Dual Optimization for Kolmogorov Model Learning Using Enhanced Gradient Descent

TL;DR

This paper introduces a scalable dual optimization algorithm with enhanced gradient descent for learning the Kolmogorov model, significantly improving computational efficiency and interpretability in big data applications.

Contribution

It proposes a novel scalable KM learning algorithm using regularized dual optimization and enhanced gradient descent, with acceleration schemes for large-dimensional problems.

Findings

Achieves roughly 100x reduction in training time compared to existing methods.

Maintains comparable predictive performance on big data.

Exceeds 80% accuracy in logical relation mining for interpretability.

Abstract

Data representation techniques have made a substantial contribution to advancing data processing and machine learning (ML). Improving predictive power was the focus of previous representation techniques, which unfortunately perform rather poorly on the interpretability in terms of extracting underlying insights of the data. Recently, the Kolmogorov model (KM) was studied, which is an interpretable and predictable representation approach to learning the underlying probabilistic structure of a set of random variables. The existing KM learning algorithms using semi-definite relaxation with randomization (SDRwR) or discrete monotonic optimization (DMO) have, however, limited utility to big data applications because they do not scale well computationally. In this paper, we propose a computationally scalable KM learning algorithm, based on the regularized dual optimization combined with enhanced gradient descent (GD) method. To make our method more scalable to large-dimensional problems, we propose two acceleration schemes, namely, the eigenvalue decomposition (EVD) elimination strategy and an approximate EVD algorithm. Furthermore, a thresholding technique by exploiting the error bound analysis and leveraging the normalized Minkowski $\ell_1$-norm, is provided for the selection of the number of iterations of the approximate EVD algorithm. When applied to big data applications, it is demonstrated that the proposed method can achieve compatible training/prediction performance with significantly reduced computational complexity; roughly two orders of magnitude improvement in terms of the time overhead, compared to the existing KM learning algorithms. Furthermore, it is shown that the accuracy of logical relation mining for interpretability by using the proposed KM learning algorithm exceeds $80\%$.