通过重参数化解决镜面下降在边界处的KKT收敛难题
Establishing Boundary KKT Convergence of Mirror Descent through Reparameterization
- 用重参数化方法缓解勒让德核在边界处梯度爆炸问题
- 证明在特定条件下重参数序列有界且收敛，原序列也收敛到KKT点
- 适用于熵、幂核等典型非凸结构，解释实际应用有效性
非凸镜面下降在使用勒让德核时，其序列收敛至边界KKT点的问题长期未解。困难源于勒让德核在边界处梯度发散。近期工作表明，镜面下降可能在目标值下降的同时累积于非KKT边界点，一般无法保证收敛到KKT点。尽管如此，镜面下降在众多实际任务中仍表现有效。本文针对此矛盾，直接处理边界难题，建立了一类结构化非凸问题下镜面下降的KKT收敛性。通过在重参数变量中分析镜面下降，使海森度量在逼近边界时保持非退化。在目标函数、勒让德核与可行域共同满足扩展性与可定义性条件的前提下，重参数序列具有有限长度并收敛，从而恢复原序列对KKT点的收敛。该通用框架适用于若干具体实例：香农熵、费米-狄拉克熵及多面体上的幂核。
原文摘要 · Abstract (English)
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.
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