adaptive learning rates and momentum, demonstrating its effectiveness and linear speedup in training deep neural networks.

## Contribution

The paper offers the first convergence rate analysis of AdaSAM, integrating adaptive learning rate and momentum with SAM in stochastic non-convex optimization.

## Key findings

- AdaSAM achieves a convergence rate of O(1/√(bT)) with linear speedup.
- AdaSAM outperforms SGD, AMSGrad, and SAM on NLP tasks.
- Theoretical analysis introduces delayed second-order momentum for decoupling steps.

## Abstract

Sharpness aware minimization (SAM) optimizer has been extensively explored as it can generalize better for training deep neural networks via introducing extra perturbation steps to flatten the landscape of deep learning models. Integrating SAM with adaptive learning rate and momentum acceleration, dubbed AdaSAM, has already been explored empirically to train large-scale deep neural networks without theoretical guarantee due to the triple difficulties in analyzing the coupled perturbation step, adaptive learning rate and momentum step. In this paper, we try to analyze the convergence rate of AdaSAM in the stochastic non-convex setting. We theoretically show that AdaSAM admits a $\mathcal{O}(1/\sqrt{bT})$ convergence rate, which achieves linear speedup property with respect to mini-batch size $b$. Specifically, to decouple the stochastic gradient steps with the adaptive learning rate and perturbed gradient, we introduce the delayed second-order momentum term to decompose them to make them independent while taking an expectation during the analysis. Then we bound them by showing the adaptive learning rate has a limited range, which makes our analysis feasible. To the best of our knowledge, we are the first to provide the non-trivial convergence rate of SAM with an adaptive learning rate and momentum acceleration. At last, we conduct several experiments on several NLP tasks, which show that AdaSAM could achieve superior performance compared with SGD, AMSGrad, and SAM optimizers.