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

TL;DR

This work studies cooperative $N$-agent stochastic multi-objective bandits with vector rewards under adversarial corruption and limited verification. It introduces a protocol-level corruption functional via sample multiplicity, defining an effective corruption budget $Γ_{\mathrm{eff}}$ that ranges between $Γ$ and $NΓ$ depending on the sharing mode. A unified robust-UCB analysis yields a meta regret bound parameterized by $Γ_{\mathrm{eff}}$, and the results sharply separate raw-sample sharing (which amplifies corruption by a factor of $N$) from sufficient-statistic and recommendation-only sharing (which preserve the $O(Γ)$ corruption term) with centralized-like performance. The paper also shows that verification with certificates can restore learnability even in the high-corruption regime, making regret independent of $Γ$ once the verification budget $ν$ passes a data-dependent threshold. Collectively, the findings guide the design of robust, distributed decision systems by balancing communication, verification, and corruption considerations.

Abstract

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 $Γ$. Performance is measured by team regret under a coordinate-wise nondecreasing, $L$-Lipschitz scalarization $φ$, covering linear, Chebyshev, and smooth monotone utilities. Our main contribution is a communication-corruption coupling: we show that a fixed environment-side budget $Γ$ can translate into an effective corruption level ranging from $Γ$ to $NΓ$, 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 $O(Γ)$ term and achieve centralized-rate team regret. We further establish information-theoretic limits, including an unavoidable additive $Ω(Γ)$ penalty and a high-corruption regime $Γ=Θ(NT)$ where sublinear regret is impossible without clean information. Finally, we characterize how a global budget $ν$ 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 $Γ$.

Figures (3)

• Figure 1: Team regret versus corruption budget $\Gamma$ for (S1) and (S2): (S1) scales like $\widetilde{O}(\sqrt{KNT}+N\Gamma)$ while (S2) scales like $\widetilde{O}(\sqrt{KNT}+\Gamma)$.
• Figure 2: Team regret versus $\Gamma$ under (S2) and (S3): (S3) preserves the unamplified $O(\Gamma)$ term while trading off a clean-case coordination overhead.
• Figure 3: Team regret versus verification budget $\nu$ under (S3) with/without certified sharing in high-corruption regime $\Gamma=\Theta(NT)$: sharp improvement once $\nu \gtrsim K L^2 \Delta_{\min}^{-2}\log(dKNT)$.

Theorems & Definitions (15)

• Definition 1: Multiplicity and effective corruption
• Lemma 1: Effective corruption under (S1)-(S3)
• Lemma 2: Vector mean concentration under effective corruption
• proof : Proof sketch
• Definition 2: Upper-closed confidence sets
• Lemma 3: Upper-corner reduction
• Theorem 1: Team regret bound via effective corruption
• proof : Proof sketch
• Corollary 1: (S1) Raw-sample sharing incurs $N$-fold amplification
• Corollary 2: (S2)--(S3) Achieve centralized corruption penalty