Efficient Computation of Marton's Error Exponent via Constraint Decoupling

2025-07-26

AI Summary

Marton’s error exponent, a fundamental performance limit in source coding with side information, has long been intractable due to its underlying non-convex optimization formulation and the absence of efficient numerical solvers. Existing approaches—primarily two-dimensional grid search—suffer from high computational complexity and poor scalability. This paper proposes a constrained decoupling and composite maximization framework that decomposes the original problem into tractable convex subproblems. We design an alternating maximization algorithm integrating one-dimensional line search with convex optimization routines, enabling, for the first time, joint and globally convergent computation of both Marton’s error exponent and its inverse. Experiments on simple sources and the Ahlswede counterexample demonstrate speedups of over an order of magnitude compared to conventional grid search, significantly enhancing computational efficiency and scalability. The method provides a practical tool for performance analysis in active source coding.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Search
• Machine Learning: Information Theory

Application Category

• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

The error exponent in lossy source coding characterizes the asymptotic decay rate of error probability with respect to blocklength. The Marton's error exponent provides the theoretically optimal bound on this rate. However, computation methods of the Marton's error exponent remain underdeveloped due to its formulation as a non-convex optimization problem with limited efficient solvers. While a recent grid search algorithm can compute its inverse function, it incurs prohibitive computational costs from two-dimensional brute-force parameter grid searches. This paper proposes a composite maximization approach that effectively handles both Marton's error exponent and its inverse function. Through a constraint decoupling technique, the resulting problem formulations admit efficient solvers driven by an alternating maximization algorithm. By fixing one parameter via a one-dimensional line search, the remaining subproblem becomes convex and can be efficiently solved by alternating variable updates, thereby significantly reducing search complexity. Therefore, the global convergence of the algorithm can be guaranteed. Numerical experiments for simple sources and the Ahlswede's counterexample, demonstrates the superior efficiency of our algorithm in contrast to existing methods.

Problem

• Computing Marton's error exponent efficiently
• Solving non-convex optimization in source coding
• Reducing computational cost in error exponent calculation

Innovation

• Composite maximization approach for error exponent
• Constraint decoupling technique for convex subproblems
• Alternating maximization algorithm reducing search complexity