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.

此论文分析了AdaSAM在随机非凸环境下的收敛速度，证明了AdaSAM的收敛速度为O（1/√（bT）），具有线性加速性质，随机梯度步骤与自适应学习率和扰动梯度分别分解分析，证明了自适应学习率具有有限的范围，为SAM和自适应学习率与动量加速提供了非平凡的收敛率，实验表明AdaSAM可以获得比SGD，AMSGrad和SAM优化器更好的性能。

AdaSAM: 自适应学习率和动量的锐度感知最小化，用于深度神经网络训练