A common feature of these applications is that there are multiple decision makers interacting with each other in a common environment.

In this section, we prove Theorem 3.1, which says that it suffices to the augmented state space First, we have the following lemma. Lemma A.2. W e have D Now, we prove Theorem 3.1. By Lemma A.2, we have F Theorem 3.1 follows straightforwardly from this result. Consider the same setup as in Lemma B.1. Lemma B.1, we have F Consider the same setup as in Lemma B.1.

Stochastic Convex Optimization (SCO) setting, which assumes that the learning objective is convex.

The prompt-based paradigm has fostered two recent human-AI interaction trends.

Then, the desired lemma is obtained.

When the losses are stochastically generated, [Simchowitz and Jamieson, 2019, Y ang et al., 2021]

The proof is based on Proposition A.1 and Proposition A.2, which will be introduced as Proposition A.1 shows that the set The proof follows Theorem 14.12 in [87], with non-trivial modifications for our setting. In Claim A.4 we will show that with overwhelming probability in m, sup Then we use Talagrand's inequality [83] to show that Pr null Z In order to apply Talagrand's inequality, we need to bound σ Proposition A.1 shows that the set Consider the indicator random variable associated with the complement of the above event. This implies that there exist a subset of batches J [ M ] with |J | 0. 95M such that 1 b nullA Note that this a subset of all vectors in the span of W that have norm at most null . The third line follows from the Cauchy-Schwartz inequality, and the fourth line follows from Jensen's This implies that there at least 0.95 M batches in which all samples are well behaved. Consider the setting of Lemma 5.3 with measurements satisfying Proposition A.1 shows that the set Note that this a subset of all vectors in the span of W that have norm at most null .