Classification in a Large Network

We construct and analyze the communication cost of protocols (interactive and one-way) for classifying ${\mathbf X}=(X_1,X_2,\ldots,X_n) \in [0,1)^n \subset \mathbb{R}^n$, in a network with $n\geq 2$ nodes, with $X_i$ known only at node $i$. The classifier takes the form $\sum_{i=1}^nh_iX_i \gtrless a$, with weights $h_i \in \{-1,+1\}$. The interactive protocol (a zero-error protocol) exchanges a variable number of messages depending on the input ${\mathbf X}$ and its sum rate is directly proportional to its mean stopping time. An exact analysis, as well as an approximation of the mean stopping time is presented and shows that it depends on $\gamma=\alpha+(1/2-\beta)$, where $\alpha=a/n$ and $\beta=m/n$, with $m$ being the number of positive weights. In particular, the mean stopping time grows logarithmically in $n$ when $\gamma=0$, and is bounded in $n$ otherwise. Comparisons show that the sum rate of the interactive protocol is smaller than that of the one-way protocol when the error probability for the one-way protocol is small, with the reverse being true when the error probability is large. Comparisons of the interactive protocol are also made with lower bounds on the sum rate.