# Non-KKT Accumulation in Entropic Mirror Descent

arXiv: https://arxiv.org/abs/2608.01658 · HTML: https://mlsanity.com/paper/2608.01658/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 36 references that are in the map.
- First posted on arXiv 3 Aug 2026.

## Builds on (most cited first, 17 of 17)

- [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)
- [Learning Latent Permutations with Gumbel-Sinkhorn Networks](https://mlsanity.com/paper/1802.08665/) (2018)
- [Policy Mirror Descent for Reinforcement Learning: Linear Convergence, New Sampling Complexity, and Generalized Problem Classes](https://mlsanity.com/paper/2102.00135/) (2021)
- [Relatively-Smooth Convex Optimization by First-Order Methods, and Applications](https://mlsanity.com/paper/1610.05708/) (2016)
- [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)
- [Taming Nonconvex Stochastic Mirror Descent with General Bregman Divergence](https://mlsanity.com/paper/2402.17722/) (2024)
- [Convergence of the Iterates in Mirror Descent Methods](https://mlsanity.com/paper/1805.01526/) (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 Proximal Gradient Method Revisited: Kernel Conditioning and Painless Variance Reduction](https://mlsanity.com/paper/2401.03155/) (2024)
- [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)

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

- [Establishing Boundary KKT Convergence of Mirror Descent through Reparameterization](https://mlsanity.com/paper/2608.07248/) (2026)

## Abstract (arXiv)

For mirror descent generated by a Legendre kernel, perhaps one of the most basic question in optimization is this: must every accumulation point of a bounded mirror descent sequence be Karush--Kuhn--Tucker (KKT) stationary under proper stepsizes? We show that the answer is no. A longstanding obstacle to resolving this question is the boundary blow-up of the Legendre gradient: it keeps every mirror step in the interior, while at a boundary limit, the inverse entropy metric vanishes on active coordinates and can erase the dual-feasibility in the KKT system. We construct C^∞ objectives and bounded sequences generated by the Shannon-entropic mirror descent on the nonnegative orthant R_+n, for every n≥ 3, and on the probability simplex Deltan, for every n≥ 4, such that, in each case, the set of accumulation points is a smooth boundary circle containing a nonempty relatively open arc of non-KKT points. The steps satisfy αkasymp k-β with β∈(1/2,1), the objective values are nonincreasing, and the objectives are entropy-relatively smooth. Hence the pathology stems from the degeneracy of the Bregman geometry at the boundary, rather than from failure of descent, or improper stepsizes. To the best of our knowledge, these provide the first counterexamples to KKT accumulation for bounded mirror descent sequences with nonincreasing objective values.