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The gradient noise (GN) in the stochastic gradient descent (SGD) algorithm is often considered to be Gaussian in the large data regime by assuming that the classical central limit theorem (CLT) kicks in. This assumption is often made for mathematical convenience, since it enables SGD to be analyzed as a stochastic differential equation (SDE) driven by a Brownian motion. We argue that the Gaussianity assumption might fail to hold in deep learning settings and hence render the Brownian motion-based analyses inappropriate. Inspired by non-Gaussian natural phenomena, we consider the GN in a more general context and invoke the generalized CLT (GCLT), which suggests that the GN converges to a heavy-tailed -stable random variable. Accordingly, we propose to analyze SGD as an SDE driven by a Lévy motion. Such SDEs can incur ‘jumps’, which force the SDE transition from narrow minima to wider minima, as proven by existing metastability theory. To validate the -stable assumption, we conduct extensive experiments on common deep learning architectures and show that in all settings, the GN is highly non-Gaussian and admits heavy-tails. We further investigate the tail behavior in varying network architectures and sizes, loss functions, and datasets. Our results open up a different perspective and shed more light on the belief that SGD prefers wide minima. 

Despite the phenomenal success of deep neural networks in a broad range of learning tasks, there is a lack of theory to understand the way they work. In particular, Convolutional Neural Networks (CNNs) are known to perform much better than Fully-Connected Networks (FCNs) on spatially structured data: the architectural structure of CNNs benefits from prior knowledge on the features of the data, for instance their translation invariance. The aim of this work is to understand this fact through the lens of dynamics in the loss landscape. We introduce a method that maps a CNN to its equivalent FCN (denoted as eFCN). Such an embedding enables the comparison of CNN and FCN training dynamics directly in the FCN space. We use this method to test a new training protocol, which consists in training a CNN, embedding it to FCN space at a certain relax time'', then resuming the training in FCN space. We observe that for all relax times, the deviation from the CNN subspace is small, and the final performance reached by the eFCN is higher than that reachable by a standard FCN of same architecture. More surprisingly, for some intermediate relax times, the eFCN outperforms the CNN it stemmed, by combining the prior information of the CNN and the expressivity of the FCN in a complementary way. The practical interest of our protocol is limited by the very large size of the highly sparse eFCN. However, it offers interesting insights into the persistence of architectural bias under stochastic gradient dynamics. It shows the existence of some rare basins in the FCN loss landscape associated with very good generalization. These can only be accessed thanks to the CNN prior, which helps navigate the landscape during the early stages of optimization. 

Abstract Academic researchers, government agencies, industry groups, and individuals have produced forecasts at an unprecedented scale during the COVID-19 pandemic. To leverage these forecasts, the United States Centers for Disease Control and Prevention (CDC) partnered with an academic research lab at the University of Massachusetts Amherst to create the US COVID-19 Forecast Hub. Launched in April 2020, the Forecast Hub is a dataset with point and probabilistic forecasts of incident cases, incident hospitalizations, incident deaths, and cumulative deaths due to COVID-19 at county, state, and national, levels in the United States. Included forecasts represent a variety of modeling approaches, data sources, and assumptions regarding the spread of COVID-19. The goal of this dataset is to establish a standardized and comparable set of short-term forecasts from modeling teams. These data can be used to develop ensemble models, communicate forecasts to the public, create visualizations, compare models, and inform policies regarding COVID-19 mitigation. These open-source data are available via download from GitHub, through an online API, and through R packages.