A scaled gradient projection method for Bayesian learning in dynamical systems

TL;DR

This paper introduces a scaled gradient projection method for Bayesian system identification, efficiently solving nonconvex optimization problems to select optimal model classes with high accuracy and computational speed.

Contribution

It proposes a novel scaled gradient projection algorithm with a tailored scaling matrix and step size for Bayesian model selection in dynamical systems.

Findings

Method achieves solutions in a fraction of a second.

Results are comparable to state-of-the-art approaches.

Algorithm is adaptable to various machine learning and signal processing problems.

Abstract

A crucial task in system identification problems is the selection of the most appropriate model class, and is classically addressed resorting to cross-validation or using asymptotic arguments. As recently suggested in the literature, this can be addressed in a Bayesian framework, where model complexity is regulated by few hyperparameters, which can be estimated via marginal likelihood maximization. It is thus of primary importance to design effective optimization methods to solve the corresponding optimization problem. If the unknown impulse response is modeled as a Gaussian process with a suitable kernel, the maximization of the marginal likelihood leads to a challenging nonconvex optimization problem, which requires a stable and effective solution strategy. In this paper we address this problem by means of a scaled gradient projection algorithm, in which the scaling matrix and the steplength parameter play a crucial role to provide a meaning solution in a computational time comparable with second order methods. In particular, we propose both a generalization of the split gradient approach to design the scaling matrix in the presence of box constraints, and an effective implementation of the gradient and objective function. The extensive numerical experiments carried out on several test problems show that our method is very effective in providing in few tenths of a second solutions of the problems with accuracy comparable with state-of-the-art approaches. Moreover, the flexibility of the proposed strategy makes it easily adaptable to a wider range of problems arising in different areas of machine learning, signal processing and system identification.