Same Sample or Fresh? The Coupling Geometry of Stochastic Extragradient

Stochastic extragradient calls its oracle twice per step, and the second call may reuse the first sample or draw a fresh one; neither dominates. With both marginals fixed, a sampler is a joint law of the two indices, a point of the Birkhoff polytope: reuse is a vertex and fresh sampling the barycenter. On linear games with multiplicative noise this point acts on the mean step only through the cross-covariance of the two oracle errors and, around a conservative mean with distinct frequencies, on the small-step mean-square rate only through its trace on each rotation plane. Reuse contributes the plane's signature, its dilational minus its rotational noise energy; fresh sampling contributes zero. For skew noise, as in stochastic bilinear games, reuse is an optimal coupling to leading order; for symmetric noise it is the worst, and the best couplings anti-correlate the two calls. Some optimal coupling mixes at most one pairing per plane, a budget attained with two and three planes. We prove a mean-square law for arbitrary noise (a convex Perron root), an almost-sure law for plane-preserving noise, and exact rates for commuting blocks at every step; an algorithm designs the coupling. On a nonlinear game with nonmonotone components, reuse fails as the noise turns dilational, while the designed coupling converges in every run and has the best predicted rate; on a group-robust digits game dominated by additive noise, reuse beats fresh sampling.