In this paper, we study distributed constrained online convex optimization (OCO) problem in a system consisting of a parameter server and clients. Each client is associated with a local constraint function and time-varying local loss functions, which are disclosed sequentially. The clients seek to minimize the accumulated total loss subject to the total constraint by choosing sequential decisions based on causal information of the loss functions. Existing distributed constrained OCO algorithms require clients to send their raw decisions to the server, leading to large communication overhead unaffordable in many applications. To reduce the communication cost, we devise an adaptive quantization method, where the center and the radius of the quantizer are adjusted in an adaptive manner as the OCO algorithm progresses. We first examine the scenario of full information feedback, where the complete information of the loss functions is revealed at each time. We propose a distributed online saddle point algorithm with adaptive quantization, which can reduce the communication overhead considerably. The performance of this algorithm is analyzed, and an regret bound and an constraint violation bound are established, which are the same as (in order sense) those for existing algorithm transmitting raw decisions without quantization. We further extend the adaptive quantization method to the scenario of bandit feedback, where only the values of the local loss functions at two points are revealed at each time. A bandit OCO algorithm with adaptive quantization is developed and is shown to possess the same (in order sense) regret and constraint violation bounds as in the full information feedback case. Finally, numerical results on distributed online rate control problem are presented to corroborate the efficacy of the proposed algorithms.

中文翻译：

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具有自适应量化的分布式约束在线凸优化


在本文中，我们研究由参数服务器和客户端组成的系统中的分布式约束在线凸优化（OCO）问题。每个客户端都与局部约束函数和时变局部损失函数相关联，这些函数将按顺序公开。客户通过基于损失函数的因果信息选择顺序决策，寻求在总约束下最小化累积总损失。现有的分布式约束 OCO 算法要求客户端将其原始决策发送到服务器，从而导致许多应用程序无法承受巨大的通信开销。为了降低通信成本，我们设计了一种自适应量化方法，随着 OCO 算法的进展，以自适应方式调整量化器的中心和半径。我们首先检查完整信息反馈的场景，其中每次都会显示损失函数的完整信息。我们提出了一种具有自适应量化的分布式在线鞍点算法，可以大大减少通信开销。分析了该算法的性能，并建立了后悔边界和约束违反边界，它们与（在顺序意义上）传输未经量化的原始决策的现有算法相同。我们进一步将自适应量化方法扩展到老虎机反馈的场景，其中每次仅显示两个点的局部损失函数的值。开发了具有自适应量化的老虎机 OCO 算法，并证明其具有与全信息反馈情况相同的（按顺序意义）遗憾和约束违反界限。 最后，给出了分布式在线速率控制问题的数值结果，以证实所提出算法的有效性。