Multi-Resolution Online Deterministic Annealing: A Hierarchical and Progressive Learning Architecture

Hierarchical learning algorithms that gradually approximate a solution to a data-driven optimization problem are essential to decision-making systems, especially under limitations on time and computational resources. In this study, we introduce a general-purpose hierarchical learning architecture that is based on the progressive partitioning of a possibly multi-resolution data space. The optimal partition is gradually approximated by solving a sequence of optimization sub-problems that yield a sequence of partitions with increasing number of subsets. We show that the solution of each optimization problem can be estimated online using gradient-free stochastic approximation updates. As a consequence, a function approximation problem can be defined within each subset of the partition and solved using the theory of two-timescale stochastic approximation algorithms. This simulates an annealing process and defines a robust and interpretable heuristic method to gradually increase the complexity of the learning architecture in a task-agnostic manner, giving emphasis to regions of the data space that are considered more important according to a predefined criterion. Finally, by imposing a tree structure in the progression of the partitions, we provide a means to incorporate potential multi-resolution structure of the data space into this approach, significantly reducing its complexity, while introducing hierarchical feature extraction properties similar to certain classes of deep learning architectures. Asymptotic convergence analysis and experimental results are provided for clustering, classification, and regression problems.