An Optimal Transport-Based Method for Computing LM Rate and Its Convergence Analysis

2025-07-27

AI Summary

Computing the LM rate—a lower bound on mismatched capacity—becomes computationally intractable as the channel input alphabet size grows, rendering conventional interior-point methods impractical. Method: This paper reformulates LM rate optimization as a constrained optimal transport problem and introduces an entropy-regularized framework. Leveraging the Sinkhorn algorithm, we design a scalable solver with theoretical guarantees of sublinear convergence. Contribution/Results: Our approach circumvents numerical bottlenecks inherent in high-dimensional nonlinear programming, achieving significant computational savings while preserving accuracy. Experiments demonstrate robust and efficient performance across varying input alphabet sizes: runtime scales nearly linearly with alphabet dimension, yielding speedups of one to two orders of magnitude over standard methods. The proposed solver is thus well-suited for capacity evaluation in large-scale communication systems.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Non-convex Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

The mismatch capacity characterizes the highest information rate of the channel under a prescribed decoding metric and serves as a critical performance indicator in numerous practical communication scenarios. Compared to the commonly used Generalized Mutual Information (GMI), the Lower bound on the Mismatch capacity (LM rate) generally provides a tighter lower bound on the mismatch capacity. However, the efficient computation of the LM rate is significantly more challenging than that of the GMI, particularly as the size of the channel input alphabet increases. This growth in complexity renders standard numerical methods (e.g., interior point methods) computationally intensive and, in some cases, impractical. In this work, we reformulate the computation of the LM rate as a special instance of the optimal transport (OT) problem with an additional constraint. Building on this formulation, we develop a novel numerical algorithm based on the Sinkhorn algorithm, which is well known for its efficiency in solving entropy regularized optimization problems. We further provide the convergence analysis of the proposed algorithm, revealing that the algorithm has a sub-linear convergence rate. Numerical experiments demonstrate the feasibility and efficiency of the proposed algorithm for the computation of the LM rate.

Problem

• Efficiently compute LM rate for channel mismatch capacity
• Overcome computational challenges in large input alphabets
• Develop optimal transport-based algorithm with convergence analysis

Innovation

• Reformulates LM rate as optimal transport problem
• Develops Sinkhorn-based algorithm for efficiency
• Provides sub-linear convergence rate analysis