Title: Provable Sharpness-Aware Minimization with Adaptive Learning Rate . Abstract: Sharpness aware minimization (SAM) optimizer has been extensively explored as it can converge fast and train deep neural networks efficiently via introducing extra perturbation steps to flatten the landscape of deep learning models. A combination of SAM with adaptive learning rate (AdaSAM) has also been explored to train large-scale deep neural networks without theoretical guarantee due to the dual difficulties in analyzing the perturbation step and the coupled adaptive learning rate. In this paper, we try to analyze the convergence rate of AdaSAM in the stochastic non-convex setting. We theoretically show that AdaSAM admit a $\mathcal{O}(1/\sqrt{bT})$ convergence rate and show linear speedup property with respect to mini-batch size b. To best of our knowledge, we are the first to provide the non-trivial convergence rate of SAM with an adaptive learning rate.  To decouple the two stochastic gradient steps with the adaptive learning rate, we first introduce the delayed second-order momentum during the convergence to decompose them to make them independent while taking an expectation. Then we bound them by showing the adaptive learning rate has a limited range, which makes our analysis feasible.  At last, we conduct experiments on several NLP tasks and they show that AdaSAM could achieve superior performance compared with SGD, AMSGrad, and SAM optimizer.