Avoiding strict saddle points of nonconvex regularized problems

In this paper, we consider a class of non-convex and non-smooth sparse optimization problems, which encompass most existing nonconvex sparsity-inducing terms. We show the second-order optimality conditions only depend on the nonzeros of the stationary points. We propose two damped iterative reweighted algorithms including the iteratively reweighted $\ell_1$ algorithm (DIRL$_1$) and the iteratively reweighted $\ell_2$ (DIRL$_2$) algorithm, to solve these problems. For DIRL$_1$, we show the reweighted $\ell_1$ subproblem has support identification property so that DIRL$_1$ locally reverts to a gradient descent algorithm around a stationary point. For DIRL$_2$, we show the solution map of the reweighted $\ell_2$ subproblem is differentiable and Lipschitz continuous everywhere. Therefore, the map of DIRL$_1$ and DIRL$_2$ and their inverse are Lipschitz continuous, and the strict saddle points are their unstable fixed points. By applying the stable manifold theorem, these algorithms are shown to converge only to local minimizers with randomly initialization when the strictly saddle point property is assumed.