Abstract: Deep learning has enjoyed huge empirical success in recent years. Although training a deep neural network is a highly nonconvex optimization problem,simple (stochastic) gradient methods are able to produce good solutions that minimize the training error, and more surprisingly, can generalize well to out-of sample data, even when the number of parameters is significantly larger than t...
Abstract: We prove the mean field limit and quantitative estimates for many-particle systems with singular attractive interactions between particles. As an important example, a full rigorous derivation (with quantitative estimates) of the Patlak-Keller-Segel model in optimal subcritical regimes is obtained for the first time. To give an answer to this longstanding problem, we take advantage of ...