In this paper, we consider decentralized optimization problems where agents have individual cost functions to minimize subject to subspace constraints that require the minimizers across the network to lie in low-dimensional subspaces. This constrained formulation includes consensus or single-task optimization as special cases, and allows for more general task relatedness models such as multitask smoothness and coupled optimization. In order to cope with communication constraints, we propose and study an adaptive decentralized strategy where the agents employ differential randomized quantizers to compress their estimates before communicating with their neighbors. The analysis shows that, under some general conditions on the quantization noise, and for sufficiently small step-sizes $\mu$, the strategy is stable both in terms of mean-square error and average bit rate: by reducing $\mu$, it is possible to keep the estimation errors small (on the order of $\mu$) without increasing indefinitely the bit rate as $\mu\rightarrow 0$. Simulations illustrate the theoretical findings and the effectiveness of the proposed approach, revealing that decentralized learning is achievable at the expense of only a few bits.

翻译：本文考虑去中心化优化问题，其中智能体需在子空间约束下最小化各自代价函数——这种约束要求网络中各极小值点位于低维子空间中。该约束形式将共识优化或单任务优化作为特例，并能刻画更广义的任务相关性模型，如多任务平滑性与耦合优化。为应对通信限制，我们提出并研究了一种自适应去中心化策略：智能体在向邻居发送估计值前，采用差分随机量化器进行压缩。分析表明，在量化噪声满足一般性条件且步长$\mu$足够小时，该策略在均方误差与平均比特率方面均保持稳定：通过减小$\mu$，可在$\mu\rightarrow 0$时避免比特率无限增长的同时，将估计误差控制在$\mu$量级。仿真结果验证了理论发现与所提方法的有效性，揭示出仅需数比特开销即可实现去中心化学习。