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 \emph{polyhedral instability}: the number of changes of the active region. Under full information feedback and fixed partition assumptions, if $\mathrm{RS}_T$ denotes the number of region switches and $V_{\max}$ the maximum number of vertices per region, we prove $\Regret_T= Θ(\sqrt{(1+\mathrm{RS}_T)\,T\,\log V_{\max}})$ interpolating between experts-like and dimension-dependent OCO rates. For online submodular--concave games under Lovász co...
| Authors | Yuetai Li, Fengqing Jiang, Yichen Feng et al. |
|---|---|
| Primary category | cs.LG |
| Published | 13 May 2026 |
| Last updated at source | 13 May 2026 |
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