  Information Bottleneck (IB) is a technique to extract information about one
target random variable through another relevant random variable. This technique
has garnered significant interest due to its broad applications in information
theory and deep learning. Hence, there is a strong motivation to develop
efficient numerical methods with high precision and theoretical convergence
guarantees. In this paper, we propose a semi-relaxed IB model, where the Markov
chain and transition probability condition are relaxed from the
relevance-compression function. Based on the proposed model, we develop an
algorithm, which recovers the relaxed constraints and involves only closed-form
iterations. Specifically, the algorithm is obtained by analyzing the Lagrangian
of the relaxed model with alternating minimization in each direction. The
convergence property of the proposed algorithm is theoretically guaranteed
through descent estimation and Pinsker's inequality. Numerical experiments
across classical and discrete distributions corroborate the analysis. Moreover,
our proposed algorithm demonstrates notable advantages in terms of
computational efficiency, evidenced by significantly reduced run times compared
to existing methods with comparable accuracy.
