NBDT: NEURAL-BACKED DECISION TREE
Year: 2021

Abstract:
Machine learning applications such as finance and medicine demand accurate and justifiable predictions, barring most deep learning methods from use. In response, previous work combines decision trees with deep learning, yielding models that (1) sacrifice interpretability for accuracy or (2) sacrifice accuracy for interpretability. We forgo this dilemma by jointly improving accuracy and interpretability using Neural-Backed Decision Trees (NBDTs). NBDTs replace a neural network’s final linear layer with a differentiable sequence of decisions and a surrogate loss. This forces the model to learn high-level concepts and lessens reliance on highlyuncertain decisions, yielding (1) accuracy: NBDTs match or outperform modern neural networks on CIFAR, ImageNet and better generalize to unseen classes by up to 16%. Furthermore, our surrogate loss improves the original model’s accuracy by up to 2%. NBDTs also afford (2) interpretability: improving human trust by clearly identifying model mistakes and assisting in dataset debugging. Code and pretrained NBDTs are at github.com/alvinwan/neural-backed-decision-trees.

heading: 1 INTRODUCTION
text: Many computer vision applications (e.g. medical imaging and autonomous driving) require insight into the model’s decision process, complicating applications of deep learning which are traditionally black box. Recent efforts in explainable computer vision attempt to address this need and can be grouped into one of two categories: (1) saliency maps and (2) sequential decision processes. Saliency maps retroactively explain model predictions by identifying which pixels most affected the prediction. However, by focusing on the input, saliency maps fail to capture the model’s decision making process. For example, saliency offers no insight for a misclassification when the model is “looking” at the right object for the wrong reasons. Alternatively, we can gain insight into the model’s decision process by breaking up predictions into a sequence of smaller semantically meaningful decisions as in rule-based models like decision trees. However, existing efforts to fuse deep learning and decision trees suffer from (1) significant accuracy loss, relative to contemporary models (e.g., residual networks), (2) reduced interpretability due to accuracy optimizations (e.g., impure leaves and ensembles), and (3) tree structures that offer limited insight into the model’s credibility.
To address these, we propose Neural-Backed Decision Trees (NBDTs) to jointly improve both (1) accuracy and (2) interpretability of modern neural networks, utilizing decision rules that preserve (3) properties like sequential, discrete decisions; pure leaves; and non-ensembled predictions. These properties in unison enable unique insights, as we show. We acknowledge that there is no universally-accepted definition of interpretability (Lundberg et al., 2020; Doshi-Velez & Kim, 2017; Lipton, 2016), so to show interpretability, we adopt a definition offered by Poursabzi-Sangdeh et al. (2018): A model is interpretable if a human can validate its prediction, determining when the model has made a sizable mistake. We picked this definition for its importance to downstream benefits we can evaluate, specifically (1) model or dataset debugging and (2) improving human trust. To accomplish this, NBDTs replace the final linear layer of a neural network with a differentiable oblique decision tree and, unlike its predecessors (i.e. decision trees, hierarchical classifiers), uses a hierarchy derived from model parameters, does not employ a hierarchical softmax, and can be created from any existing classification neural network without architectural modifications. These improvements
⇤denotes equal contribution
tailor the hierarchy to the network rather than overfit to the feature space, lessens the decision tree’s reliance on highly uncertain decisions, and encourages accurate recognition of high-level concepts. These benefits culminate in joint improvement of accuracy and interpretability. Our contributions:
1. We propose a tree supervision loss, yielding NBDTs that match/outperform and outgeneralize modern neural networks (WideResNet, EfficientNet) on ImageNet, TinyImageNet200, and CIFAR100. Our loss also improves the original model by up to 2%.
2. We propose alternative hierarchies for oblique decision trees – induced hierarchies built using pre-trained neural network weights – that outperform both data-based hierarchies (e.g. built with information gain) and existing hierarchies (e.g. WordNet), in accuracy.
3. We show NBDT explanations are more helpful to the user when identifying model mistakes, preferred when using the model to assist in challenging classification tasks, and can be used to identify ambiguous ImageNet labels.

heading: 2 RELATED WORKS
text: Saliency Maps. Numerous efforts (Springenberg et al., 2014; Zeiler & Fergus, 2014; Simonyan et al., 2013; Zhang et al., 2016; Selvaraju et al., 2017; Ribeiro et al., 2016; Petsiuk et al., 2018; Sundararajan et al., 2017) have explored the design of saliency maps identifying pixels that most influenced the model’s prediction. White-box techniques (Springenberg et al., 2014; Zeiler & Fergus, 2014; Simonyan et al., 2013; Selvaraju et al., 2017; Sundararajan et al., 2017) use the network’s parameters to determine salient image regions, and black-box techniques (Ribeiro et al., 2016; Petsiuk et al., 2018) determine pixel importance by measuring the prediction’s response to perturbed inputs. However, saliency does not explain the model’s decision process (e.g. Was the model confused early on, distinguishing between Animal and Vehicle? Or is it only confused between dog breeds?).
Transfer to Explainable Models. Prior to the recent success of deep learning, decision trees were state-of-the-art on a wide variety of learning tasks and the gold standard for interpretability. Despite this recency, study at the intersection of neural network and decision tree dates back three decades, where neural networks were seeded with decision tree weights (Banerjee, 1990; 1994; Ivanova & Kubat, 1995a;b), and decision trees were created from neural network queries (Krishnan et al., 1999; Boz, 2000; Dancey et al., 2004; Craven & Shavlik, 1996; 1994), like distillation (Hinton et al., 2015). The modern analog of both sets of work (Humbird et al., 2018; Siu, 2019; Frosst & Hinton, 2017) evaluate on feature-sparse, sample-sparse regimes such as the UCI datasets (Dua & Graff, 2017) or MNIST (LeCun et al., 2010) and perform poorly on standard image classification tasks.
Hybrid Models. Recent work produces hybrid decision tree and neural network models to scale up to datasets like CIFAR10 (Krizhevsky, 2009), CIFAR100 (Krizhevsky, 2009), TinyImageNet (Le & Yang, 2015), and ImageNet (Deng et al., 2009). One category of models organizes the neural network into a hierarchy, dynamically selecting branches to run inference (Veit & Belongie, 2018; McGill & Perona, 2017; Teja Mullapudi et al., 2018; Redmon & Farhadi, 2017; Murdock et al., 2016). However, these models use impure leaves resulting in uninterpretatble, stochastic paths. Other approaches fuse deep learning into each decision tree node: an entire neural network (Murthy et al., 2016), several layers (Murdock et al., 2016; Roy & Todorovic, 2016), a linear layer (Ahmed et al., 2016), or some other parameterization of neural network output (Kontschieder et al., 2015). These models see reduced interpretability by using k-way decisions with large k (via depth-2 trees) (Ahmed et al., 2016; Guo et al., 2018) or employing an ensemble (Kontschieder et al., 2015; Ahmed et al., 2016), which is often referred to as a “black box” (Carvalho et al., 2019; Rudin, 2018).
Hierarchical Classification (Silla & Freitas, 2011). One set of approaches directly uses a preexisting hierarchy over classes, such as WordNet (Redmon & Farhadi, 2017; Brust & Denzler, 2019; Deng et al.). However conceptual similarity is not indicative of visual similarity. Other models build a hierarchy using the training set directly, via a classic data-dependent metric like Gini impurity (Alaniz & Akata, 2019) or information gain (Rota Bulo & Kontschieder, 2014; Biçici et al., 2018). These models are instead prone to overfitting, per (Tanno et al., 2019). Finally, several works introduce hierarchical surrogate losses (Wu et al., 2017; Deng et al., 2012), such as hierarchical softmax (Mohammed & Umaashankar, 2018), but as the authors note, these methods quickly suffer from major accuracy loss with more classes or higher-resolution images (e.g. beyond CIFAR10). We demonstrate hierarchical classifiers attain higher accuracy without a hierarchical softmax.

heading: 3 METHOD
text: Neural-Backed Decision Trees (NBDTs) replace a network’s final linear layer with a decision tree. Unlike classical decision trees or many hierarchical classifiers, NBDTs use path probabilities for inference (Sec 3.1) to tolerate highly-uncertain intermediate decisions, build a hierarchy from pretrained model weights (Sec 3.2 & 3.3) to lessen overfitting, and train with a hierarchical loss (Sec 3.4) to significantly better learn high-level decisions (e.g., Animal vs. Vehicle).

heading: 3.1 INFERENCE
text: Our NBDT first featurizes each sample using the neural network backbone; the backbone consists of all neural network layers before the final linear layer. Second, we run the final fully-connected layer as an oblique decision tree. However, (a) a classic decision tree cannot recover from a mistake early in the hierarchy and (b) just running a classic decision tree on neural features drops accuracy significantly, by up to 11% (Table 2). Thus, we present modified decision rules (Figure 1, B):
1. Seed oblique decision rule weights with neural network weights. An oblique decision tree supports only binary decisions, using a hyperplane for each decision. Instead, we associate a weight vector ni with each node. For leaf nodes, where i = k 2 [1,K], each ni = wk is a row vector from the fully-connected layer’s weights W 2 RD⇥K . For all inner nodes, where i 2 [K+1, N ], find all leaves k 2 L(i) in node i’s subtree and average their weights: ni = P k2L(i) wk/|L(i)|.
2. Compute node probabilities. Child probabilities are given by softmax inner products. For each sample x and node i, compute the probability of each child j 2 C(i) using p(j|i) = SOFTMAX(h~ni, xi)[j], where ~ni = (hnj , xi)j2C(i). 3. Pick a leaf using path probabilities. Inspired by Deng et al. (2012), consider a leaf, its class k and its path from the root Pk. The probability of each node i 2 Pk traversing the next node in the path Ck(i) 2 Pk \ C(i) is denoted p(Ck(i)|i). Then, the probability of leaf and its class k is
p(k) = ⇧i2Pkp(Ck(i)|i) (1)
In soft inference, the final class prediction k̂ is defined over these class probabilities,
k̂ = argmaxkp(k) = argmaxk⇧i2Pkp(Ck(i)|i) (2)
Our inference strategy has two benefits: (a) Since the architecture is unchanged, the fully-connected layer can be run regularly (Table 5) or as decision rules (Table 1), and (b) unlike decision trees and
other conditionally-executed models (Tanno et al., 2019; Veit & Belongie, 2018), our method can recover from a mistake early in the hierarchy with sufficient uncertainty in the incorrect path (Figure 1 C, Appendix Table 7). This inference mode bests classic tree inference (Appendix C.2).

heading: 3.2 BUILDING INDUCED HIERARCHIES
text: Existing decision-tree-based methods use (a) hierarchies built with data-dependent heuristics like information gain or (b) existing hierarchies like WordNet. However, the former overfits to the data, and the latter focuses on conceptual rather than visual similarity: For example, by virtue of being an animal, Bird is closer to Cat than to Plane, according to WordNet. However, the opposite is true for visual similarity: by virtue of being in the sky, Bird is more visually similar to Plane than to Cat. Thus, to prevent overfitting and reflect visual similarity, we build a hierarchy using model weights.
Our hierarchy requires pre-trained model weights. Take row vectors wk : k 2 [1,K], each representing a class, from the fully-connected layer weights W . Then, run hierarchical agglomerative clustering on the normalized class representatives wk/kwkk2. Agglomerative clustering decides which nodes and groups of nodes are iteratively paired. As described in Sec 3.1, each leaf node’s weight is a row vector wk 2 W (Figure 2, Step B) and each inner node’s weight ni is the average of its leaf node’s weights (Figure 2, Step C). This hierarchy is the induced hierarchy (Figure 2).

heading: 3.3 LABELING DECISION NODES WITH WORDNET
text: WordNet is a hierarchy of nouns. To assign WordNet meaning to nodes, we compute the earliest common ancestor for all leaves in a subtree: For example, say Dog and Cat are two leaves that share a parent. To find WordNet meaning for the parent, find all ancestor concepts that Dog and Cat share, like Mammal, Animal, and Living Thing. The earliest shared ancestor is Mammal, so we assign Mammal to the parent of Dog and Cat. We repeat for all inner nodes.
However, the WordNet corpus is lacking in concepts that are not themselves objects, like object attributes (e.g., Pencil and Wire are both cylindrical) and (b) abstract visual ideas like context (e.g., fish and boat are both aquatic). Many of these which are littered across our induced hierarchies (Appendix Figure 14). Despite this limitation, we use WordNet to assign meaning to intermediate decision nodes, with more sophisticated methods left to future work.

heading: 3.4 FINE-TUNING WITH TREE SUPERVISION LOSS
text: Even though standard cross entropy loss separates representatives for each leaf, it is not trained to separate representatives for each inner node (Table 3, “None”). To amend this, we add a tree super-
vision loss, a cross entropy loss over the class distribution of path probabilities Dnbdt = {p(k)}Kk=1 (Eq. 1) from Sec 3.1, with time-varying weights !t, t where t is the epoch count:
L = t CROSSENTROPY(Dpred,Dlabel)| {z } Loriginal +!t CROSSENTROPY(Dnbdt,Dlabel)| {z } Lsoft
(3)
Our tree supervision loss Lsoft requires a pre-defined hierarchy. We find that (a) tree supervision loss damages learning speed early in training, when leaf weights are nonsensical. Thus, our tree supervision weight !t grows linearly from !0 = 0 to !T = 0.5 for CIFAR10, CIFAR100, and to !T = 5 for TinyImageNet, ImageNet; t 2 [0, 1] decays linearly over time. (b) We re-train where possible, fine-tuning with Lsoft only when the original model accuracy is not reproducible. (c) Unlike hierarchical softmax, our path-probability cross entropy loss Lsoft disproportionately upweights decisions earlier in the hierarchy, encouraging accurate high-level decisions; this is reflected our out-generalization of the baseline neural network by up to 16% to unseen classes (Table 6).

heading: 4 EXPERIMENTS
text: NBDTs obtain state-of-the-art results for interpretable models and match or outperform modern neural networks on image classification. We report results on different models (ResNet, WideResNet, EfficientNet) and datasets (CIFAR10, CIFAR100, TinyImageNet, ImageNet). We additionally conduct ablation studies to verify the hierarchy and loss designs, find that our training procedure improves the original neural network’s accuracy by up to 2%, and show that NBDTs improve generalization to unseen classes by up to 16%. All reported improvements are absolute.

heading: 4.1 RESULTS
text: Small-scale Datasets. Our method (Table 1) matches or outperforms recently state-of-the-art neural networks. On CIFAR10 and TinyImageNet, NBDT accuracy falls within 0.15% of the baseline neural network. On CIFAR100, NBDT accuracy outperforms the baseline by ⇠1%. Large-scale Dataset. On ImageNet (Table 3), NBDTs obtain 76.60% top-1 accuracy, outperforming the strongest competitor NofE by 15%. Note that we take the best competing results for any decision-tree-based method, but the strongest competitors hinder interpretability by using ensembles of models like a decision forest (DNDF, DCDJ) or feature shallow trees with only depth 2 (NofE).

heading: 4.2 ANALYSIS
text: Analyses show that our NBDT improvements are dominated by significantly improved ability to distinguish higher-level concepts (e.g., Animal vs. Vehicle).
Comparison of Hierarchies. Table 2 shows that our induced hierarchies outperform alternatives. In particular, data-dependent hierarchies overfit, and the existing WordNet hierarchy focuses on conceptual rather than visual similarity.
Comparisons of Losses. Previous work suggests hierarchical softmax (Appendix C.1) is necessary for hierarchical classifiers. However, our results suggest otherwise: NBDTs trained with hierarchical softmax see ⇠3% less accuracy than with tree supervision loss on TinyImageNet (Table 3). Original Neural Network. Per Sec 3.1, we can run the original neural network’s fully-connected layer normally, after training with tree supervision loss. Using this, we find that the original neural network’s accuracy improves by up to 2% on CIFAR100, TinyImageNet (Table 5).
Zero-Shot Superclass Generalization. We define a “superclass” to be the hypernym of several classes. (e.g. Animal is a superclass of Cat and Dog). Using WordNet (per Sec 3.2), we (1) identify which superclasses each NBDT inner node is deciding between (e.g. Animal vs. Vehicle). (2) We find unseen classes that belong to the same superclass, from a different dataset. (e.g. Pull Turtle images from ImageNet). (3) Evaluate the model to ensure the unseen class is classified into the correct superclass (e.g. ensure Turtle is classified as Animal). For an NBDT, this is straightforward: one of the inner nodes classifies Animal vs. Vehicle (Sec 3.3). For a standard neural network, we consider the superclass that the final prediction belongs to. (i.e. When evaluating Animal vs. Vehicle on a Turtle image, the CIFAR-trained model may predict any CIFAR Animal class). See Appendix B.2 for details. Our NBDT consistently bests the original neural network by 8%+ (Table 6). When discerning Carnivore vs. Ungulate, NBDT outperforms the original neural network by 16%.
Mid-Training Hierarchy: We test NBDTs without using pre-trained weights, instead constructing hierarchies during training from the partially-trained network’s weights. Tree supervision loss with mid-training hierarchies reliably improve the original neural network’s accuracy, up to ⇠0.6%, and the NBDT itself can match the original neural network’s accuracy (Table 4). However, this underperforms NBDT (Table 1), showing fully-trained weights are still preferred for hierarchy construction.

heading: 5 INTERPRETABILITY
text: By breaking complex decisions into smaller intermediate decisions, decision trees provide insight into the decision process. However, when the intermediate decisions are themselves neural network
predictions, extracting insight becomes more challenging. To address this, we adopt benchmarks and an interpretability definition offered by Poursabzi-Sangdeh et al. (2018): A model is interpretable if a human can validate its prediction, determining when the model has made a sizable mistake. To assess this, we adapt Poursabzi-Sangdeh et al. (2018)’s benchmarks to computer vision and show (a) humans can identify misclassifications with NBDT explanations more accurately than with saliency explanations (Sec 5.1), (b) a way to utilize NBDT’s entropy to identify ambiguous labels (Sec. 5.4), and (c) that humans prefer to agree with NBDT predictions when given a challenging image classification task (Sec. 5.2 & 5.3). Note that these analyses depend on three model properties that NBDT preserves: (1) discrete, sequential decisions, so that one path is selected; (2) pure leaves, so that one path picks one class; and (3) non-ensembled predictions, so that path to prediction attribution is discrete. In all surveys, we use CIFAR10-trained models with ResNet18 backbones.

heading: 5.1 SURVEY: IDENTIFYING FAULTY MODEL PREDICTIONS
text: In this section we aim to answer a question posed in (Poursabzi-Sangdeh et al., 2018) ”How well can someone detect when the model has made a sizable mistake?”. In this survey, each user is given 3 images, 2 of which are correctly classified and 1 is mis-classified. Users must predict which image was incorrectly classified given a) the model explanations and b) without the final prediction. For saliency maps, this is a near-impossible task as saliency usually highlights the main object in the image, regardless of wrong or right. However, hierarchical methods provide a sensible sequence of
intermediate decisions that can be checked. This is reflected in the results: For each explainability technique, we collected 600 survey responses. When given saliency maps and class probabilities, only 87 predictions were correctly identified as wrong. In comparison, when given the NBDT series of predicted classes and child probabilities (e.g., “Animal (90%) ! Mammal (95%)”, without the final leaf prediction) 237 images were correctly identified as wrong. Thus, respondents can better recognize mistakes in NBDT explanations nearly 3 times better.
Although NBDT provides more information than saliency maps about misclassification, a majority – the remaining 363 NBDT predictions – were not correctly identified. To explain this, we note that ⇠ 37% of all NBDT errors occur at the final binary decision, between two leaves; since we provide all decisions except the final one, these leaf errors would be impossible to distinguish.

heading: 5.2 SURVEY: EXPLANATION-GUIDED IMAGE CLASSIFICATION
text: In this section we aim to answer a question posed in (Poursabzi-Sangdeh et al., 2018) “To what extent do people follow a model’s predictions when it is beneficial to do so?”. In this first survey, each user is asked to classify a severely blurred image (Fig 4). This survey affirms the problem’s difficulty, decimating human performance to not much more than guessing: 163 of 600 responses are correct (27.2% accuracy).
In the next survey, we offer the blurred image and two sets of predictions: (1) the original neural network’s predicted class and its saliency map, and (2) the NBDT predicted class and the sequence of decisions that led up to it (“Animal, Mammal, Cat”). For all examples, the two models predict different classes. In 30% of the examples, NBDT is right and the original model is wrong. In another 30%, the opposite is true. In the last 40%, both models are wrong. As shown in Fig. 4, the image is extremely blurry, so the user must rely on the models to inform their prediction. When offered model predictions, in this survey, 255 of 600 responses are correct (42.5% accuracy), a 15.3 point improvement over no model guidance. We observe that humans trust NBDT-explained prediction more often than the saliency-explained predictions. Out of 600 responses, 312 responses agreed with the NBDT’s prediction, 167 responses agreed with the base model’s prediction, and 119 responses disagreed with both model’s predictions. Note that a majority of user decisions (⇠ 80%) agreed with either model prediction, even though neither model prediction was correct in 40% of examples, showing our images were sufficiently blurred to force reliance on the models. Furthermore, 52% of responses agreed with NBDT (against saliency’s 28%), even though only 30% of NBDT predictions were correct, showing improvement in model trust.

heading: 5.3 SURVEY: HUMAN-DIAGNOSED LEVEL OF TRUST
text: The explanation of an NBDT prediction is the visualization of the path traversed. We then compare these NBDT explanations to other explainability methods in human studies. Specifically, we ask participants to pick an expert to trust (Appendix, Figure 13), based on the expert’s explanation – a saliency map (ResNet18, GradCAM), a decision tree (NBDT), or neither. We only use samples where ResNet18 and NBDT predictions agree. Of 374 respondents that picked one method over the other, 65.9% prefer NBDT explanations; for misclassified samples, 73.5% prefer NBDT. This supports the previous survey’s results, showing humans trust NBDTs more than current saliency techniques when explicitly asked.

heading: 5.4 ANALYSIS: IDENTIFYING FAULTY DATASET LABELS
text: There are several types of ambiguous labels (Figure 5), any of which could hurt model performance for an image classification dataset like ImageNet. To find these images, we use entropy in NBDT
decisions, which we find is a much stronger indicator of ambiguity than entropy in the original neural network prediction. The intuition is as follows: If all intermediate decisions have high certainty except for a few decisions, those decisions are deciding between multiple equally plausible cases. Using this intuition, we can identify ambiguous labels by finding samples with high “path entropy” – or highly disparate entropies for intermediate decisions on the NBDT prediction path.
Per Figure 6, the highest “path entropy” samples in ImageNet contain multiple objects, where each object could plausibly be used for the image class. In contrast, samples that induce the highest entropy in the baseline neural network do not suggest ambiguous labels. This suggests NBDT entropy is more informative compared to that of a standard neural network.

heading: 6 CONCLUSION
text: In this work, we propose Neural-Backed Decision Trees that see (1) improved accuracy: NBDTs out-generalize (16%+), improve (2%+), and match (0.15%) or outperform (1%+) state-of-the-art neural networks on CIFAR10, CIFAR100, TinyImageNet, and ImageNet. We also show (2) improved interpretability by drawing unique insights from our hierarchy, confirming that humans trust NBDT’s over saliency and illustrate how path entropy can be used to identify ambiguous labels. This challenges the conventional supposition of a dichotomy between accuracy and interpretability, paving the way for jointly accurate and interpretable models in real-world deployments.

Review summaries:
The paper proposes a method to make neural networks more accurate and interpretable by replacing their final layers with a probabilistic decision tree. As a result, the network can produce a sequence of decisions that leads to the final classification result, given an input image. The method is trained with soft decisions by assigning probabilities to each leaf, which are associated with a single class. The tree decision hyperplanes are constructed automatically from the backbone networks final dense layer and finetuned. The fact that decisions are soft solves the differentiablility problem of decisions as in various other similar papers, cited or uncited (more below).

Aim to improve the interpretability and the accuracy of the neural network, this paper takes a step further on the integration of NN with a decision tree. It will replace the final linear layer of the NN with a decision tree induced by pre-trained model weights. It takes advantage of both hard and soft decision trees and designs suitable tree supervision loss thereon. Extensive experiments verify the design choice of the proposed components. On both small-scale and large-scale datasets, it beats the decision tree counterparts. Also, on the aspects of generalization and interpretability, it shows the strength compared to NN.