A Dynamical Systems Approach for Convergence of the Bayesian EM Algorithm

• URL: http://arxiv.org/abs/2006.12690v2
• Date: Fri, 12 Feb 2021 16:23:54 GMT
• Title: A Dynamical Systems Approach for Convergence of the Bayesian EM Algorithm
• Authors: Orlando Romero, Subhro Das, Pin-Yu Chen, S\'ergio Pequito
• Abstract summary: We show how (discrete-time) Lyapunov stability theory can serve as a powerful tool to aid, or even lead, in the analysis (and potential design) of optimization algorithms that are not necessarily gradient-based. The particular ML problem that this paper focuses on is that of parameter estimation in an incomplete-data Bayesian framework via the popular optimization algorithm known as maximum a posteriori expectation-maximization (MAP-EM) We show that fast convergence (linear or quadratic) is achieved, which could have been difficult to unveil without our adopted S&C approach.
• Score: 59.99439951055238
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Out of the recent advances in systems and control (S\&C)-based analysis of optimization algorithms, not enough work has been specifically dedicated to machine learning (ML) algorithms and its applications. This paper addresses this gap by illustrating how (discrete-time) Lyapunov stability theory can serve as a powerful tool to aid, or even lead, in the analysis (and potential design) of optimization algorithms that are not necessarily gradient-based. The particular ML problem that this paper focuses on is that of parameter estimation in an incomplete-data Bayesian framework via the popular optimization algorithm known as maximum a posteriori expectation-maximization (MAP-EM). Following first principles from dynamical systems stability theory, conditions for convergence of MAP-EM are developed. Furthermore, if additional assumptions are met, we show that fast convergence (linear or quadratic) is achieved, which could have been difficult to unveil without our adopted S\&C approach. The convergence guarantees in this paper effectively expand the set of sufficient conditions for EM applications, thereby demonstrating the potential of similar S\&C-based convergence analysis of other ML algorithms.

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