Efficient first-order methods for structured sparse optimization and their financial applications

Abstract

This thesis aims to propose several algorithms for structured sparse optimization problems based on the iterative thresholding framework, where some specific struc­tural patterns are considered with sparsity in the solution. For instance, group sparse optimization incorporates intra-group sparsity with provided group informa­tion. The convergence guarantees of the proposed algorithms are investigated, and several numerical experiments demonstrate their efficiencies. Finally, the proposed algorithms are applied to solve the structured sparse optimization problems arising in financial applications. The empirical results illustrate the excellent performance of our methods. For the structured sparse optimization in compressed sensing, we propose two algorithms based on the iterative thresholding framework to find a structured sparse solution constrained in a polyhedral set. Specifically, the proposed algorithms in­clude a structured thresholding operator to cope with the combinatorial constraints. This thresholding operator comprises a hard thresholding operator and a group hard thresholding operator, which are successively performed or vice versa to achieve the structured sparsity. Also, a subproblem for debiasing the estimation is employed to address the polyhedral constraint, which also can be achieved by projection. Under the assumption of restricted isometry property, we demonstrate that the sequences generated by the proposed algorithms approximate the ground true solution with a high probability. Numerical results demonstrate the strongly structured-sparse­-promoting capability of the proposed algorithms by comparing them with state-of-the-art algorithms. The application to enhanced indexation shows the excellent performance of the proposed algorithms with high annualized excess returns. We generalize the structured sparse optimization to the scenarios with a convex and smooth function and propose two algorithms under the same framework to solve it. The convergence guarantees of the proposed algorithms are illustrated under the assumption of restricted strong convexity and smoothness. Specifically, we consider the application to the logistic regression and the mean-variance portfolio selection with a high probability of achieving convergence. Numerical experiments illustrate that the proposed algorithms perform better than state-of-the-art algorithms in the logistic regression problem on simulated data. The application to mean-variance portfolio selection indicates the superiority of considering the sector/industry infor­mation. Finally, we propose a structured sparse graphical model to infer the financial contagion network based on the geo-information. The proposed optimization problem consists of the combinatorial constraints associated with the number of edges inside and outside the community. We develop two algorithms with a community-based thresholding operator to meet these combinatorial constraints. The debiasing step is considered to ensure the positive definiteness of the numerical solution, which can also be achieved by modifying its diagonal entries. Generalizing the restricted strong convexity and smoothness for the graphical loss, we prove that the sequences generated by the proposed algorithms approximate the ground true solution with error bound under the assumption. Numerical experiments show the effectiveness of the proposed algorithms in network inference. Furthermore, the empirical results of the financial contagion network reveal the channels of risk transmission during and after the crisis. Banks are affected more by those banks in the same community, and such an effect is enlarged after the crisis.