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 of the algorithm in terms of the quantization intervals. We show that asymptotic optimality can be achieved if the quantization intervals approach zero. On the other hand, if the quantization intervals are fixed and nonzero, then non-diminishing performance gaps exist, which indicate a tradeoff between optimization performance and communication overhead. We further extend the algorithm and analysis to the scenario of bandit feedback, where samples of the random variables are not available and only the values of the cost function at two random points close to the current decision are disclosed. We show that the performance of the algorithm does not degrade in order sense compared to its counterpart with sample feedback. Finally, two numerical examples, namely decentralized quadratically constrained quadratic program and decentralized logistic regression, are presented to corroborate the efficacy of the proposed algorithms.

中文翻译：

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具有成对约束和量化通信的分散式多智能体随机优化

分散优化方法通常需要邻居之间的信息交换。在很多情况下，由于通信带宽有限，交换的信息必须被量化。在本文中，我们研究了量化对分散随机优化性能的影响。我们考虑一个多代理网络，其中每个节点都与一个随机成本相关联，而每对邻居都与一个约束相关联。网络的目标是在所有约束条件下最小化总预期成本。我们首先针对样本反馈的场景开发了一种具有量化通信的分散式随机鞍点算法，其中随机成本函数中的随机变量的样本每次都显示给每个节点。我们根据量化间隔为算法的预期成本次优和预期约束违反建立了性能界限。我们表明，如果量化间隔接近零，则可以实现渐近最优性。另一方面，如果量化间隔是固定的且非零，则存在非递减的性能差距，这表明优化性能和通信开销之间的权衡。我们进一步将算法和分析扩展到强盗反馈的场景，其中随机变量的样本不可用，并且只公开了接近当前决策的两个随机点的成本函数值。我们表明，与具有样本反馈的对应算法相比，该算法的性能在顺序意义上不会降低。最后，