MEAN FIELD ASYMPTOTICS IN HIGH-DIMENSIONAL STATISTICS: FROM EXACT RESULTS TO EFFICIENT ALGORITHMS

Modern data analysis challenges require building complex statistical models with massive numbers of parameters. It is nowadays commonplace to learn models with millions of parameters by using iterative optimization algorithms. What are typical properties of the estimated models? In some cases, the high-dimensional limit of a statistical estimator is analogous to the thermodynamic limit of a certain (disordered) statistical mechanics system. Building on mathematical ideas from the mean-field theory of disordered systems, exact asymptotics can be computed for high-dimensional statistical learning problems. This theory suggests new practical algorithms and new procedures for statistical inference. Also, it leads to intriguing conjectures about the fundamental computational limits for statistical estimation.