# BENK: The Beran Estimator with Neural Kernels for Estimating the Heterogeneous Treatment Effect
## 8 Conclusions  
A new method called BENK for solving the CATE problem under the condition of censored data has been presented. It extends the idea behind TNW-CATE proposed in [16] to the case of censored data. In spite of many similar parts of TNW-CATE and BENK, they are different because BENK is based on using the Beran estimator for training and can be successfully applied to survival analysis of controls and treatments. However, TNW-CATE and BENK use the same idea to train neural kernels: implementation as neural networks instead of using standard kernels. It is also interesting to point out that BENK does not require oneto have a large dataset for training, even though the neural network is used for implementing the kernels. This is due to a special way that is proposed to train the network, which considers pairs of examples from the control group for training, as in Siamese neural networks. Our \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Generating Functions**} \\ \hline
**Model** & **Spiral** & **Bell-Shaped** & **Circular** & \(p\)**-Value** \\ \hline T-NW & 5.876 & 4.868 & 5.713 & 0.0457 \\ \hline S-NW & 5.759 & 5.868 & 5.946 & 0.0121 \\ \hline X-NW & 4.985 & 5.090 & 6.317 & 0.0149 \\ \hline T-Cox & 6.198 & 6.518 & 6.126 & 0.0128 \\ \hline S-Cox & 5.959 & 5.941 & 5.963 & 0.0112 \\ \hline X-Cox & 6.331 & 7.396 & 8.357 & 0.0178 \\ \hline T-SF & 3.721 & 5.563 & 6.460 & 0.0401 \\ \hline S-SF & 5.959 & 5.900 & 5.882 & 0.0035 \\ \hline X-SF & 4.853 & 6.339 & 7.176 & 0.0154 \\ \hline BENK & 2.373 & 3.288 & 3.570 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The PEHE values of CATE for different models obtained via different generating functions and the corresponding \(p\)-values. Figure 12: The PEHE metric as a function of the amount of censored observations in the training dataset when the circular function is used for generating examples. experiments have illustrated the outperforming characteristics of BENK. At the same time, we have to point out some disadvantages of BENK. First, it has many tuning parameters, including parameters of the neural network and parameters of training \(n\) and \(N\), such that the training time may be significantly increased in comparison with other methods of solving the CATE problem. Second, BENK assumes that the feature vector domains are similar for controls and treatments. This does not mean that they have to totally coincide, but the corresponding difference in domains should not be very large. A method which could take into account a possible difference between the feature vector domains for controls and treatments can be regarded as a direction for further research. An idea behind the method is to combine the domain adaptation models and BENK. Another direction for further research is to study robust versions of BENK when there are anomalous observations that may impact training the neural network. An idea behind the robust version is to use attention weights for feature vectors and also to introduce additional attention weights for predictions. It should be noted that the Beran estimator is one of several estimators that are used in survival analysis. Moreover, we have studied only the difference in expected lifetimes as a definition of the CATE in the case of censored data. There are other definitions, for instance, the difference in SFs and the hazard ratio, which may lead to more interesting models. Therefore, BENK implementations and studies using other estimators and definitions of the CATE can be also considered as directions for further research. The proposed method can be used in applications that are different from medicine. For example, it can be applied to selection and control of the most efficient regimes in the Internet of Things. This is also an interesting direction for further research. Conceptualization, S.K., L.U. and A.K. ; methodology, L.U. and V.M. ; software, S.K. and A.K. ; validation, S.K., V.M. and A.K. ; formal analysis, A.K. and L.U. ; investigation, L.U., A.K. and V.M. ; resources, A.K. and V.M.