Establishing Boundary KKT Convergence of Mirror Descent through Reparameterization
Abstract
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~\cite{dingtoh2026nonkkt} 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.
Related
- PL conditions do not guarantee convergence of gradient descent-ascent dynamics
- A Rate Transfer Method: Convergence Rates in Stochastic Gradient Descent via Stochastic Calculus
- Convergence rate of stochastic gradient descent under Łojasiewicz type condition and polynomially controlled noise
- Exact Asymptotics for the Harmonic Descent Chain and Extended Convergence Bounds for Recursive Statistics
- Large-Step Last-Iterate Convergence of Random-Reshuffling Stochastic Gradient Descent
BibTeX
@misc{ding2026establishing,
title = {{Establishing Boundary KKT Convergence of Mirror Descent through Reparameterization}},
author = {Kuangyu Ding and Kim-Chuan Toh},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.07248},
eprint = {2608.07248},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.07248},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Establishing Boundary KKT Convergence of Mirror Descent through Reparameterization
AU - Kuangyu Ding
AU - Kim-Chuan Toh
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.07248
UR - https://arxiv.org/abs/2608.07248
AB - 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~\cite{dingtoh2026nonkkt} 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.
ER -