Online Learning with Ordered Experts (via Two-point Feedback)

We study adversarial online learning with expert recommendations in a totally ordered action space, where the experts' recommendations may change and be reordered arbitrarily. Our focus is two-point feedback: on each round the learner plays two current recommendations, observes both, and pays their average. We study two problems. In the separated monotone problem, the loss is the sum of a hidden nondecreasing component and a hidden nonincreasing component, and a query reveals the two components separately. In the BV problem, the losses are bounded and have bounded variation (BV) along the order, and a query reveals only the loss. In the separated monotone problem, two paid observations achieve

$$O\bigl(\log(|\cF|T)\sqrt{T\log|\cF|}\bigr)$$

regret against the best of $|\cF|$ experts, within a logarithmic factor of the classical full-information bound. For the BV problem, we obtain $\wtO(\sqrt T+|\cF|)$ regret. We give an application in dynamic pricing with expert advice, showing that two-option menus matches the full information rate up to $\log$ factors. The main technical ingredient is a marginal-preserving endpoint flow estimator. Two observations reveal a loss increment along the order; random paths supply telescoping identities that turn this local information into unbiased loss estimates for all current recommendations. A reversible pair sampler preserves the learner's intended marginal at both played actions. The estimator's second-moment and stability bounds support online updates for both feedback models.