Extend the lower-bound construction to randomized first-order methods

Develop a robust saddle zero-chain, potentially using the probabilistic zero-chain mechanism of Arjevani et al., to extend the deterministic first-order lower-bound construction for smooth nonconvex–Polyak–Łojasiewicz minimax optimization to randomized first-order methods.

Background

The paper establishes an optimal deterministic first-order oracle lower bound for smooth nonconvex–P minimax optimization by constructing a saddle zero-chain whose coordinates are revealed sequentially. The authors note that extending this argument to randomized algorithms requires a more robust zero-chain mechanism that remains effective under randomization.

The paper identifies the probabilistic zero-chain mechanism of Arjevani et al. as a natural starting point, but does not provide the required randomized construction or a corresponding lower bound.

References

Two directions remain open. First, extending the construction to randomized first-order methods requires a robust saddle zero-chain; the probabilistic zero-chain mechanism of \citet{arjevani2023lower} is the natural starting point.

— Lower Bounds for Nonconvex-PŁ Minimax Optimization  (2608.26799 - Pan et al., 27 Aug 2026) in Section 6, Closing Remarks

Extending the lower bound to arbitrary randomized algorithms, stochastic gradients of differentiable sample losses, or an unconstrained dual domain remains open for this construction.

— Tight Stochastic Condition-Number Dependence in Nonconvex-Strongly-Concave Minimax Optimization  (2609.30877 - Zhou, 25 Sep 2026) in Section Conclusion

Several questions remain open. First, it would be useful to extend the lower bounds beyond zero-respecting algorithms to more general, possibly randomized, first-order methods.

— Tight Lower Bounds for Stochastic Nonconvex-Strongly-Concave Minimax Optimization  (2609.35206 - Zhang et al., 28 Sep 2026) in Section Conclusion and Future Work

Finally, our zero-chain hard instances require a dimension that grows with the target accuracy; whether comparable lower bounds hold in fixed or low dimensions, in the spirit of \citet{chewi2023complexity}, remains open.

— Tight Lower Bounds for Stochastic Nonconvex-Strongly-Concave Minimax Optimization  (2609.35206 - Zhang et al., 28 Sep 2026) in Section Conclusion and Future Work

An important next step is to determine whether these barriers extend beyond zero-respecting algorithms. Resolving this question would clarify the extent to which the observed complexity is intrinsic to stochastic minimax optimization rather than tied to coordinate-revelation restrictions.

— Lower Bounds for Stochastic First-Order Algorithms with Variance Reduction in Nonconvex--Concave Minimax Optimization  (2610.01662 - Song et al., 1 Oct 2026) in Section Conclusion