Efficient and Provably Convergent Computation of Information Bottleneck

Core Concepts

The core message of this work is to propose an accurate, efficient and convergence-guaranteed algorithm for computing the relevance-compression (RI) function of the Information Bottleneck (IB) problem by introducing a semi-relaxed IB model.

Abstract

The paper proposes a semi-relaxed IB model by relaxing the Markov chain and transition probability constraints from the original IB formulation. Based on this semi-relaxed model, the authors develop an Alternating Bregman Projection (ABP) algorithm that recovers the relaxed constraints through an alternating minimization framework. The key highlights of the paper are: The semi-relaxed IB model simplifies the structure of the mutual information constraints and the objective function, while proving the equivalence of the solution to the original IB model. The ABP algorithm involves only closed-form iterations in updating the primal variables, ensuring computational efficiency. The descent of the objective function can be precisely estimated in each iteration, leading to provable convergence guarantees. Numerical experiments on classical distributions and a real-world dataset demonstrate that the proposed ABP algorithm outperforms existing methods in terms of computational efficiency and accuracy, especially in cases with phase transition phenomena. The convergence analysis shows that the sequence generated by the ABP algorithm converges to a local minimum of the semi-relaxed IB model.

Stats

The paper does not contain any explicit numerical data or statistics to support the key claims. The results are presented in the form of computational time comparisons and convergence trajectories.

Quotes

"To address the aforementioned difficulty, in this work, we propose an accurate, efficient and convergence-guaranteed algorithm for computing the RI function." "The proposed algorithm is based on a new IB problem formulation by relaxing the Markov chain and the inherent one-side marginal distribution representations, for which we term it a semi-relaxed IB model." "The descent value in each iteration is precisely calculated and estimated through the Pinsker's inequality with our proposed algorithm."

Key Insights Distilled From

Efficient and Provably Convergent Computation of Information Bottleneck

by Lingyi Chen,... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.04862.pdf

Deeper Inquiries

How can the proposed semi-relaxed IB model be extended to handle continuous distributions or high-dimensional data

The extension of the proposed semi-relaxed IB model to handle continuous distributions or high-dimensional data involves adapting the formulation and algorithm to accommodate the characteristics of such data types. For continuous distributions, the discretization of the continuous space into intervals or bins can be a viable approach. This discretization allows for the application of the same principles used in the discrete case, with adjustments made to account for the continuous nature of the data. In the case of high-dimensional data, the extension would involve scaling the algorithm to handle a larger number of dimensions efficiently. This may require optimizations in the computational processes, memory management, and convergence analysis to ensure that the algorithm remains effective and scalable in high-dimensional spaces. Additionally, techniques like dimensionality reduction or feature selection may be employed to reduce the complexity of the data representation while preserving relevant information. Overall, the extension to continuous distributions or high-dimensional data would involve modifications in the formulation, algorithm design, and convergence analysis to suit the specific characteristics and challenges posed by these types of data.

What are the potential limitations or drawbacks of the relaxation approach used in the semi-relaxed IB model

While the relaxation approach used in the semi-relaxed IB model offers advantages in terms of simplifying the problem structure and reducing computational complexity, there are potential limitations and drawbacks to consider: Loss of Precision: Relaxing the Markov chain and transition probability constraints may lead to a loss of precision in the extracted information. By relaxing these constraints, the model may not capture the full complexity of the relationships between variables, potentially affecting the quality of the extracted information. Feasibility Concerns: The relaxation approach may result in solutions that do not fully satisfy the original constraints of the IB problem. This could lead to suboptimal solutions or solutions that do not align with the desired properties of the information bottleneck. Generalization Challenges: The relaxation approach may not generalize well to all types of data distributions or scenarios. The simplified constraints may not be suitable for complex data structures or relationships, limiting the applicability of the model in diverse settings. Convergence Issues: Relaxing constraints could impact the convergence properties of the algorithm. The trade-off between relaxation and convergence guarantees needs to be carefully balanced to ensure that the algorithm converges to meaningful solutions reliably. Overall, while the relaxation approach offers computational advantages, it is essential to consider these limitations and drawbacks when applying the semi-relaxed IB model in practice.

Can the ideas behind the ABP algorithm be applied to solve other information-theoretic optimization problems beyond the IB framework

The ideas behind the Alternating Bregman Projection (ABP) algorithm can indeed be applied to solve other information-theoretic optimization problems beyond the Information Bottleneck (IB) framework. The ABP algorithm's key features, such as alternating minimization, Bregman projection, and convergence analysis, are general optimization techniques that can be adapted to various information-theoretic problems. Some potential applications of the ABP algorithm in other contexts include: Rate-Distortion Problems: The ABP algorithm can be utilized to solve rate-distortion optimization problems, where the goal is to minimize the distortion in representing a source signal subject to a rate constraint. By formulating the problem appropriately and applying the ABP algorithm, efficient solutions can be obtained. Channel Coding: In the context of channel coding, the ABP algorithm can be used to optimize coding schemes that maximize information transmission rates while ensuring reliable communication over noisy channels. By incorporating channel constraints and utilizing the ABP framework, optimal coding strategies can be developed. Source Coding: For source coding tasks, where the goal is to compress data efficiently while preserving relevant information, the ABP algorithm can aid in finding optimal compression schemes. By adapting the algorithm to handle source coding constraints, it can facilitate the design of effective coding strategies. Machine Learning: The ABP algorithm's principles can be applied to various machine learning problems that involve information-theoretic constraints, such as feature selection, model compression, and representation learning. By leveraging the ABP framework, these machine learning tasks can be optimized effectively. In essence, the ABP algorithm's versatility and effectiveness make it a valuable tool for solving a wide range of information-theoretic optimization problems beyond the IB framework. Its adaptability and robustness make it a promising approach for addressing diverse challenges in information theory and related fields.