发表机构
School of Industrial Engineering, Purdue University; National University of Singapore（普渡大学工业工程学院; 新加坡国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对有界镜像下降序列的KKT积累问题，构造反例证明其积累点可为非KKT点，揭示边界Bregman几何退化是该问题的根源。
AI 中文摘要
对于由勒让德核生成的镜像下降，优化领域最基础的问题之一是：在合适的步长下，有界镜像下降序列的每个积累点是否必须是Karush-Kuhn-Tucker（KKT）平稳点？我们证明答案是否定的。解决该问题的一个长期障碍是勒让德梯度的边界爆破：它使每次镜像步都保持在内部，而在边界极限下，逆熵度量会在活跃坐标上消失，可能消除KKT系统中的对偶可行性。我们针对每个n≥3的非负卦限ℝ₊ⁿ，以及每个n≥4的概率单纯形Δₙ，构造C^∞目标函数和由香农熵镜像下降生成的有界序列，使得在每种情况下，积累点集合是一条光滑边界圆，包含非空的非KKT点相对开弧。步长满足αₖ∝k^(-β)（其中β∈(1/2,1)），目标值非递增，且目标函数是熵相对光滑的。因此，这种病态源于边界处Bregman几何的退化，而非下降失效或步长不当。据我们所知，这些是针对具有非递增目标值的有界镜像下降序列的KKT积累问题的首批反例。
英文摘要
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.
