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The distributed subgradient method (DSG) is a widely discussed algorithm tocope with large-scale distributed optimization problems in the arising machinelearning applications. Most exisiting works on DSG focus on ideal communicationbetween the cooperative agents such that the shared information between agentsis exact and perfect. This assumption, however, could lead to potential privacyconcerns and is not feasible when the wireless transmission links are not ofgood quality. To overcome the challenge, a common approach is to quantize thedata locally before transmission, which avoids exposure of raw data andsignificantly reduces the size of data. Compared with perfect data,quantization poses fundamental challenges on loss of data accuracy, whichfurther impacts the convergence of the algorithms. To settle the problem, wepropose a generalized distributed subgradient method with random quantization,which can be intepreted as a two time-scale stochastic approximation method. Weprovide comprehensive results on the convergence of the algorithm and deriveupper bounds on the convergence rates in terms of the quantization bit,stepsizes and the number of network agents. Our results extend the existingresults, where only special cases are considered and general conclusions forthe convergence rates are missing. Finally, numerical simulations are conductedon linear regression problems to support our theoretical results.

Convergence Theory of Generalized Distributed Subgradient Method with Random Quantization