## 5 Experimental Results  
We apply Algorithm 4.1 to data streams generated from the Reuters Corpus RCV1-V2. The data is available from [16] and consists of \(781,265\) tokenized documents with DID (document ID) ranging from 2651 to 810596. The methodology described below attempts to follow that presented in [8]. We simulate \(n\) streams by arranging the feature vectors in ascending order with respect to DID, and selecting feature vectors for the stream in the round robin fashion. In the Reuters Corpus RCV1-V2 each document is labeled as belonging to one or more categories. We label a vector as \"relevant\" if it belongs to the \"CORPORATE/INDUSTRIAL\" (\"CCAT\") category, and \"spam\" otherwise. Following [9] we focus on three features: \"bosnia\", \"ipo\", and \"febru\". Each experiment was performed with 10 nodes, where each node holds a sliding window containing the last 6700 documents it received. First we use \(67,000\) documents to generate initial sliding windows. The remaining \(714,265\) documents are used to generate data streams, hence the selected feature information gain is computed \(714,265\) times. Based on all the documents contained in the sliding window at each one of the \(714,266\) time instances, we compute and graph \(714,266\) information gain values for the feature \"bosnia\" (see Figure 5). For the experiments described below the threshold value \(r\) is predefined, and the goal is to monitor the inequality \(f(\mathbf{v})-r>0\) while minimizing communication between the nodes. From now on we shall assume simultaneous arrival of a new text at each node. Figure 4: failed cover by \(l_{1}\) balls. As new texts arrive, the local constraint (_i.e._, inequalities \(\|{\bf v}_{j}-{\bf v}_{j}(t_{i})\|<\delta\), \(j=1,\ldots,n\)) at each node is verified. If at least one node violates the local constraint, the average \({\bf v}(t_{i})\) is updated. Our numerical experiment with the feature \"bosnia\", the \(l_{2}\) norm, and the threshold \(r=0.0025\) (reported in [8] as the threshold for feature \"bosnia\" incurring the highest communication cost) shows overall \(4006\) computation of the mean vector. An application of Equation (14) yields \(240,360\) messages. We repeat this experiment with \(l_{\infty}\), and \(l_{1}\) norms. The results obtained and collected in Table 2 show that the smallest number of the mean updates is required for the \(l_{1}\) norm. Throughout the iterations the mean \({\bf v}(t_{i})\) goes through a sequence of updates, and the values \(f({\bf v}(t_{i}))\) may be larger than, equal to, or less than the threshold \(r\). We monitor the case \(f({\bf v})\leq r\) the same way as that of \(f({\bf v})>r\). In addition to the number of mean computations, we collect statistics concerning \"crossings\" (or lack of thereof), _i.e._, number of instances when the location of the mean \({\bf v}\) and its update \({\bf v}^{\prime}\) relative to the surface \(\left\{{\bf x}\ :\ {\bf x}\in{\bf R}^{d},\ f({\bf x})=r\right\}\) are either identical or different. Specifically over the monitoring period we denote by: \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Distance** & **Mean Comps** & **Messages** & **LL** & **LG** & **GL** & **GG** \\ \hline \(l_{2}\) & 4006 & 240,360 & 959 & 2 & 2 & 3043 \\ \(l_{\infty}\) & 3801 & 228,060 & 913 & 2 & 2 & 2884 \\ \(l_{1}\) & 3053 & 183,180 & 805 & 2 & 2 & 2244 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: number of mean computations, messages, and crossings per norm for feature “bosnia” with threshold \(r=0.0025\). Figure 5: information gain values for the feature “bosnia”. 1. \"LL\" the number of instances when \(f(\mathbf{v})<r\) and \(f(\mathbf{v}^{\prime})<r\),
2. \"LG\" the number of instances when \(f(\mathbf{v})<r\) and \(f(\mathbf{v}^{\prime})>r\),
3. \"GL\" the number of instances when \(f(\mathbf{v})>r\) and \(f(\mathbf{v}^{\prime})<r\),
4. \"GG\" the number of instances when \(f(\mathbf{v})>r\) and \(f(\mathbf{v}^{\prime})>r\).