Polyhedral Instability Governs Regret in Online Learning

Many online decision problems over combinatorial actions are addressed via convex relaxations, leading to online convex optimization with piecewise linear objectives and induced polyhedral structure. We show that regret in such problems is governed by polyhedral instability: the number of changes of the active region. Under full information feedback and fixed partition assumptions, if RST denotes the number of region switches and Vmax the maximum number of vertices per region, we prove RegretT= Theta(√(1+RST),T,log Vmax) interpolating between experts-like and dimension-dependent OCO rates. For online submodular--concave games under Lov'asz convexification, this reduces to the permutation-switch count SCT, yielding the matching rate RegretT= Theta(√(1+SCT),T,log n). Experiments on synthetic and real combinatorial problems (shortest path, influence maximization) validate the predicted scaling and indicate that low-instability regimes can arise in practice without explicit enumeration of actions.