We consider a hierarchical inference system with multiple clients connected to a server via a shared communication resource. When necessary, clients with low-accuracy machine learning models can offload classification tasks to a server for processing on a high-accuracy model. We propose a distributed online offloading algorithm which maximizes the accuracy subject to a shared resource utilization constraint thus indirectly realizing accuracy-delay tradeoffs possible given an underlying network scheduler. The proposed algorithm, named Lyapunov-EXP4, introduces a loss structure based on Lyapunov-drift minimization techniques to the bandits with expert advice framework. We prove that the algorithm converges to a near-optimal threshold policy on the confidence of the clients’ local inference without prior knowledge of the system’s statistics and efficiently solves a constrained bandit problem with sublinear regret. We further consider settings where clients may employ multiple thresholds, allowing more aggressive optimization of overall accuracy at a possible loss in fairness. Extensive simulation results on real and synthetic data demonstrate convergence of Lyapunov-EXP4, and show the accuracy-delay-fairness tradeoffs achievable in such systems.