Two Analog Neural Models with the Controllability on Number of Assets for Sparse Portfolio Design

Abstract

This paper explores the use of continuous time neural networks for the sparse portfolio problem. The objective function of the sparse portfolio problem contains an ℓ1-norm regularization term. We devise two continuous time neural models for optimizing the portfolio based on the Lagrange programming neural network (LPNN) framework. In our formulation, we can control the number of the assets in the resultant portfolio, and the weighting between risk and return. To handle the non-differentiable ℓ1-norm term, the first model uses a differentiable ℓ1-norm approximation and is called LPNN-Approximation. In the second model, we combine the locally competitive algorithm (LCA) concept with the LPNN approach to handle the non-differentiable sparse portfolio optimization problem. This model is called LPNNLCA. For the LPNN-Approximation, we prove that it is global stable, and that the state of the network converges to the optimal solution of the sparse portfolio problem. For the LPNN-LCA, we prove that the equilibrium point of its dynamics corresponds to the optimal solution of the sparse portfolio problem, and that the equilibrium point is asymptotically stable. In addition, the realization of the two models is discussed. Especially, the detailed circuit realization of the thresolding element in LPNN-LCA is presented. The effectiveness of the proposed approaches is verified by the numerical experiments on three real-world datasets. Simulation results show that the two proposed models is superior to two state-of-art analog models.

Keywords: Convergence analysis, Analog Neural Network, Stability