Heuristic parameter-choice rules for convex variational regularization based on error estimates

In this paper, we are interested in heuristic parameter choice rules for general convex variational regularization which are based on error estimates. Two such rules are derived and generalize those from quadratic regularization, namely the Hanke-Raus rule and quasi-optimality criterion. A posteriori error estimates are shown for the Hanke-Raus rule, and convergence for both rules is also discussed. Numerical results for both rules are presented to illustrate their applicability

A new approach to nonlinear constrained Tikhonov regularization

We present a novel approach to nonlinear constrained Tikhonov regularization from the viewpoint of optimization theory. A second-order sufficient optimality condition is suggested as a nonlinearity condition to handle the nonlinearity of the forward operator. The approach is exploited to derive convergence rates results for a priori as well as a posteriori choice rules, e.g., discrepancy principle and balancing principle, for selecting the regularization parameter. The idea is further illustrated on a general class of parameter identification problems, for which (new) source and nonlinearity conditions are derived and the structural property of the nonlinearity term is revealed. A number of examples including identifying distributed parameters in elliptic differential equations are presented.Comment: 21 pages, to appear in Inverse Problem

On the Regularizing Property of Stochastic Gradient Descent

Stochastic gradient descent is one of the most successful approaches for solving large-scale problems, especially in machine learning and statistics. At each iteration, it employs an unbiased estimator of the full gradient computed from one single randomly selected data point. Hence, it scales well with problem size and is very attractive for truly massive dataset, and holds significant potentials for solving large-scale inverse problems. In the recent literature of machine learning, it was empirically observed that when equipped with early stopping, it has regularizing property. In this work, we rigorously establish its regularizing property (under \textit{a priori} early stopping rule), and also prove convergence rates under the canonical sourcewise condition, for minimizing the quadratic functional for linear inverse problems. This is achieved by combining tools from classical regularization theory and stochastic analysis. Further, we analyze the preasymptotic weak and strong convergence behavior of the algorithm. The theoretical findings shed insights into the performance of the algorithm, and are complemented with illustrative numerical experiments.Comment: 22 pages, better presentatio

On the regularizing property of stochastic gradient descent

Stochastic gradient descent (SGD) and its variants are among the most successful approaches for solving large-scale optimization problems. At each iteration, SGD employs an unbiased estimator of the full gradient computed from one single randomly selected data point. Hence, it scales well with problem size and is very attractive for handling truly massive dataset, and holds significant potentials for solving large-scale inverse problems. In this work, we rigorously establish its regularizing property under a priori early stopping rule for linear inverse problems, and also prove convergence rates under the canonical sourcewise condition. This is achieved by combining tools from classical regularization theory and stochastic analysis. Further, we analyze its preasymptotic weak and strong convergence behavior, in order to explain the fast initial convergence typically observed in practice. The theoretical findings shed insights into the performance of the algorithm, and are complemented with illustrative numerical experiments