A New Interpretable Neural Network-Based Rule Model for Healthcare
  Decision Making
In healthcare applications, understanding how machine/deep learning models
make decisions is crucial. In this study, we introduce a neural network
framework, $\textit{Truth Table rules}$ (TT-rules), that combines the global
and exact interpretability properties of rule-based models with the high
performance of deep neural networks. TT-rules is built upon $\textit{Truth
Table nets}$ (TTnet), a family of deep neural networks initially developed for
formal verification. By extracting the necessary and sufficient rules
$\mathcal{R}$ from the trained TTnet model (global interpretability) to yield
the same output as the TTnet (exact interpretability), TT-rules effectively
transforms the neural network into a rule-based model. This rule-based model
supports binary classification, multi-label classification, and regression
tasks for small to large tabular datasets. After outlining the framework, we
evaluate TT-rules' performance on healthcare applications and compare it to
state-of-the-art rule-based methods. Our results demonstrate that TT-rules
achieves equal or higher performance compared to other interpretable methods.
Notably, TT-rules presents the first accurate rule-based model capable of
fitting large tabular datasets, including two real-life DNA datasets with over
20K features.
Adrien Benamira, Tristan Guerand, Thomas Peyrin
# A New Interpretable Neural Network-Based Rule Model for Healthcare Decision Making

###### Abstract

In healthcare applications, understanding how machine/deep learning models make decisions is crucial. In this study, we introduce a neural network framework, _Truth Table rules_ (TT-rules), that combines the global and exact interpretability properties of rule-based models with the high performance of deep neural networks. TT-rules is built upon _Truth Table nets_ (TTnet), a family of deep neural networks initially developed for formal verification. By extracting the necessary and sufficient rules \(\mathcal{R}\) from the trained TTnet model (global interpretability) to yield the same output as the TTnet (exact interpretability), TT-rules effectively transforms the neural network into a rule-based model. This rule-based model supports binary classification, multi-label classification, and regression tasks for small to large tabular datasets.

After outlining the framework, we evaluate TT-rules' performance on healthcare applications and compare it to state-of-the-art rule-based methods. Our results demonstrate that TT-rules achieves equal or higher performance compared to other interpretable methods. Notably, TT-rules presents the first accurate rule-based model capable of fitting large tabular datasets, including two real-life DNA datasets with over 20K features.

## 1 Related Work

Traditional rule-based models, such as decision trees [5], rule lists [20, 3, 9], linear models, and rule sets [13, 7, 8, 18, 24], are commonly used for interpretable classification and regression tasks. However, these models face limitations in handling large datasets, binary classification tasks, and capturing complex feature relationships, which can result in reduced accuracy and limited practicality [25, 23]. To overcome these challenges, recent work by Benamira _et al._ introduced an architecture encoded into CNF formulas, demonstrating scalability on large datasets [4, 6]. Our objective is to extend this approach to handle diverse classification tasks and regression on tabular datasets of varying feature dimensions.

There have been investigations into the connection between deep neural networks (DNNs) and rule-based models. Notable works include DNF-net [1], which focuses on the activation function, and RRL [23], which addresses classification tasks but raises concerns about interpretability due to its complexity and time-consuming training process. Another architecture, Neural Additive Models (NAMs) [2], combines the flexibility of DNNs with the interpretability of additive models but deviates from the strict rule-based model paradigm, posing challenges in interpretation, especially with a large number of features.

## 2 Methodology

This paper introduces a novel neural network framework that effectively combines the interpretability of rule-based models with the high performance of DNNs. Our framework, called TT-rules, builds upon the advancements made by Benamira _et al._[4] Benamira _et al._[4] introduced a new Convolutional Neural Network (CNN) filter function called the Learning Truth Table (LTT) block. The LTT block has the unique property of its complete distribution being computable in constant and practical time, regardless of the architecture. This allows the transformation of the LTT block from weights into an exact mathematical Boolean formula. Since an LTT block is equivalent to a CNN filter, **the entire neural network model, known as Truth Table net (TTnet), can itself be represented as a Boolean formula.** We then optimize our formula set \(\mathcal{R}\) in two steps. We automatically integrate human logic into the truth tables. This reduces the size of each rule in the set \(\mathcal{R}\). Then we analyze the correlation to decrease the number of rules in \(\mathcal{R}\). These optimizations, specific to the TT-rules framework, automatically and efficiently transform the set \(\mathcal{R}\) into an optimized set in constant time. To enhance the interpretability of the model, we convert all boolean formulas into Reduced Ordered Binary Decision Diagrams. An example is given Figure 1.

## 3 Experiments

### Datasets

We utilized a variety of healthcare datasets for our study, including the Diabetes 130 US-Hospitals dataset for multi-classification1[10], two single-cell RNA-seq analysis datasets (head and neck cancer2[17] and melanoma3[21]), the Breast Cancer Wisconsin (Original) dataset4, and the TCGA lung cancer dataset for regression5[15].

Footnote 1: [https://bit.ly/diabetes_130_uci](https://bit.ly/diabetes_130_uci)

Footnote 2: [https://bit.ly/neck_head_rna](https://bit.ly/neck_head_rna)

Footnote 3: [https://bit.ly/meloma_rna](https://bit.ly/meloma_rna)

Footnote 4: [https://archive.ics.uci.edu/dataset/15/breast+cancer+wisconsin+original](https://archive.ics.uci.edu/dataset/15/breast+cancer+wisconsin+original)

Footnote 5: [https://bit.ly/tcga_lung_rna](https://bit.ly/tcga_lung_rna)

Our TT-rules framework's scalability is demonstrated using two DNA datasets. These include single-cell RNA-seq analysis datasets for head and neck cancer, melanoma cancer [17, 21], and the TCGA lung cancer dataset [15]. These datasets contain 23689 and 20530 features, respectively, and are commonly used in real-life machine learning applications [14, 11, 19, 22]. In the melanoma cancer setup, we trained on the head and neck dataset [17] and tested on the melanoma dataset [21] following established literature [14, 11, 19, 22].
### Performance Comparison

Table 1 presents a comparison of various rule-based models, including ours, on the datasets introduced before, in terms of RMSE, AUC and Accuracy. Our proposed model outperforms the others in terms of accuracy on the Diabetes dataset and on the Breast Cancer dataset. XGBoost and DNNs performs better on Diabetes but worse on bigger datasets as shown in the next section. Although GL provides a better tradeoff between performance and complexity, we highlight that GL does not support multi-class classification tasks and is not scalable for larger datasets such as DNA datasets, as shown in the next section.

### Scalability Comparison

Our TT-rules framework demonstrated excellent scalability to real-life datasets with up to 20K features. This result is not surprising, considering the original TTnet paper [4] showed the architecture's ability to scale to ImageNet. Furthermore, our framework's superiority was demonstrated by outperforming other rule-based models that failed to converge to such large datasets (GL [24]). Regarding performance, the TT-rules framework outperforms all other methods. Our approach not only scales but also reduces the input feature set, acting as a feature selection method. We generated a set of 1064 rules out of 20530 features for the regression problem, corresponding to a drastic reduction in complexity. For the binary classification dataset, we generated 9472 rules, which more then halved the input size from 23689 to 9472.

## 4 Conclusion

In conclusion, our proposed TT-rules framework provides a new and optimized approach for achieving global and exact interpretability in regression and classification tasks. With its ability to scale on large datasets and its potential for feature reduction, the TT-rules framework appears as a valuable tool towards explainable artificial intelligence for healthcare applications.

\begin{table}
\begin{tabular}{l|c|c c|c} \hline \hline  & **Regression (RMSE)** & **Binary classification (AUC)** & **Multi-classification (Accuracy)** \\ \hline  & TCCA Cancer & Melanoma & Breast Cancer & Diabetes \\ continuous/binary \# & 0/20530 & 0/23689 features & 0/81 features & 43/296 features \\ \hline Linear/ log & 0.092 & 0.833 & 0.985 & 0.581 \\ DT & - & - & 0.908 & 0.572 \\ GL & - & - & 0.984 & - \\ TT-rules (Ours) & 0.029 & 0.835 & 0.986 & 0.584 \\ \hline Random Forest & 0.42 & 0.729 & 0.950 & 0.587 \\ DNNs & 0.028 & 0.725 & 0.982 & 0.603 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison machine learning dataset of our method to Linear/Logistic Regression [16], Decision Trees (DT) [16], GL [24], Random Forest [12] and DNNs. Means and standard deviations are reported from 5-fold cross validation. Our TT-rules models were trained with a final linear regression with weights as floating points for the Breast Cancer and Diabetes dataset for better performances. The two other datasets were trained with a sparse binary linear regression to reduce the number of final features. The lower RMSE the better, the higher AUC/Accuracy the better.

Figure 1: Our neural network model trained on the Breast Cancer dataset in the form of Boolean decision trees: the output of the DNN and the output of these decision trees are the same, reaching 99.30% AUC. On the same test set, Random Forest reaches 95.08% AUC, Decision Tree 90.36% AUC and XGBoost 97.79% AUC. Each rule \(r_{i}\) is a function \(r_{i}:\{0,1\}^{n}\mapsto\{-1,0,1\}\), i.e for each data sample I we associate for each rule \(r_{i}\) a score which is in \(\{-1,0,1\}\). The prediction of our classifier is then as stated above. As our model has 24 rules, we have only reported two positive rules out of the 24 to provide an example of the type of rules obtained.