Adaptive Alternating Minimization Algorithms

TL;DR

This paper introduces an adaptive version of the alternating minimization algorithm, providing convergence conditions applicable to dynamic, real-world problems where parameters change over time.

Contribution

It establishes minimal sufficient conditions for the convergence of adaptive alternating minimization algorithms, extending classical fixed-parameter results to dynamic settings.

Findings

Provided a general set of convergence conditions for adaptive algorithms

Applied results to adaptive mixture decomposition, portfolio selection, and filter design

Demonstrated the minimality of the convergence conditions

Abstract

The classical alternating minimization (or projection) algorithm has been successful in the context of solving optimization problems over two variables. The iterative nature and simplicity of the algorithm has led to its application to many areas such as signal processing, information theory, control, and finance. A general set of sufficient conditions for the convergence and correctness of the algorithm is quite well-known when the underlying problem parameters are fixed. In many practical situations, however, the underlying problem parameters are changing over time, and the use of an adaptive algorithm is more appropriate. In this paper, we study such an adaptive version of the alternating minimization algorithm. As a main result of this paper, we provide a general set of sufficient conditions for the convergence and correctness of the adaptive algorithm. Perhaps surprisingly, these conditions seem to be the minimal ones one would expect in such an adaptive setting. We present applications of our results to adaptive decomposition of mixtures, adaptive log-optimal portfolio selection, and adaptive filter design.