Communication-Corruption Coupling and Verification in Cooperative Multi-Objective Bandits

We study cooperative stochastic multi-armed bandits with vector-valued rewards under adversarial corruption and limited verification. In each of $T$ rounds, each of $N$ agents selects an arm, the environment generates a clean reward vector, and an adversary perturbs the observed feedback subject to a global corruption budget $\Gamma$. Performance is measured by team regret under a coordinate-wise nondecreasing, $L$-Lipschitz scalarization $\phi$, covering linear, Chebyshev, and smooth monotone utilities. Our main contribution is a communication-corruption coupling: we show that a fixed environment-side budget $\Gamma$ can translate into an effective corruption level ranging from $\Gamma$ to $N\Gamma$, depending on whether agents share raw samples, sufficient statistics, or only arm recommendations. We formalize this via a protocol-induced multiplicity functional and prove regret bounds parameterized by the resulting effective corruption. As corollaries, raw-sample sharing can suffer an $N$-fold larger additive corruption penalty, whereas summary sharing and recommendation-only sharing preserve an unamplified $O(\Gamma)$ term and achieve centralized-rate team regret. We further establish information-theoretic limits, including an unavoidable additive $\Omega(\Gamma)$ penalty and a high-corruption regime $\Gamma=\Theta(NT)$ where sublinear regret is impossible without clean information. Finally, we characterize how a global budget $\nu$ of verified observations restores learnability. That is, verification is necessary in the high-corruption regime, and sufficient once it crosses the identification threshold, with certified sharing enabling the team's regret to become independent of $\Gamma$.