Distributed Consensus Optimization under Zeroth-Order Oracles and Uniform Quantization

We consider a constrained multi-agent optimization problem where the bit rate of communication in the network is limited. This problem arises in a network with time-varying connectivity where all the agents try to minimize a sum of non smooth but Lipschitz continuous functions, and the estimates of each agent are restricted to lie in the same convex set. We design a uniform quantizer and present a distributed zeroth-order method, which relies on the functional evaluations and quantized estimates. We establish conditions on the hit rate and the initial quantization intervals that ensures the convergence of the algorithm. In particular, we provide convergence analysis results, and highlight the dependence on the smoothing parameters and the quantization resolution.