Tight Stochastic Condition-Number Dependence in Nonconvex-Strongly-Concave Minimax Optimization

@misc{zhou2026tight,
  title = {{Tight Stochastic Condition-Number Dependence in Nonconvex-Strongly-Concave Minimax Optimization}},
  author = {Qihao Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30877},
  eprint = {2609.30877},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30877},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Tight Stochastic Condition-Number Dependence in Nonconvex-Strongly-Concave Minimax Optimization
AU  - Qihao Zhou
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.30877
UR  - https://arxiv.org/abs/2609.30877
AB  - We study whether the linear condition-number dependence in the stochastic complexity of SAPD+ is necessary for nonconvex-strongly-concave minimax optimization. For jointly L-smooth objectives with dual strong-concavity parameter μ, we prove a lower bound that matches the SAPD+ upper bound under the same Moreau-envelope stationarity criterion and the same primal-dual initialization gap. Specifically, when σ≥ε, the worst-case complexity of zero-respecting algorithms is Θ(κLGσ^2ε^−4) in the stated accuracy regime, where κ=L/μ, G bounds the initial primal-dual gap, and σ^2 bounds the variance of a general unbiased first-order oracle. The lower bound is realized on a smooth problem class with a bounded dual box. Our construction routes each link of a nonconvex zero-chain through a dual gradient of magnitude proportional to ε/√κ, while an undiscovered primal coordinate prevents stationarity. It also yields the primal-gradient lower bound Ω(LΔ(√κε^−2+κσ^2ε^−4)) after combination with the known deterministic bound, where Δ bounds the initial primal function gap.
ER  -