Adaptive Alternating Minimization Algorithms

The classical alternating minimization (or projection) algorithm has been successful in the context of solving optimization problems over two variables or equivalently of finding a point in the intersection of two sets. 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. Our result is a generalization of the work by Csiszar and Tusnady on alternating minimization procedures. We present applications of our results to adaptive decomposition of mixtures, adaptive log-optimal portfolio selection, and adaptive filter design.