The Surrogate Is Not the Reward: Post-Surrogate Primary-Outcome Acquisition in Contextual Bandits

Kyungbok Lee, Michael R. Kosorok

We study contextual bandits in which a surrogate is observed after the action but before the learner decides whether to acquire the primary outcome that defines action value and regret. The value of acquiring the primary outcome depends on both decision relevance (how much the current outcome matters for comparing policies) and the residual uncertainty after observing the surrogate. The Audited Surrogate Bandit (ASB) learns a contextual policy while allocating a budget of $B$ primary-outcome acquisitions over $T$ rounds. ASB sets a pre-surrogate acquisition level from current decision relevance and, after observing the surrogate, redistributes that level using an estimate of that residual uncertainty. For a finite class of $N$ policies over $K$ actions, ASB incurs $\widetilde O[\sqrt{KT\log N}\{1+\sqrt{T/B}\}]$ regret relative to the best policy in the class. In a two-action family where the surrogate does not reveal the better action, a learner that observes the surrogate before deciding whether to acquire can achieve bounded regret, whereas any learner that must decide before seeing the surrogate incurs $Ω(T/B)$ worst-case regret under the same budget. Synthetic experiments…