Non-KKT Accumulation in Entropic Mirror Descent

2026-08-02

AI Summary

This study investigates whether accumulation points of bounded mirror descent sequences must necessarily be KKT stationary points. Focusing on Shannon entropy-induced mirror descent over the non-negative orthant (ℝ₊ⁿ, n≥3) and the probability simplex (Δₙ, n≥4), the authors construct smooth objective functions and corresponding iterates that yield, for the first time, counterexamples where accumulation points include non-KKT points—thereby challenging the prevailing assumption that all accumulation points are KKT points. By leveraging entropy-based relative smoothness, a Bregman geometric boundary degeneracy mechanism, and asymptotic step sizes αₖ ∼ k⁻ᵝ with β∈(1/2,1), they generate sequences whose objective values are monotonically non-increasing and whose accumulation sets form smooth boundary arcs containing non-stationary points. This reveals that mirror descent may converge to non-stationary points near the boundary of the feasible region.

Abstract

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 $C^\infty$ objectives and bounded sequences generated by the Shannon-entropic mirror descent on the nonnegative orthant $\R_+^n$, for every $n\geq 3$, and on the probability simplex $Δ_n$, for every $n\geq 4$, 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 $α_k\asymp k^{-β}$ with $β\in(1/2,1)$, 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.

Problem

• mirror descent
• KKT conditions
• accumulation points
• Legendre kernel
• Bregman geometry

Innovation

• mirror descent
• KKT conditions
• entropy regularization
• Bregman geometry
• non-KKT accumulation