Title: Rethinking the Structure of Stochastic Gradients: Empirical and Statistical Evidence. Abstract: It is well known that stochastic gradients significantly improve both optimization and generalization of deep neural networks (DNNs). Some works attempted to explain the success of stochastic optimization for deep learning by the arguably heavy-tail properties of gradient noise, while other works presented theoretical and empirical evidence against the heavy-tail hypothesis on gradient noise. Unfortunately, formal statistical tests for analyzing the structure and heavy tails of stochastic gradients in deep learning are still under-explored. In this paper, we mainly make two contributions. First, we conduct formal statistical tests on the distribution of stochastic gradients and gradient noise across both parameters and iterations. Our statistical tests reveal that dimension-wise gradients usually exhibit power-law heavy tails, while iteration-wise gradients and stochastic gradient noise caused by minibatch training usually do not exhibit power-law heavy tails. Second, we further discover that the covariance spectra of stochastic gradients have the power-law structures in deep learning. While previous papers believed that the anisotropic structure of stochastic gradients matters to deep learning, they did not expect the gradient covariance can have such an elegant mathematical structure. Our work challenges the existing belief and provides novel insights on the structure of stochastic gradients. The novel structure of stochastic gradients may help understand the success of stochastic optimization for deep learning.