Derandomizing the slingshot schedule for convex–concave problems

Develop a deterministic rule for choosing, within each two-step pair of the slingshot stepsize schedule for smooth convex–concave objectives, which variable (x or y) uses the negative step so that the resulting deterministic GDA retains the convergence guarantees established for the randomized schedule in Definition 5.

Background

For general smooth convex–concave min–max problems, the paper’s slingshot schedule (Definition 5) uses randomness to break symmetry between the minimization and maximization variables: in each pair of iterations, one variable takes a negative step while the other does not, with the choice randomized. This symmetry breaking is crucial to avoid cycling and to obtain last-iterate convergence guarantees.

The authors conjecture that these results can be achieved without randomness, i.e., via a deterministic intra-pair sign-selection rule. Establishing such a rule would simplify implementation, remove stochastic variability, and potentially strengthen theoretical guarantees by eliminating expectation-based analyses.