Succinct Explanations with Cascading Decision Trees

1 INTRODUCTION . Binary classification is the process of classifying the given input set into two classes based on some classification criteria . Binary classification is widely used in everyday life : for example , a typical application for binary classification is determining whether a patient has some disease by analyzing their comprehensive medical record . Existing work on binary classification mainly uses the accuracy of prediction as the main criterion for evaluating model performance . However , in order for a model to be useful in real-world applications , it is imperative that users are able to understand and explain the logic underlying model predictions . Model comprehensibility1 in some real-world applications , especially in the medical and scientific domains , is of the utmost importance . In these cases , users need to understand the classification model to scientifically explain the reasons behind the classification or even rely on the model itself to discover the possible solution to the target problem . It is difficult to provide explainability without sacrificing classification accuracy using current models . “ Black-box ” models such as deep neural network , random forests , and ensembles of classifiers tend to have the highest accuracy in binary classification Freitas ( 2014 ) ; Doilovi et al . ( 2018 ) . However , their opaque structure hinders understandability , making the logic behind the predictions difficult to trace . This lack of transparency may further discourage users from using the model Augasta & Kathirvalavakumar ( 2012 ) ; Van Assche & Blockeel ( 2007 ) . Decision tree models , on the other hand , have transparent decision making steps . A traversal of features on the decision path from the root to the leaf node is presented to users as a rule . Therefore , compared to other models , the decision tree model has historically been characterized as having 1In this paper , comprehensibility and interpretability are used interchangeably . high comprehensibility Freitas ( 2014 ) ; Doilovi et al . ( 2018 ) . However , whether models generated by classic decision trees provide enough comprehensibility has been challenged : “ decision trees [ ... ] can grow so large that no human can understand them or verify their correctness “ Caruana et al . ( 1999 ) , or they may contain subtrees with redundant attribute conditions , resulting in potential misinterpretation of the root cause for model predictions Freitas ( 2014 ) . The work presented in this paper introduces an algorithm for deriving succinct explanations for positive classifications , while maintaining the overall prediction accuracy . To this end , we introduce a novel cascading decision tree model . A cascading decision tree is a sequence of several smaller decision trees ( subtrees ) with the predefined tree depth . After every subtree sequentially follows another subtree , mimicking in this way a cascading waterfall , thus the name . The sequence ends when the subtree does not contain any leaves describing positively classified samples . Fig . 2 depicts one such cascading decision tree . The main idea behind cascading decision trees is that , while most algorithms for constructing decision trees are greedy and they try to classify as many samples as soon as possible , such classification results in large explanation paths for the samples in the lower levels of the tree . Instead , we construct a subtree of the predefined depth . That subtree contains a short explanation for the samples it managed to classify . However , the subtree with a short depth will misclassify samples . We next repeat the process on the training set with the samples that were previously classified positively removed . This way , the samples classified as positive in the second subtree will have a much shorter explanation path than they would in the original decision tree . In the cascading decision tree model , an explanation path for positively classified sample is the path that starts in the root of the corresponding subtree . We target explanations for only positive classifications , based on real-world motivation . In the medical domain , a positive classification result indicates that a person has the disease for which the test is being done NIH ( 2020 ) . The positive classification is also combined with additional testings needed for a full diagnosis CDC ( 2020 ) . Note that , if a practical application arises , our cascading decision trees model could easily be changed to target the negative classifications . Reducing the size and the depth of a decision tree to improve comprehensibility has been studied , both from a theoretical and a practical perspective . However , constructing such optimally small decision trees is an NP-complete problem Hyafil & Rivest ( 1976 ) , and the main drawback of these approaches is that the model is computationally too expensive to train . Even when using the stateof-the-art libraries Aglin et al . ( 2020 ) ; Verwer & Zhang ( 2019 ) , we observed that complexity . To illustrate this on an example , to learn a model on the Ionosphere dataset ( from the UCI Machine Learning Repository ) , the BinOCT tool needs approximately 10 minutes , while our approach completes this task in 1.1 seconds . We demonstrate the applicability of the cascading decision tree model in two ways . First , we use our model to perform standard binary classification on three numerical datasets from the UCI Machine Learning Repository . Second , we apply our model to a new application of binary classification , namely continuous integration ( CI ) build status prediction Santolucito et al . ( 2018 ) . Overall , we report that compared to the classical decision tree algorithm , our approach shortens the explanation depth for positive classifications by more than 40.8 % while maintaining the prediction accuracy . 2 MOTIVATING EXAMPLES . In this section we demonstrate how cascading decision trees can generate a shorter and more succinct explanation . The following simple synthetic example is contrived to illustrate our tool ’ s basic functionality . Given the dataset in Table1 , a classical decision tree will construct a model shown in Fig . 1 . Using the same dataset , our cascading decision trees algorithm generates a model with three subtrees in shown in Fig . 2 . Let ’ s assume that there is a new sample , “ Sample11 ” , with the feature vector ( F , F , F , T ) . Both models classify “ Sample11 ” with the same prediction result , “ Positive ” . However , the explanations extracted from these two models are different . In the classic decision tree model ( Fig . 1 ) , “ Sample11 ” falls into node ( 9 ) . Thus , the explanation path here is “ Feature1 = F ” , “ Feature2 = F ” and “ Feature3 = F ” , with the explanation depth of three . Using the cascading decision tree model ( Fig . 2 ) , “ Sample11 ” first falls into node ( 7 ) in the first subtree . Node ( 7 ) is not classifying positive samples , so “ Sample11 ” is passed to the second subtree and eventually falls into node ( 10 ) . This way , the explanation path for “ Sample11 ” is “ Feature4 = T ” , with an explanation depth of only one . 3 RELATED WORK . Ensemble Methods . The cascading decision trees algorithm pursues a different goal than existing ensemble methods , such as Bagging Breiman ( 1996 ) and Boosting Schapire ( 1990 ) . Ensemble methods for decision trees combine multiple weaker classifiers into a larger classifier with the goal of increasing overall accuracy , while cascading decision trees focus on shortening explanations . Bagging divides the dataset into multiple , smaller datasets , generates the classifier for each dataset and produces the final classification via a voting process . Boosting algorithms iteratively learn weak classifiers , gives them weights , and adds them one after another to generate a final strong classifier . The combination strategies of these ensemble methods obfuscate the traceability of a classic decision tree , which is the source of simple explanations for classifications Zhao & Sudha Ram ( 2004 ) . The limited model comprehensibility of Bagging and Boosting is also shared by other ensemble methods , which makes their output difficult to interpret . Oblique Decision Trees . Oblique decision trees Murthy et al . ( 1994 ) extend the classic decision tree model by allowing each decision node to combine checks against multiple features . This series of work contains a rich family of models such as multivariate decision trees Brodley & Utgoff ( 1995 ) , loose and tight coupling Gama & Brazdil ( 2000 ) , and constrained cascade generalization of decision trees Zhao & Sudha Ram ( 2004 ) . However , compared to cascading decision trees , this oblique design is not only the training stage computationally expensive Lee & Jaakkola ( 2020 ) , but also makes the explaining a classification in the decision-making stage opaque and hard to interpret Zhao & Sudha Ram ( 2004 ) . Optimal Decision Trees . Optimal decision tree algorithms learn models with the optimal prediction accuracy under the constraints of the predefined depth of the tree . However , learning an fixed depth optimal decision tree without heuristics is an NP-complete problem Hyafil & Rivest ( 1976 ) . The main drawback of using this approach in practice is that the model is too computationally expensive to train . For example , the state-of-the-art optimal decision trees implementations Aglin et al . ( 2020 ) ; Verwer & Zhang ( 2019 ) need ten minutes to train the ionosphere dataset2 . On the same dataset , we show in our evaluation ( Sec . 5 ) that our cascading decision trees learning process terminated in seconds with competitive accuracy ( cf . Verwer & Zhang ( 2019 ) ) . In addition , instead of being a direct competitor , an optimal decision tree learner could be used as the base decision tree inducer in our algorithm to potentially improve the prediction accuracy , though at the cost of training time . Model Interpretability . The comprehensibility of classification models , including decision tree models , has been extensively explored Doilovi et al . ( 2018 ) . Specifically , Quinlan ( 1987 ) demonstrated that small models are more interpretable than larger ones in the decision tree . Similarly , Huysmans et al . ( 2011 ) ran a user study to illustrate that larger representations result in a decrease in both the accuracy of user ’ s answers and users confidence in the model itself . This work supports the motivation that minimizing explanations is a valuable direction to explore . 4 METHODOLOGY . Formalism . For interpretability , we take inspiration from the LIME system Ribeiro et al . ( 2016 ) , which states that interpretable explanations should “ provide qualitative understanding between joint values of input variables and the resulting predicted response value ” . In our work , we specifically focus on binary classification , and only target explanations for positive classifications . An explanation in our system should capture the essence of the positive classification . That is , the explanation identifies the key input values that contributed to the positive classification . The classification should still be the same , no matter the state of the input values not included in the explanation . It is possible the sample will be classified positively in a different way , but as long as the input values in the explanation are unchanged , the classification will still be positive . To define a valid explanation , we consider a dataset of samples with binary labels . Without loss of generality ( as we consider only the decision tree setting ) , let ( x , y ) : ( 2n , C ) denote a sample as an n-dimensional Boolean vector and its ground-truth classification respectively . The number of classes is |C| - in our application we consider only binary classification , where |C| = 2 , though 2https : //archive.ics.uci.edu/ml/datasets/Ionosphere the definition of valid explanations generalizes to larger |C| . Given a test set T : { ( x , y ) } , we can learn a model M : 2n → C which maps input vectors of samples to a classification . An explanatory model , M+ : 2n → E expands upon the modelM to additionally map input vectors to explanations , E . An explantation E : 2n is a Boolean vector , where every true element means that the feature at the corresponding index in the input vector is a necessary part of the classification result . The size of an explanation , |E| = ∑ i∈n E [ i ] , is the number of true elements in the explanation vector . We define the validity of an explanation via array multiplication . An explanation , E , is valid over an input vector x , when any part of the input vector not captured by the explanation may be modified without changing the classification result of the model . isV alid ( E , x ) ⇐⇒ ∀z ∈ 2n . M ( x ) =M ( E ∗ x+ z ∗ ¬ E ) Example 1 A trivial explanation is always available for any sample - the explanation that every element of the sample vector is important . From Fig . 1 , for input vector “ Sample5 ” , a valid explanation is E = [ 1 , 1 , 1 , 1 ] . This means that as long as no features change , the classification remains the same . Example 2 In general , a shorter explanation can be extracted from classic decision trees . As an example , the explanation of “ Sample5 ” from the decision tree in Fig . 1 is the explanation that keeps only the elements in the decision path of the tree : E = [ 1 , 1 , 1 , 0 ] . This is the commonly accepted approach to explanations of decision tree classifications . Example 3 An even shorter explanation is possible with cascading decision trees . Looking again to “ Sample5 ” as classified by the cascading decision tree in Fig . 2 , a valid explanation keeps only “ Feature 4 ” - E = [ 0 , 0 , 0 , 1 ] . Although changes to “ Feature 1-3 ” may change the decision path , the classification remains the same as long as “ Feature 4 ” is unchanged . A explanatory model is valid if and only if the provided explanation is valid for all possible inputs , that is : ∀x ∈ 2n . isV alid ( M+ ( x ) , x ) . We note that a valid explanatory model only needs to correctly explain its classifications for all inputs - not correctly classify all inputs . The above definition of a valid explanation is always true of the decision path of a classic decision tree . This aligns with the notion that the decision path of a classification in a classic decision tree is a valid explanation for the resultant classification of a sample . As we will show , this definition of a valid explanation is also always true of the shortened “ explanation path ” we propose with our cascading decision trees . Cascading decision trees provide explanations only over samples classified as positive , though the definition is more general . Building Cascading Decision Trees . The cascading decision trees algorithm is described through pseudocode in Algorithm 1 . Our insight is in the training process - it is the goal of each cascading decision subtree to identify the smallest set of features that can classify as many Positive samples as possible , without misclassifying any Negative samples . Specifically , the procedure Fit builds a classic decision tree , clf , on our training set , T R and adds it to the cascading tree list - initially this list is empty . For every leaf node we compute the mixed value , which is the percentage of samples from T R classified by this leaf node that are also Positive samples . Positive nodes are leaf nodes with a mixed value greater than the threshold ( we use 80 % in this paper ) . If clf has no Positive nodes , it means we have learned a sufficiently good classifier and we stop ( Line 11 ) . Otherwise , we first remove samples truly classified by Positive nodes in this clf from T R , and then obtain a new T R to use in the next iteration of the loop ( Line 14 ) . Our cascading decision trees algorithm1 presented here has no pruning phase . However , our algorithm is generic enough to be combined with any pruning techniques Esposito et al . ( 1997 ) and different goodness measurement for decision nodes split , such as entropy Shannon ( 1948 ) and gini impurity Havrda et al . ( 1967 ) . The time complexity of cascading decision tree algorithm is bounded by the size of the training set . Suppose the training time for building classic decision trees is a function of the number of the training samples , n and the number of features k. We use the decision tree module in scikit-learn3 as our base classic decision tree inducer , which is built upon the CART Breiman et al . ( 1984 ) method . Since features are recursively reused in every decision node based on a numerical splitting criterion , 3https : //scikit-learn.org/stable/ Algorithm 1 Build Cascading Decision Trees Classifier Input : T R : Labelled Training Set Input : θ : Threshold Input : depth : Maximum depth of each decision subtree Output : cascadingTree : A list of cascading decision subtrees 1 : procedure FIT ( T R , θ , depth ) 2 : done = False 3 : cascadingTree = [ ] 4 : while done = False do 5 : clf = classicDecisionTree.fit ( T R , depth ) 6 : cascadingTree.append ( clf ) 7 : done = True 8 : for leafNode in clf do 9 : if ( mixed ( leafNode ) > = θ ) then 10 : done = False 11 : if ( done = True ) then 12 : Break 13 : else 14 : T R = filter ( clf .predict ( sample , θ ) = Positive , T R ) return cascadingTree the depth of the decision tree is bounded by the number of the training samples n in our model . Therefore , the time complexity for building one base decision tree is bounded by O ( n2k ) . According to our cascading decision trees algorithm1 , after building one decision subtree , samples that are classified as True Positive are removed from the next round of decision subtree construction . In the worst case , every time , only one True Positive is classified in the current decision tree . The time complexity for building the next cascading decision subtree is in O ( ( n− ) 2k ) . Therefore , the overall cascading decision trees training time T is bounded by : O ( ∑n m=1m 2k ) = O ( n ( n+1 ) ( 2n+1 ) 6 k ) = O ( n 3k ) . Testing Cascading Decision Trees . Our cascading decision trees testing process is described in Algorithm 2 . In the procedure Test , we run all decision trees sequentially , and we report the decision path of that tree only as our explanation path . Algorithm 2 Test Cascading Decision Trees Classifier Input : cascadingTree : A list of cascading decision tree classfier Input : θ : Threshold Output : Prediction Result : A boolean variable , Postive or Negative . Output : Explanation Paths : A conjunction of boolean statements to explain the decisions . 1 : procedure TEST ( cascadingTree , θ ) 2 : for clf in cascadingTree do 3 : if clf.predict ( sample , θ ) = Positive then 4 : return ( Postive , clf .path ( sample ) ) return ( Negative )

Summary: This paper introduces the Cascading Decision Tree, a novel variant of decision trees with permits to extract short explanations for a class of interest. The idea is to realize a cascade of small decision trees: at a certain level, the tree is built using all points except the positive ones correctly classified by trees in previous levels. The method has been tested using three standard datasets and a novel application.