Distributed constrained online convex optimization with adaptive quantization

Abstract

In this paper, we study distributed constrained online convex optimization (OCO) problem in a system consisting of a parameter server and $n$ 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 $O\left(\sqrt{T}\right)$ regret bound and an $O\left(T^{\frac{3}{4}}\right)$ 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.