关键词：局部最优；理论模型；目标函数；线性模型
摘 要：We establish theoretical results concerning all local optima of various regularized M- estimators, where both loss and penalty functions are allowed to be nonconvex. Our results show that as long as the loss function satis es restricted strong convexity and the penalty function satis es suitable regularity conditions, any local optimum of the composite objective function lies within statistical precision of the true parameter vector. Our theory covers a broad class of nonconvex objective functions, including corrected versions of the Lasso for error-in-variables linear models; regression in generalized linear models using nonconvex regularizers such as SCAD and MCP; and graph and inverse covariance matrix estimation. On the optimization side, we show that a simple adaptation of composite gradient descent may be used to compute a global optimum up to the statistical precision stat in log(1=stat) iterations, which is the fastest possible rate of any rst-order method. We provide a variety of simulations to illustrate the sharpness of our theoretical predictions.