Polyhedral Instability Governs Regret in Online Learning

TL;DR

This paper demonstrates that regret in online convex optimization with polyhedral structure is primarily determined by the number of active region changes, linking regret bounds to polyhedral instability measures.

Contribution

It introduces the concept of polyhedral instability as a key factor governing regret, providing new bounds that interpolate between classical rates and dimension-dependent rates.

Findings

Regret scales with the square root of the number of region switches and maximum vertices per region.

In submodular-concave games, regret depends on permutation-switch count, matching known rates.

Experiments confirm the theoretical scaling and show low-instability regimes can occur in practice.

Abstract

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 $\Regret_T= \Theta(\sqrt{(1+\mathrm{RS}_T)\,T\,\log V_{\max}})$ interpolating between experts-like and dimension-dependent OCO rates. For online submodular--concave games under Lov\'{a}sz convexification, this reduces to the permutation-switch count $\mathrm{SC}_T$, yielding the matching rate $\Regret_T= \Theta(\sqrt{(1+\mathrm{SC}_T)\,T\,\log n})$. 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.