Descent--Ascent Structure of Uniform Nondegeneracy
Abstract
Classical models such as smooth strongly convex minimization and strongly convex--strongly concave optimization combine two types of curvature information: curvature remains uniformly separated from zero, and its positive and negative orientations are prescribed in advance. Uniform nondegeneracy disentangles these roles, retaining the separation from zero while allowing the curvature orientation to vary across the domain. We identify a family of descent--ascent inequalities (DAI) that characterizes uniform nondegeneracy and naturally gives rise to Alternating Descent Ascent (ADA), a simple explicit method that alternates full-gradient descent and ascent. ADA converges globally and linearly to the unique stationary point from arbitrary initialization, with dimension-free first-order complexity. The framework recovers the smooth strongly convex setting as a limit and encompasses the strongly convex--strongly concave and linearly constrained settings, highlighting uniform nondegeneracy as a common first-order structure across various classical regimes.
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BibTeX
@misc{su2026descentascent,
title = {{Descent--Ascent Structure of Uniform Nondegeneracy}},
author = {Hua Su and Lei Zhang and Jin Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06115},
eprint = {2610.06115},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06115},
}
RIS
TY - UNPB
TI - Descent--Ascent Structure of Uniform Nondegeneracy
AU - Hua Su
AU - Lei Zhang
AU - Jin Zhao
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06115
UR - https://arxiv.org/abs/2610.06115
AB - Classical models such as smooth strongly convex minimization and strongly convex--strongly concave optimization combine two types of curvature information: curvature remains uniformly separated from zero, and its positive and negative orientations are prescribed in advance. Uniform nondegeneracy disentangles these roles, retaining the separation from zero while allowing the curvature orientation to vary across the domain. We identify a family of descent--ascent inequalities (DAI) that characterizes uniform nondegeneracy and naturally gives rise to Alternating Descent Ascent (ADA), a simple explicit method that alternates full-gradient descent and ascent. ADA converges globally and linearly to the unique stationary point from arbitrary initialization, with dimension-free first-order complexity. The framework recovers the smooth strongly convex setting as a limit and encompasses the strongly convex--strongly concave and linearly constrained settings, highlighting uniform nondegeneracy as a common first-order structure across various classical regimes.
ER -