  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 establish an inner bound of error exponent
regions. 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.
