Minibatch Starvation: A Spurious Absorbing State in Group DRO’s Multiplicative-Weights Dynamics, with a Closed-Form Collapse Law and a Fix

Group Distributionally Robust Optimization (Group DRO) runs the multiplicative-weights (Hedge) update on a repeated zero-sum game between a model and an adversary choosing a distribution $q$ over groups, exponentiating $q$ toward whichever group is currently worst. Treated as a discrete-time dynamical system on the simplex, this game is meant to have every group's weight bounded away from the boundary; we show the implemented dynamics instead admit a spurious absorbing state at $q_{\min} = 0$ that has nothing to do with the game being solved. The cause is an estimator artifact: the standard minibatch implementation zero-fills a batch-absent group's loss to 0, and the exponentiated update then drains that group's mass through the normalizer on every batch it misses, driving its per-class accuracy to as low as 0\% while aggregate metrics barely move. We prove this artifact alone suffices for this absorbing state via an equal-loss construction in which the exact (non-minibatched) dynamics provably keep the simplex interior fixed, yet minibatched play still collapses. Integrating the induced drift yields a closed-form collapse-time law, \begin{equation} T_{collapse} = \frac{\Delta^\circ}{\eta \bar\ell (1-f)^B} \quad \text{with} \quad f = \frac{p}{1+R}, \end{equation} that reproduces the measured phenomenology from one calibrated scale, including an epoch-time illusion whereby collapse looks size-dependent under one clock and is size-invariant under the correct one. Renormalizing over only the present groups removes this spurious drain (an order-of-magnitude suppression, weakening to $\approx 1.5\times$ on real data where a slower, genuine channel competes); the remaining dynamics instead cross a phase boundary $n_c^*(d)$ separating survival from legitimate collapse. The law retrodicts collapse on seven of eight real datasets (a one-dataset, directional margin over a prior heuristic at 6/8), and the fix is a three-line change to standard implementations that narrows, but does not close, the worst-group accuracy gap; the failure is not special to Group DRO, but generic to no-regret dynamics run on subsampled, possibly-absent payoffs.