Ordering as Control: Faster, Single-Call, and Bias-Cancelling Stochastic Extragradient

The order in which stochastic extragradient (SEG) visits the components of a finite sum changes where it settles under a constant step. We treat this order as a control input, for strongly monotone finite-sum variational inequalities. Expanding one epoch shows that the order acts through a single prefix-sum term, $X(\pi )$, beside an order-free correction and a higher-order remainder. Four actions follow. Herded orders shrink $X(\pi )$. Alternating an order with its reverse cancels its leading contribution across epochs (affine model). Reusing the records herding already keeps removes the second operator call: SC-HERD makes one call per step and, for a fixed order herding the ideal offsets, matches the two-call herded plateau up to a higher-order term. An order can even null the whole leading bias of single-call past extragradient on an explicit affine family; BC-SPEG searches for such orders with estimated Jacobians, under a conditional a-posteriori certificate. Against an instance-wise lower bound for random reshuffling, herding improves the worst case by $\sqrt{n}$ on rotation-dominated instances at the largest admissible step, up to conditioning and component-dominance factors; on generic instances in fixed dimension the leading coefficient improves by a factor $n$ under per-node split accuracy, with finite-step gains still conditioning-dependent. Experiments cover affine, nonlinear, multi-player and real-data problems. At two frozen adversarial ResNet checkpoints, $⟨g,X(\pi )⟩$ ranks the one-epoch outcomes of unseen random orders at partial Spearman $0.97$; the size of the prefix sums alone does not.