Online Convex Optimization with Dueling Feedback

Noisy binary comparison between two candidates is a common interface between human and learning systems, especially in modern large language model (LLM) post-training alignment. We study online convex optimization with dueling (pairwise comparison) feedback, where the learner observes only a binary preference between two queried points. We consider adversarial sequences of convex losses and measure regret with the loss at both queried points, under a comparison link with a known nonzero slope at the origin. We propose a simple reduction that converts dueling feedback into approximate gradients, enabling the use of standard first-order methods. We show that regret guarantees transfer under this reduction, yielding $\mathcal{O}(T^{3/4})$ static and adaptive regret, and $\mathcal{O}(T^{3/4}\sqrt{1+P_T/D})$ dynamic regret with unknown comparator path length $P_T$. For strongly convex losses, the static and adaptive bounds improve to $\tilde{\mathcal{O}}(T^{2/3})$. For smooth losses, we presents unified dueling ellipsoidal FTRL, and proves $\tilde{\mathcal{O}}(T^{2/3})$ static regret, which improves to $\tilde{\mathcal{O}}(\sqrt{T})$ under additional strong convexity.