上海科技大学知识管理系统

We consider a class of l1 -regularized optimization problems and the associated smooth “over -parameterized” optimization problems built upon the Hadamard parametrization, or equivalently, the Hadamard difference parametrization (HDP). We characterize the set of second -order stationary points of the HDP-based model and show that they correspond to some stationary points of the corresponding l1 -regularized model. More importantly, we show that the Kurdyka- Lojasiewicz (KL) exponent of the HDP-based model at a second -order stationary point can be inferred from that of the corresponding l1 -regularized model under suitable assumptions. Our assumptions are general enough to cover a wide variety of loss functions commonly used in l1 -regularized models, such as the least squares loss function and the logistic loss function. Since the KL exponents of many l1 -regularized models are explicitly known in the literature, our results allow us to leverage these known exponents to deduce the KL exponents at second -order stationary points of the corresponding HDP-based models, which were previously unknown. Finally, we demonstrate how these explicit KL exponents at second -order stationary points can be applied to deducing the explicit local convergence rate of a standard gradient descent method for solving the HDP-based model.