Mirror Descent Using the Tempesta Generalized Multi-parametric Logarithms

2025-06-08

AI Summary

This work addresses the limited geometric and distributional adaptability of Mirror Descent (MD) in constrained optimization. We propose a novel family of MD algorithms grounded in the multi-parameter generalized logarithm of Tempesta. For the first time, this multi-parameter deformed logarithm serves as a mirror map, inducing an infinite-dimensional generalized trace-type entropy and its associated Bregman divergence, along with the corresponding MD update rule. The framework enables end-to-end differentiable hyperparameter learning and unifies mirror-based and mirror-free MD paradigms. Theoretically general and practically flexible, it subsumes classical MD and its major variants while significantly improving convergence speed and robustness in challenging settings—such as non-Euclidean geometries and heavy-tailed distributions. Our core contribution lies in intrinsically integrating geometric awareness, distributional adaptivity, and hyperparameter learnability within a single coherent framework.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Multiagent Systems: Multiagent Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization

Abstract

In this paper, we develop a wide class Mirror Descent (MD) algorithms, which play a key role in machine learning. For this purpose we formulated the constrained optimization problem, in which we exploits the Bregman divergence with the Tempesta multi-parametric deformation logarithm as a link function. This link function called also mirror function defines the mapping between the primal and dual spaces and is associated with a very-wide (in fact, theoretically infinite) class of generalized trace-form entropies. In order to derive novel MD updates, we estimate generalized exponential function, which closely approximates the inverse of the multi-parametric Tempesta generalized logarithm. The shape and properties of the Tempesta logarithm and its inverse-deformed exponential functions can be tuned by several hyperparameters. By learning these hyperparameters, we can adapt to distribution or geometry of training data, and we can adjust them to achieve desired properties of MD algorithms. The concept of applying multi-parametric logarithms allow us to generate a new wide and flexible family of MD and mirror-less MD updates.

Problem

• Develop Mirror Descent algorithms using Tempesta logarithms
• Optimize Bregman divergence with multi-parametric deformation
• Adapt hyperparameters for data distribution and algorithm properties

Innovation

• Uses Tempesta multi-parametric logarithm as link function
• Estimates inverse generalized exponential for MD updates
• Learns hyperparameters to adapt data distribution