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

arXiv: https://arxiv.org/abs/2609.30877 · HTML: https://mlsanity.com/paper/2609.30877/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 13 references that are in the map.
- First posted on arXiv 25 Sep 2026.

## Builds on (most cited first, 12 of 12)

- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/1906.00331/) (2019)
- [Lower Bounds for Finding Stationary Points I](https://mlsanity.com/paper/1710.11606/) (2017)
- [Near-Optimal Algorithms for Minimax Optimization](https://mlsanity.com/paper/2002.02417/) (2020)
- [Stochastic Recursive Gradient Descent Ascent for Stochastic Nonconvex-Strongly-Concave Minimax Problems](https://mlsanity.com/paper/2001.03724/) (2020)
- [The Complexity of Nonconvex-Strongly-Concave Minimax Optimization](https://mlsanity.com/paper/2103.15888/) (2021)
- [Complexity Lower Bounds for Nonconvex-Strongly-Concave Min-Max Optimization](https://mlsanity.com/paper/2104.08708/) (2021)
- [SAPD+: An Accelerated Stochastic Method for Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/2205.15084/) (2022)
- [Lower Complexity Bounds for Nonconvex-Strongly-Convex Bilevel Optimization with First-Order Oracles](https://mlsanity.com/paper/2511.19656/) (2025)
- [On the Condition Number Dependency in Bilevel Optimization](https://mlsanity.com/paper/2511.22331/) (2025)
- [Lower Bounds for Nonconvex-Concave Minimax Optimization](https://mlsanity.com/paper/2609.14233/) (2026)
- [Lower Bounds for Nonconvex-PL Minimax Optimization](https://mlsanity.com/paper/2608.26799/) (2026)

## Built on by (most cited first, 1 of 1)

- [Near-Optimal Complexity of Finite-Sum Nonconvex-Strongly-Concave Minimax Optimization](https://mlsanity.com/paper/2610.04944/) (2026)

## Abstract (arXiv)

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≥varepsilon, the worst-case complexity of zero-respecting algorithms is Theta(kappa LGσ2varepsilon-4) in the stated accuracy regime, where kappa=L/μ, G bounds the initial primal-dual gap, and σ2 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 varepsilon/√kappa, while an undiscovered primal coordinate prevents stationarity. It also yields the primal-gradient lower bound Omega(LDelta(√kappavarepsilon-2+kappaσ2varepsilon-4)) after combination with the known deterministic bound, where Delta bounds the initial primal function gap.