A Levenberg-Marquardt method for nonsmooth regularized least squares

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We develop a Levenberg-Marquardt method for minimizing the sum of a smooth nonlinear least-squares term \(f(x) = \tfrac{1}{2} \|F(x)\|_2^2\) and a nonsmooth term \(h\). Both \(f\) and \(h\) may be nonconvex. Steps are computed by minimizing the sum of a regularized linear least-squares model and a model of \(h\) using a first-order method such as the proximal gradient method. We establish global convergence to a first-order stationary point of both a trust-region and a regularization variant of the Levenberg-Marquardt method under the assumptions that \(F\) and its Jacobian are Lipschitz continuous and \(h\) is proper and lower semi-continuous. In the worst case, both methods perform \(O(\epsilon^{-2})\) iterations to bring a measure of stationarity below \(\epsilon \in (0, 1)\). We report numerical results on three examples: a group-lasso basis-pursuit denoise example, a nonlinear support vector machine, and parameter estimation in neuron firing. For those examples to be implementable, we describe in detail how to evaluate proximal operators for separable \(h\) and for the group lasso with trust-region constraint. In all cases, the Levenberg-Marquardt methods perform fewer outer iterations than a proximal-gradient method with adaptive step length and a quasi-Newton trust-region method, neither of which exploit the least-squares structure of the problem. Our results also highlight the need for more sophisticated subproblem solvers than simple first-order methods.

, 20 pages

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