🤖 AI Summary
This study addresses whether the linear dependence of stochastic complexity on the condition number is necessary in nonconvex-strongly-concave minimax optimization. By constructing nonconvex zero-chains, dual gradient routing, and Moreau envelope stationarity criteria, this work develops lower bound instances that match the upper bound of the SAPD+ algorithm. It provides the first rigorous proof that this linear dependence cannot be improved, establishing a worst-case complexity of Θ(κLGσ²ε⁻⁴) for zero-respecting algorithms. Furthermore, it derives a combined stochastic-deterministic lower bound of Ω(LΔ(√κε⁻²+κσ²ε⁻⁴)), verifying optimal convergence rates and closing existing theoretical gaps.
📝 Abstract
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 $σ\ge\varepsilon$, the worst-case complexity of zero-respecting algorithms is $Θ(κLGσ^2\varepsilon^{-4})$ 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 $\varepsilon/\sqrtκ$, while an undiscovered primal coordinate prevents stationarity. It also yields the primal-gradient lower bound $Ω(LΔ(\sqrtκ\varepsilon^{-2}+κσ^2\varepsilon^{-4}))$ after combination with the known deterministic bound, where $Δ$ bounds the initial primal function gap.