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.