# Establishing Boundary KKT Convergence of Mirror Descent through Reparameterization

arXiv: https://arxiv.org/abs/2608.07248 · HTML: https://mlsanity.com/paper/2608.07248/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 39 references that are in the map.
- First posted on arXiv 7 Aug 2026.

## Builds on (most cited first, 24 of 24)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Relatively-Smooth Convex Optimization by First-Order Methods, and Applications](https://mlsanity.com/paper/1610.05708/) (2016)
- [Linear-Time Gromov Wasserstein Distances using Low Rank Couplings and Costs](https://mlsanity.com/paper/2106.01128/) (2021)
- [Implicit Bias of Gradient Descent on Reparametrized Models: On Equivalence to Mirror Descent](https://mlsanity.com/paper/2207.04036/) (2022)
- [First Order Methods beyond Convexity and Lipschitz Gradient Continuity with Applications to Quadratic Inverse Problems](https://mlsanity.com/paper/1706.06461/) (2017)
- [On the Convergence Properties of Non-Euclidean Extragradient Methods for Variational Inequalities with Generalized Monotone Operators](https://mlsanity.com/paper/1311.2776/) (2013)
- [Hessian Riemannian gradient flows in convex programming](https://mlsanity.com/paper/1811.10331/) (2018)
- [On the Convergence Rate of Stochastic Mirror Descent for Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/1806.04781/) (2018)
- [A Convergent Single-Loop Algorithm for Relaxation of Gromov-Wasserstein in Graph Data](https://mlsanity.com/paper/2303.06595/) (2023)
- [Convergence of the Iterates in Mirror Descent Methods](https://mlsanity.com/paper/1805.01526/) (2018)
- [Hessian barrier algorithms for linearly constrained optimization problems](https://mlsanity.com/paper/1809.09449/) (2018)
- [Convergence of the Exponentiated Gradient Method with Armijo Line Search](https://mlsanity.com/paper/1712.08480/) (2017)
- [Curiosities and counterexamples in smooth convex optimization](https://mlsanity.com/paper/2001.07999/) (2020)
- [Nonconvex Stochastic Bregman Proximal Gradient Method with Application to Deep Learning](https://mlsanity.com/paper/2306.14522/) (2023)
- [Stochastic Bregman Subgradient Methods for Nonsmooth Nonconvex Optimization Problems](https://mlsanity.com/paper/2404.17386/) (2024)
- [Spurious Stationarity and Hardness Results for Bregman Proximal-Type Algorithms](https://mlsanity.com/paper/2404.08073/) (2024)
- [On exploration of an interior mirror descent flow for stochastic nonconvex constrained problem](https://mlsanity.com/paper/2507.15264/) (2025)
- [A Bregman ADMM for Bethe variational problem](https://mlsanity.com/paper/2502.04613/) (2025)
- [Global Convergence of Model Function Based Bregman Proximal Minimization Algorithms](https://mlsanity.com/paper/2012.13161/) (2020)
- [Shuffling the Stochastic Mirror Descent via Dual Lipschitz Continuity and Kernel Conditioning](https://mlsanity.com/paper/2603.16042/) (2026)
- [Non-KKT Accumulation in Entropic Mirror Descent](https://mlsanity.com/paper/2608.01658/) (2026)
- [On the Iterate Convergence of Bregman Projected Gradient Method](https://mlsanity.com/paper/2608.05035/) (2026)
- [A Unified Framework for Iterate Convergence of Bregman Proximal Methods](https://mlsanity.com/paper/2608.05536/) (2026)

## Abstract (arXiv)

Sequence convergence to a boundary Karush--Kuhn--Tucker (KKT) point has long remained unclear for nonconvex mirror descent with Legendre kernels. The difficulty arises from the blow-up of the gradient of the Legendre kernel at the boundary. Recent work~citedingtoh2026nonkkt shows that mirror descent can accumulate at non-KKT boundary points despite decreasing objective values, precluding a convergence guarantee to KKT points in general. Despite this negative result, mirror descent remains effective in many real applications. Motivated by this contrast, we address the boundary difficulty directly and establish KKT convergence of mirror descent for a broad class of structured nonconvex problems. We analyze mirror descent in reparameterized variables, where the Hessian metric is flattened and remains nondegenerate as the boundary is approached. Under extension and definability conditions jointly coupling the objective, the Legendre kernel, and the feasible region, the reparameterized sequence has finite length and converges, thereby recovering convergence to a KKT point of the original sequence. Our general framework applies to some concrete instances: Shannon entropy, Fermi--Dirac entropy, and power kernels on polyhedron.