Abstract: The effectiveness of stochastic gradient descent (SGD) in neural network optimization is significantly influenced by stochastic gradient noise (SGN). Following the central limit theorem, SGN was initially described as Gaussian, but recently Simsekli et al (2019) demonstrated that the $S\alpha S$ Lévy distribution provides a better fit for the SGN. This assertion was purportedly debunked and rebounded to the Gaussian noise model that had been previously proposed. This study provides robust, comprehensive empirical evidence that SGN is heavy-tailed and is better represented by the $S\alpha S$ distribution. Our experiments include several datasets and multiple models, both discriminative and generative. Furthermore, we argue that different network parameters preserve distinct SGN properties. We develop a novel framework based on a Lévy-driven stochastic differential equation (SDE), where one-dimensional Lévy processes describe each parameter. This leads to a more accurate characterization of the dynamics of SGD around local minima. We use our framework to study SGD properties near local minima; these include the mean escape time and preferable exit directions.