Nonconvex Theory of $M$-estimators with Decomposable Regularizers

High-dimensional inference addresses scenarios where the dimension of the data approaches, or even surpasses, the sample size. In these settings, the regularized $M$-estimator is a common technique for inferring parameters. (Negahban et al.,2009) establish a unified framework for establishing convergence rates in the context of high-dimensional scaling, demonstrating that estimation errors are confined within a restricted set, and revealing fast convergence rates. The key assumption underlying their work is the convexity of the loss function. However, many loss functions in high-dimensional contexts are nonconvex. This leads to the question: if the loss function is nonconvex, do estimation errors still fall within a restricted set? If yes, can we recover convergence rates of the estimation error under nonconvex situations? This paper provides affirmative answers to these critical questions.