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Decentralized optimization methods often entail information exchange between neighbors. In many circumstances, due to the limited communication bandwidth, the exchanged information has to be quantized. In this paper, we investigate the impact of quantization on the performance of decentralized stochastic optimization. We consider a multi-agent network, in which each node is associated with a stochastic cost and each pair of neighbors is associated with a constraint. The goal of the network is to minimize the aggregate expected cost subject to all the constraints. We first develop a decentralized stochastic saddle point algorithm with quantized communications for the scenario of sample feedback, in which a sample of the random variable in the stochastic cost function is revealed to each node at each time. We establish performance bounds for the expected cost suboptimality and expected constraint violations …