In this paper, we study a distributed learning problem constrained by constant communication bits. Specifically, we consider the distributed hypothesis testing (DHT) problem where two distributed nodes are constrained to transmit a constant number of bits to a central decoder. In such cases, we show that in order to achieve the optimal error exponents, it suffices to consider the empirical distributions of observed data sequences and encode them to the transmission bits. With such a coding strategy, we develop a geometric approach in the distribution spaces and characterize the optimal schemes. In particular, we show the optimal achievable error exponents and coding schemes for the following cases: (i) both nodes can transmit $\log_23$ bits; (ii) one of the nodes can transmit $1$ bit, and the other node is not constrained; (iii) the joint distribution of the nodes are conditionally independent given one hypothesis. Furthermore, we provide several numerical examples for illustrating the theoretical results. Our results provide theoretical guidance for designing practical distributed learning rules, and the developed approach also reveals new potentials for establishing error exponents for DHT with more general communication constraints.

中文翻译：

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具有恒定通信位的分布式学习

在本文中，我们研究了一个受恒定通信比特约束的分布式学习问题。具体来说，我们考虑分布式假设检验 (DHT) 问题，其中两个分布式节点被限制为向中央解码器传输恒定数量的比特。在这种情况下，我们表明为了获得最佳误差指数，考虑观察到的数据序列的经验分布并将它们编码为传输比特就足够了。通过这样的编码策略，我们在分布空间中开发了一种几何方法并表征了最佳方案。特别是，我们展示了以下情况的最佳可实现误差指数和编码方案：（i）两个节点都可以传输 $\log_23$ 位；(ii) 其中一个节点可以传输$1$bit，另一个节点不受约束；(iii) 给定一个假设，节点的联合分布是条件独立的。此外，我们提供了几个数值例子来说明理论结果。我们的结果为设计实用的分布式学习规则提供了理论指导，并且所开发的方法还揭示了为具有更一般通信约束的 DHT 建立误差指数的新潜力。