Theoretically exploring the advantages of neural networks might be one of the most challenging problems in the AI era. An adaptive feature program has recently been proposed to analyze the feature learning characteristic property of neural networks in a more abstract way. Motivated by the celebrated Le Cam equivalence, we advocate the over-parametrized sequence models to further simplify the analysis of the training dynamics of adaptive feature program and present several supporting evidences for the adaptive feature program. More precisely, after having introduced the feature error measure (FEM) to characterize the quality of the learned feature, we show that the FEM is decreasing during the training process of several concrete adaptive feature models including linear regression, single/multiple index models, etc. We believe that this hints at the potential successes of the adaptive feature program.

Non-convex and non-smooth optimization plays an important role in machine learning. Proximal gradient method is one of the most important methods for solving the nonconvex and non-smooth problems, where a proximal operator need to be solved exactly for each step. However, in a lot of problems the proximal operator does not have an analytic solution, or is expensive to obtain an exact solution. In this paper, we propose inexact proximal gradient methods (not only a basic inexact proximal gradient method (IPG), but also a Nesterov's accelerated inexact proximal gradient method (AIPG)) for non-convex and non-smooth optimization, which tolerate an error in the calculation of the proximal operator. Theoretical analysis shows that IPG and AIPG have the same convergence rates as in the error-free case, provided that the errors decrease at appropriate rates. Keywords: Non-convex optimization, non-smooth optimization, proximal gradient, inexact proximal operator, Nesterov's accelerated method