# Mirror Descent Using the Tempesta Generalized Multi-parametric Logarithms

arXiv: https://arxiv.org/abs/2506.13984 · HTML: https://mlsanity.com/paper/2506.13984/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 35 references that are in the map.
- First posted on arXiv 8 Jun 2025.

## Builds on (most cited first, 10 of 10)

- [Public Data-Assisted Mirror Descent for Private Model Training](https://mlsanity.com/paper/2112.00193/) (2021)
- [Stochastic Mirror Descent on Overparameterized Nonlinear Models: Convergence, Implicit Regularization, and Generalization](https://mlsanity.com/paper/1906.03830/) (2019)
- [Exponentiated Gradient Meets Gradient Descent](https://mlsanity.com/paper/1902.01903/) (2019)
- [Mirrorless Mirror Descent: A Natural Derivation of Mirror Descent](https://mlsanity.com/paper/2004.01025/) (2020)
- [Exponentiated Gradient Reweighting for Robust Training Under Label Noise and Beyond](https://mlsanity.com/paper/2104.01493/) (2021)
- [Robust Imitation via Mirror Descent Inverse Reinforcement Learning](https://mlsanity.com/paper/2210.11201/) (2022)
- [Generalized Exponentiated Gradient Algorithms and Their Application to On-Line Portfolio Selection](https://mlsanity.com/paper/2406.00655/) (2024)
- [A Mirror Descent Perspective of Smoothed Sign Descent](https://mlsanity.com/paper/2410.14158/) (2024)
- [Generalized Euler Logarithm and its Applications in Machine Learning: Natural Gradient, Backpropagation, Generalized EG, Mirror Descent and OLPS](https://mlsanity.com/paper/2502.17500/) (2025)
- [Variational Online Mirror Descent for Robust Learning in Schr"odinger Bridge](https://mlsanity.com/paper/2504.02618/) (2025)

## Built on by (most cited first, 1 of 1)

- [Group Entropies and Mirror Duality: A Class of Flexible Mirror Descent Updates for Machine Learning](https://mlsanity.com/paper/2603.08651/) (2026)

## Abstract (arXiv)

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.