In this work we derive the performance achievable by a network of distributed agents that solve, adaptively and in the presence of communication constraints, a regression problem. Agents employ the recently proposed ACTC (adapt-compress-then-combine) diffusion strategy, where the signals exchanged locally by neighboring agents are encoded with randomized differential compression operators. We provide a detailed characterization of the mean-square estimation error, which is shown to comprise a term related to the error that agents would achieve without communication constraints, plus a term arising from compression. The analysis reveals quantitative relationships between the compression loss and fundamental attributes of the distributed regression problem, in particular, the stochastic approximation error caused by the gradient noise and the network topology (through the Perron eigenvector). We show that knowledge of such relationships is critical to allocate optimally the communication resources across the agents, taking into account their individual attributes, such as the quality of their data or their degree of centrality in the network topology. We devise an optimized allocation strategy where the parameters necessary for the optimization can be learned online by the agents. Illustrative examples show that a significant performance improvement, as compared to a blind (i.e., uniform) resource allocation, can be achieved by optimizing the allocation by means of the provided mean-square-error formulas.

翻译：本文推导了在通信约束条件下，分布式代理网络自适应求解回归问题所能达到的性能。代理采用近期提出的ACTC（自适应-压缩-然后-组合）扩散策略，其中相邻代理本地交换的信号通过随机微分压缩算子进行编码。我们提供了均方估计误差的详细特征描述，该误差包含一项与代理在无通信约束下所能达到的误差相关的项，以及另一项由压缩引起的项。分析揭示了压缩损失与分布式回归问题基本属性之间的定量关系，特别是由梯度噪声引起的随机逼近误差以及网络拓扑（通过Perron特征向量）的影响。我们表明，了解这些关系对于根据代理的个体属性（如其数据质量或网络拓扑中的中心度）在代理间优化分配通信资源至关重要。我们设计了一种优化的分配策略，其中优化所需的参数可由代理在线学习。示例表明，与盲目（即均匀）资源分配相比，通过利用所提供的均方误差公式优化分配，可以显著提升性能。