# Implicit Bias and Convergence of Matrix Stochastic Mirror Descent

arXiv: https://arxiv.org/abs/2602.18997 · HTML: https://mlsanity.com/paper/2602.18997/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 15 references that are in the map.
- First posted on arXiv 22 Feb 2026.

## Builds on (most cited first, 6 of 6)

- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Characterizing Implicit Bias in Terms of Optimization Geometry](https://mlsanity.com/paper/1802.08246/) (2018)
- [Stochastic Mirror Descent on Overparameterized Nonlinear Models: Convergence, Implicit Regularization, and Generalization](https://mlsanity.com/paper/1906.03830/) (2019)
- [Stochastic Gradient/Mirror Descent: Minimax Optimality and Implicit Regularization](https://mlsanity.com/paper/1806.00952/) (2018)
- [Mirror Descent Maximizes Generalized Margin and Can Be Implemented Efficiently](https://mlsanity.com/paper/2205.12808/) (2022)
- [Stochastic Mirror Descent: Convergence Analysis and Adaptive Variants via the Mirror Stochastic Polyak Stepsize](https://mlsanity.com/paper/2110.15412/) (2021)

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

- [Dual Space Preconditioning for Gradient Descent in the Overparameterized Regime](https://mlsanity.com/paper/2603.10485/) (2026)

## Abstract (arXiv)

We investigate Stochastic Mirror Descent (SMD) with matrix parameters and vector-valued predictions, a framework relevant to multi-class classification and matrix completion problems. Focusing on the overparameterized regime, where the total number of parameters exceeds the number of training samples, we prove that SMD with matrix mirror functions psi(·) converges exponentially to a global interpolator. Furthermore, we generalize classical implicit bias results of vector SMD by demonstrating that the matrix SMD algorithm converges to the unique solution minimizing the Bregman divergence induced by psi(·) from initialization subject to interpolating the data. These findings reveal how matrix mirror maps dictate inductive bias in high-dimensional, multi-output problems.