Nonconvex sparse regularization for deep neural networks and its optimality
This work addresses a computational and theoretical bottleneck in sparse deep learning, offering an incremental improvement for practitioners needing efficient and adaptive regularization.
The paper tackles the impracticality of sparsity constraints in deep neural networks by proposing a penalized estimation method that adaptively achieves minimax convergence rates for nonparametric regression without requiring prior knowledge of the true model.
Recent theoretical studies proved that deep neural network (DNN) estimators obtained by minimizing empirical risk with a certain sparsity constraint can attain optimal convergence rates for regression and classification problems. However, the sparsity constraint requires to know certain properties of the true model, which are not available in practice. Moreover, computation is difficult due to the discrete nature of the sparsity constraint. In this paper, we propose a novel penalized estimation method for sparse DNNs, which resolves the aforementioned problems existing in the sparsity constraint. We establish an oracle inequality for the excess risk of the proposed sparse-penalized DNN estimator and derive convergence rates for several learning tasks. In particular, we prove that the sparse-penalized estimator can adaptively attain minimax convergence rates for various nonparametric regression problems. For computation, we develop an efficient gradient-based optimization algorithm that guarantees the monotonic reduction of the objective function.