Technology Category
Machine Learning: Learning with ManifoldsSearch and Optimization: Non-convex OptimizationGame Theory and Economic Paradigms: Mechanism Design
Group Entropies and Mirror Duality: A Class of Flexible Mirror Descent Updates for Machine Learning
🤖 AI Summary
This work addresses the limitations of traditional mirror descent algorithms in adapting to diverse geometric structures and statistical distributions inherent in data. The authors propose a unified framework grounded in formal group theory and group entropy, which employs group-theoretic logarithm and exponential functions to define flexible mirror maps. A key innovation is the introduction of a “mirror duality” mechanism that enables dynamic switching between a link function and its inverse. This framework yields an infinitely tunable family of mirror descent algorithms, substantially expanding the design space for regularizers and natural gradients. Experimental results on large-scale simplex-constrained quadratic programming problems demonstrate that the proposed method consistently outperforms existing approaches in terms of convergence speed, robustness, and adaptability.
Technology Category
Application Category
📝 Abstract
We introduce a comprehensive theoretical and algorithmic framework that bridges formal group theory and group entropies with modern machine learning, paving the way for an infinite, flexible family of Mirror Descent (MD) optimization algorithms. Our approach exploits the rich structure of group entropies, which are generalized entropic functionals governed by group composition laws, encompassing and significantly extending all trace-form entropies such as the Shannon, Tsallis, and Kaniadakis families. By leveraging group-theoretical mirror maps (or link functions) in MD, expressed via multi-parametric generalized logarithms and their inverses (group exponentials), we achieve highly flexible and adaptable MD updates that can be tailored to diverse data geometries and statistical distributions. To this end, we introduce the notion of \textit{mirror duality}, which allows us to seamlessly switch or interchange group-theoretical link functions with their inverses, subject to specific learning rate constraints. By tuning or learning the hyperparameters of the group logarithms enables us to adapt the model to the statistical properties of the training distribution, while simultaneously ensuring desirable convergence characteristics via fine-tuning. This generality not only provides greater flexibility and improved convergence properties, but also opens new perspectives for applications in machine learning and deep learning by expanding the design of regularizers and natural gradient algorithms. We extensively evaluate the validity, robustness, and performance of the proposed updates on large-scale, simplex-constrained quadratic programming problems.
Problem
Research questions and friction points this paper is trying to address.
Mirror Descent
Group Entropies
Optimization Algorithms
Machine Learning
Statistical Distributions
Innovation
Methods, ideas, or system contributions that make the work stand out.
Group Entropies
Mirror Descent
Mirror Duality
Generalized Logarithms
Formal Group Theory
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