An adaptive ADMM with regularized spectral penalty for sparse portfolio selection
Xin Xu
TL;DR
The paper addresses over-fitting and sparsity in mean-variance portfolio optimization by adding an $\ell_{1}$ regularization term to the MV objective. It develops an adaptive framework consisting of an initial parameter $\lambda_0$ based on $m$ and $n$ and a short-sale control that updates $\lambda_k$ via $\lambda_{k+1} = (s_m / s_n) \lambda_k$, together with a regularized Barzilai-Borwein (RBB) penalty for ADMM, where $\rho_k^{RBB} = 1/\sqrt{\alpha_k^{RBB}\beta_k^{RBB}}$. To solve the regularized problem efficiently, it introduces the RBB penalty derived from a regularized LS problem with parameter $\tau_k$, linking step size to residual behavior. Numerical experiments validate that the combined RBB penalty and adaptive regularization improve sparsity, convergence speed, and robustness of the portfolio solutions.
Abstract
The mean-variance (MV) model is the core of modern portfolio theory. Nevertheless, it suffers from the over-fitting problem due to the estimation errors of model parameters. We consider the $\ell_{1}$ regularized MV model, which adds an $\ell_{1}$ regularization term in the objective to prevent over-fitting and promote sparsity of solutions. By investigating the relationship between sample size and over-fitting, we propose an initial regularization parameter scheme in the $\ell_{1}$ regularized MV model. Then we propose an adaptive parameter tuning strategy to control the amount of short sales. ADMM is a well established algorithm whose performance is affected by the penalty parameter. In this paper, a penalty parameter scheme based on regularized Barzilai-Borwein step size is proposed, and the modified ADMM is used to solve the $\ell_{1}$ regularized MV problem. Numerical results verify the effectiveness of the two types of parameters proposed in this paper.