Oracle-Efficient Online Learning for Beyond Worst-Case Adversaries

In this paper, we study oracle-efficient algorithms for beyond worst-case analysis of online learning. We focus on two settings. First, the smoothed analysis setting of [RST11, HRS12] where an adversary is constrained to generating samples from distributions whose density is upper bounded by $1/\sigma$ times the uniform density. Second, the setting of $K$-hint transductive learning, where the learner is given access to $K$ hints per time step that are guaranteed to include the true instance. We give the first known oracle-efficient algorithms for both settings that depend only on the VC dimension of the class and parameters $\sigma$ and $K$ that capture the power of the adversary. In particular, we achieve oracle-efficient regret bounds of $ O ( \sqrt{T (d / \sigma )^{1/2} } ) $ and $ O ( \sqrt{T d K } )$ respectively for these setting. For the smoothed analysis setting, our results give the first oracle-efficient algorithm for online learning with smoothed adversaries [HRS21]. This contrasts the computational separation between online learning with worst-case adversaries and offline learning established by [HK16]. Our algorithms also imply improved bounds for worst-case setting with small domains. In particular, we give an oracle-efficient algorithm with regret of $O ( \sqrt{T(d \vert{\mathcal{X}}\vert ) ^{1/2} })$, which is a refinement of the earlier $O ( \sqrt{T\vert{\mathcal{X} } \vert })$ bound by [DS16].