Neural Basis Models for Interpretability
Authors: Filip Radenovic
Abhimanyu Dubey
Dhruv Mahajan
Venue: NeurIPS 2022 Accept (2022)
Keywords: interpretability
explainability
trustworthy AI
interpretable machine learning
generalized additive models

Abstract: Due to the widespread use of complex machine learning models in real-world applications, it is becoming critical to explain model predictions. However, these models are typically black-box deep neural networks, explained post-hoc via methods with known faithfulness limitations. Generalized Additive Models (GAMs) are an inherently interpretable class of models that address this limitation by learning a non-linear shape function for each feature separately, followed by a linear model on top. However, these models are typically difficult to train, require numerous parameters, and are difficult to scale. 
    We propose an entirely new subfamily of GAMs that utilizes basis decomposition of shape functions. A small number of basis functions are shared among all features, and are learned jointly for a given task, thus making our model scale much better to large-scale data with high-dimensional features, especially when features are sparse. We propose an architecture denoted as the Neural Basis Model (NBM) which uses a single neural network to learn these bases. On a variety of tabular and image datasets, we demonstrate that for interpretable machine learning, NBMs are the state-of-the-art in accuracy, model size, and, throughput and can easily model all higher-order feature interactions.
    Source code is available at \href{https://github.com/facebookresearch/nbm-spam}{\ttfamily github.com/facebookresearch/nbm-spam}. 

Neural Basis Models for Interpretability
Filip Radenovic
Abhimanyu Dubey
Dhruv Mahajan Meta
Meta
Meta Al
Abstract
Due to the widespread use of complex machine learning models in real-world applications, it is becoming critical explain model predictions. However, these models are typically black-box deep neural networks, explained post-hoc via meth- ods with known faithfulness limitations. Generalized Additive Models (GAMs) are an inherently interpretable class of models that address this limitation by learning a non-linear shape function for each feature separately, followed by linear model on top. However, these models are typically difficult to train, require numerous parameters, and are difficult scale. We propose an entirely new subfamily of GAMs that utilizes basis decomposition of shape functions. A small number of ba- sis functions are shared among all features, and are learned jointly for a given task, thus making our model scale much better to large-scale data with high-dimensiona. features, especially when features are sparse. We propose an architecture denoted as the Neural Basis Model (NBM) which uses single neural network to learn these bases. On variety of tabular and image datasets, we demonstrate that for interpretable machine learning, NBMs are the state-of-the-art in accuracy, model size, and, throughput and can easily model all higher-order feature interactions. Source code is available at tithub..ccm com http:///c.ccboooeerraaccc//cbbaa
Introduction
Real world machine learning models [14, 61] are mostly used as a black-box, i.e., is very difficult to analyze and understand why specific prediction made. In order to explain such black-box models, an instance-specific local interpretable model is often learned [38, 50]. However, these approaches tend to be unstable and unfaithful [1, 52], i.e., they often misrepresent the model's behavior. On the other hand, a family of models known as generalized additive models (GAMs) [24] have been used for decades
inherently interpretable alternative black-box models.
GAMs learn shape function for each feature independently, and outputs of such functions are added (with bias term) obtain the final model prediction. All models from this family share an important trait: the impact of any specific feature on the prediction does not rely on the other features, and can be understood by visualizing its corresponding shape function. Original GAMs [24] were fitted using splines, which have since been improved in explainable boosting machines (EBMs) [36] by fitting boosted decision trees, very recently in neural additive models (NAMs) [2] by fitting deep neural networks (DNNs). A drawback for all the aforementioned approaches is that for each shape function, they require either millions of decision trees [36], DNN with tens of thousands of parameters [2), making them prohibitively expensive for learning datasets with a large number of features.
In this work, we propose novel subfamily of GAMs, which, unlike previous approaches, learn to decompose each feature's shape function into small set of basis functions shared across all features. The shape functions fitted the feature-specific linear combination of these shared bases, see Figure 1. At an abstract level, our approach is motivated by signal decomposition using traditional basis functions like the Fourier basis [9] Legendre polynomials [45], where weighted
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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(hi,..., bn}
(Ai...., bnp
shared
{h p hnb
by (1
ha (11
Ap (21)
h (32)
Ag (2g
Ag (mtg
Ag
(11D)
Ag (sT
hg (
412
a22
01
a12
B2
feature specific
aDi
ADB
f((
f2
fD
112
class specific
((xt)
Figure 1: Neural Basis Model (NBM) architecture for binary classification task.
combination of few basis functions suffice to reconstruct complex signal shapes. However, in contrast to these approaches, our basis decomposition is not fixed priori. In fact, it is learnt specifically for the prediction task. Consequently, we maintain the most important feature of GAMs, L.., their interpretability, as the contribution of single feature does not depend on the other features. At the same time, we gain scalability, as the number of basis functions needed in practice is much smaller than the number of input features. Moreover, increase computational efficiency by several orders of magnitude when the input features are sparse. Additionally, propose approach to learning the basis functions using single DNN. We call this solution the Neural Basis Model (NBM). Using neural networks allows for even higher scalability, training and inference performed GPUs or other specialized hardware, and can be easily implemented in any deep learning framework using standard, already developed, building blocks.
show that the usage of basis functions can
Our contributions are as follows: (i) We propose novel subfamily of GAMs whose shape functions are composed of shared basis functions, and propose an approach to learn basis functions via DNNs, denoted as Neural Basis Model (NBM). This architecture is suitable for mini-batch gradient descent training on GPUs and easy to plug-in into any deep learning framework. (ii) We demonstrate that NBMs can be easily extended to incorporate pairwise functions, similar to GA²Ms [37], by learning another set of bases to model the higher order interactions. This approach effectively only linearly increases the parameters, while other models such EB [36, 37] and NA [2] suffer from quadratic growth of parameters, and often require heuristics and repeated training to select the most important interactions before learning [37]. (ii) Through extensive evaluation of regression, binary classification, and multi-class classification, with both tabular and computer vision datasets, we show that NBMs and 2 outperform state-of-the-art GAMs and GA²Ms, while scaling significantly better, iee., fitting much fewer parameters and having higher throughput. For datasets with more than ten features, using NBMs result in around 5 xx-00x reduction in parameters 4x-7x x better throughput. (iv) We propose an efficient extension of NBMs to sparse datasets with more than a hundred thousand features, where other GAMs do not scale at all.
NAMs [2], and
2 Related work
Shape functions GAMs [24, 64] have many different representations in the literature, including: splines [24], trees or random forests [36], deep neural networks [2]. neural oblivious decision trees [12]. Popular methods of fitting GAMs are: backfitting [24], gradient boosting [36], or mini- batch gradient descent [2, 12]. Our work falls under the GAM umbrella, however, differs from these approaches by not learning shape functions independently, but rather, learning set of shared basis functions that are used compose each shape function. The bases themselves can be complex non-linear functions that operate on one feature a time, e.g., decision trees or splines, however, to keep the scope the work concise and ensure straightforward scalability, we represent them with deep neural networks and use mini-batch stochastic gradient descent to learn the bases. This makes our work most closely related neural additive models (NAMs) [2], however, crucially, our method leams far fewer parameters by sharing bases compared to NAMs.

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Methods have been proposed to model pairwise interactions [12, 37, 58], and they are commonly de- noted as GA However, they usually require sophisticated feature selection using: heuristics [37]. pack-propagation [12], or, two-stage iterative training [58]. We note that one of the main goals of our work is to analyze the scalability of the newly proposed NBM architecture in extreme scenarios, hence, we do not apply feature selection a algorithms. That being said, before-mentioned algorithms are complementary and can be applied to NB as well.
Finally, GAMs are a popular choice in high-risk applications for healthcare [11, 56], finance [6], forensics [55], etc. Another line of work applicable these domains are interpretable surrogate models such as LIME [50], SHAP [38], tree-based surrogates [5, 57], that give a post-hoc explanation model. However, there are almost no theoretical guarantees that the simple surrogate model is highly representative of the more complex model [1]. This issue is completely resolved when using inherently transparent models such as NBMs. We hope that our approach sparks wider usage of GAMs in mission-critical applications with large-scale data, where the ability to interpret, understand, and correct the model of utmost importance.
of a high-complexin
3 Method
3.1 Background
Generalized Additive Model (GAM) [24). Given a D-dimensional interpretable input x= } = PR, a target label y, a link function (e.g., logistic function). univariate shape function f.. corresponding to the input feature Fú, bivariate shape function fúj. corresponding to the feature interaction, and a bias term fo. the prediction function in GAM and GA is expressed
GAM : g(x)) = fo + Ei(a) GA g((=)=(a=))))))))()))) = Jo (mn)) + ΣΣ (y (1) 2=1
1=1
(=1 3>1
Interpreting GAMs is straightforward as the impact of a feature on the prediction does not rely on the other features, and can be understood by visualizing its corresponding shape function, e.g., plotting Fy on the r-axis and (1,,, on the y-axis. A certain level of interpretability is sacrificed for accuracy when modeling interactions, as fuj shape functions are harder to visualize. Shape function visualization through heatmaps [12, 37] is commonly used towards that purpose. Note that, the graphs visualizations of GAMs are an exact description of how GAMs compute a prediction.
3.2 Our model architecture
We observe that, typically, input features of high-dimensional data are correlated with each other. As result, it should be possible to decompose each shape function Si into small set of basis functions shared among all the features. This the core idea behind our approach.
Neural Basis Model (NBM). We propose to represent shape functions fa as
f((x;)) Σ ha (x:)aas;
where (h1, hy,..., hg} represents set of B shared basis functions that are independent of feature indices, and coefficients a, are the projection of each feature the shared bases. We addition- ally propose to learn basis functions using a DNN, i.e., single one-input B-output multi-layer perceptron (MIP) for all (he; = 1,..., B}. The resulting architecture shown in Figure 1.
Multi-class multi-task architecture. Let l correspond to the target class y in the multi-class setting. Similar to Equation I, the prediction function gi for class yi in GAMs can be written as:
(x = for + Σ (.:))aa.
1=1
where feature shape functions f(( are shared among the classes and linearly combined using class specific weights wa. Combining Equations 2 and 3, multi-class NBM can be represented as:
D
Multi-class NBM gi (x) = ful + Σ
(4)
3

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Extension to MB Similar to NBM, we represent GA²M shape functions fúy in Equation 1 as:
B
2z) =
Σ
us (2i, xz (1ss
(5)
where {ui, U2, / represents a set of shared bi-variate basis functions that are independent of feature indices and coefficients bijk are the projection of pair-wise features to the shared bases. We learn an additional two-input B-output MLP for all {ux: k = 1....,,1. B) to learn the bases. Extension to multi-clas multi-class setting can be done in the same way for NBMs.
Sparse architecture. Typically, datasets with nigh-dimensional features are sparse in nature. For example, in the Newsgroups dataset [32], news articles represented by If-idf features, and, for a given instance, most of the features are absent due to the vocabulary being of the order of 100K words. Since NBM uses single DNN to learn all the bases, representing the absent feature to the batch, to compute the corresponding basis function values, The subsequent linear projection to feature indices via dçk is a computationally inexpensive operation.
simply append the single value
In contrast, typical GAMs (e.g., Neural Additive Model (NAM) [42]) need to pass the absent value through every shape function fi which makes it compute-intensive well as difficult implement.
Training and regularization. We use mean squared error (MSE) for regression, and cross-entropy loss for classification. To avoid overfitting, we the following regularization techniques: (i) La-normalization (weight decay) [31] of parameters; (ii) batch-norm [28] and dropout [54] on hidden layers of the basis functions network: (iii) an Lg-normalization penalty the outputs incentivize fewer strong feature contributions, as done in [2]: (iv) basis dropout to randomly drop individual basis functions in order decorrelate them. Similar techniques have been used for other GAMs (2, 12).
Selecting the number of bases. One can use the theory of Reproducing Hilbert Kernel Spaces (RKHS. (7]) devise heuristic for selecting the number of bases B. Specifically. we demonstrate that any NBM model lies on subspace within the space spanned by a complete GAM if the GAM shape functions reside within ball in RKHS. Assuming a regularity property in the data distribution, one can then demonstrate that B Olog D) bases are sufficient to obtain competitive performance. We present this formally Appendix See. A.5. This provides the alternate interpretation of NBM as learning a "principal components" decomposition in the L2 - of functions, learn a set of (preferably orthogonal) basis functions to approximate the decision boundary.
3.3 Discussion
In this section, we contrast NBMs with closely related GAMs: Neural Additive Models (NAMs) [2].
Neural Additive Model (NAM) [2]. NAMs learn a linear combination of networks that each attend to single imput feature: each fü in (1) is parametrized by a deep neural network, i.e., a one-input one-output multi-layer perceptron (MLP). These MLPs are trained jointly using backpropagation and can learn arbitrarily complex shape functions.
Number of parameters. We compare number of weight parameters needed to learn NAM vs.
70
NBM for the binsiy-classification task. See Ap-
INAM/ NNMM
60
50
pendix Section A.3 for multi-class analysis. Let us denote with M the number parameters in
MLP for each feature in NAM, and with N the number of parameters in MLP for bases in NBM.
30
20
In most experiments the optimal NAM has 3 hid- den layers with 64, 64 and 32 units (M = 6401), and, NBM has hidden layers with 256, 128, 128 units (N = 62820) and B- = 100 basis func- tions. Then the ratio of number of parameters in NAM vs. NBM is given by,
10
0
10
100
1k
10k
100k
Figure 2:
NAM vs. NBM: #parameters.
NAM
D D-
6402
(6)
NBM
N D. B+D
62820 D
101

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Figure 2 shows this ratio for different values of feature dimensionality D. For D = 10, NBMs and NAMs have roughly equal number of parameters. For most textual and vision datasets, feature dimensionality is significantly higher, thus leading to 10x-50x reduction in parameters. We also observe that as result, for many datasets, specialized regularization discussed in the previous section gives incremental gains for NBMs, while they are very crucial for NAMs to give good performance. Additional analysis on number of parameters is given in Section 4.4 and Table 2.
Throughput. One of the main challenges in approach like NAMs is the low throughput rate, that is the number of data instances processed per second, which directly affects the training speed. Since NAMs use separate MLP per dimension, it is much more challenging to implement efficiently. NBMs on the other hand are much more efficient since feature specific linear layer on the top of bases is very fast. For example, for datasets with around hundred features, the original NAM implementation [2] is around 20x slower training compared NBMs. We optimized the speed of NAMs using group convolutions (see Appendix Section A.3), but even this optimized version is around 5x slower.
Stability. The interpretability of models and their explanations is tightly coupled to their stability. For example, post-hoc explanations of black-bor models are known to be unstable, and produce different explanations for small imput changes [53]. GAMs are exactly explained by visualizing shape functions that represent the model, however, a desirable property is to have similar shape functions when repeatedly training the model on the same data while varying random initialization. Because of the aver-parameterization in NAMs, we observe that shape functions of different runs are often unstable (i.e. they often diverge), especially for feature regions where the data density is low. On the other hand, NBMs train a single set of shared bases for the entire dataset, which makes shape functions significantly more stable, see Section 4.6 and Figure
M vs. M NAMs can trivially be extended to N by learning additional zwo-input one-output MLPs for each pairwise feature interaction faj- Since the numbers of parameters grow quadratically, this setting further exaggerates the parameters and throughput discrepancies. In fact, as we show in Section 4.4, for high dimensional datasets the NA approach does not scale at all.
4 Experiments
4.1 Datasets
Tabular datasets. We report performance on CA Housing [10, 46), FICO [22], CoverType [8, 16, 20], and Newsgroups [32, 43] tabular datasets. We perform one-hot encoding for categorical features, and min-max scaling of features to [o, 1] range. Data is split to have 70/10 10/2 10/20 ratio for training, validation, and, testing, respectively: except for Newsgroups where test split is fixed, so we only split the train part to 85/15 15 ratio for train and validation.
We also report performance
MIMIC-II [41, 51]. Credit [17, 191. Click [15], Epsilon [21], Higgs [3, 26], Microsoft [40, 491, Yahoo [60], and Year [66] tabular datasets. For these datasets, we follow [12, 47] use the same training, validation, and, testing splits, and perform the same feature normalization: target encoding for categorical features, quantile transformation with 2000 bins for all features Gaussian distribution.
Additional details for all tabular datasets are given in Table and Appendix Section A.1.
Image datasets (classification). We experiment with two bird classification datasets: CUB [18, 62] and iNaturalist Birds (9). CUB images are annotated with keypoint locations of 15 bird parts, and each location is associated with one or more part-attribute labels. iNaturalist Birds contains significantly more bird classes and more challenging scenes compared to CUB, but lacks keypoint annotations. We perform the pancept-bottlennckstty.. [29] interpretable pre- pre-processing: (i) Images are randomly cropped and resized 448 x 448 size, and passed through a ResNet50 model [25] until the last pooling layer to extract 2048-D features on the 14x 14 spatial grid. (ii) Part-attribute linear classifiers are trained on the extracted features using part locations and aurt-bttribute labels from CUB training split. (iii) Max- Max-poooing of part-attribute scores over spatial locations is performed to extract 278 interpretable features (e.g., orange legs, striped wings, shapee beak) for each image in both CUB and iNaturalist, using the part-attribute classifiers trained on CUB only. Splits are preset and results are reported on the validation split, which a common practice in the computer vision community. Additional details are given in Table I and Appendix Section A.I.
5

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Table Datasets overview.
Dataset
Task
#Train
#Val
#Test
Sparse
#cat
#Class
Tabular dataset
CA Housing
Regression
14.447
2,065
4,128
No
8
FICO
Binary
7,321
1,046
2,092
No
CoverType
Multi-class
406,707
58,102
116,203
No
Newsgroups
Multi-class
9,899
1,415
7,532
99.9%
146,016
MIMIC-II
Binary
17,155
2,451
4,902
No
17
2
Credit
Binary
199.364
28,481
56,962
No
30
2
Click
Binary
800,000
100,000
100,000
No
2
Epsilon
Binary
320,000
80,000
100,000
No
2,000
2
Higgs
Binary
8,400,000
2,100,000
500,000
No
Microsoft
Regression
580,539
142,873
241,521
No
136
Yahoo
Regression
473,134
71,083
165,660
No
699
Year
Regression
370,972
92,743
51,630
No
90
Image dataset
CUB
Classification
5,994
5,794
No
278
200
iNaturalist Birds
Classification
414.847
14,860
No
278
1,486
Common Objects
Detection
2,645,488
58,525
97.0%
2,618
115
Image dataset (object detection). For this task we use a proprietary object detection dataset, denoted as Common Objects, that contains 1 114 common objects, (plus background class) with bounding box locations, 200 parts and 54 attributes. Each box for each image is pre-processed using compositions of parts and attributes to extract 2,618 interpretable features and 100k pairwise feature interactions. For the purpose of evaluating models in this paper, one data instance in this dataset equivalent to one bounding box. As with previous computer vision datasets, results reported the validation split. Additional details are given Table and Appendix Section A.I.
4.2 Baselines
We implement the following baselines in PyTorch [48], and train using mini-batch gradient descent:
Linear. Linear / logistic regression are interpretable models that make prediction decision based on the value of linear combination of the features.
NAM [2]. We experiment with two proposed NAM architectures [2]: (i) MLPs containing 3 hidden layers with 64, 64 and 32 units and ReLU [23] activation, and (ii) single hidden layer MLPs with 1,024 ExU units and ReLU-1 activation. We reimplement the original NAM implementation [42] (details in Appendix Section A.3) achieving around x x2-x 10 speedup at training, depending on the dataset. Finally, extend the implementation implementation together with the rest of our code at github. com/facebookresear www.srreeeeerrerreeeeeeeeeeeeeee wwwpcchhhnmmmeebeeeeeeee spam.
N well. We released our NAM
MLP. Multi-layer perceptron (MLP) is a non-interpretable black-box model capturing high-order interaction between the features: g(z)=f(( fx) T2, D.). For most datasets, MLP sets the upper bound on the performance, and gives an idea of the trade-off between accuracy and interpretability. We experiment with the following architectures: (i) hidden layers with 128, 128, 64, 64, 64 units; (ii) 3 hidden layers with 1024, 512, 512 units; (iii) 2 hidden layers with 2048, 1024 units. We have observed that increasing the depth by adding more layers for any of the three architectures has no additional accuracy gain. For all datasets, we report the best performing architecture across the three. For the following baselines, we use the available implementations:
Explainable Boosting Machines (EBMs) another state-of-te mate-of-the-an art GAM which use gradient boosting of millions of shallow trees to learn a shape function for each input feature. These models support automatic pairwise interactions through their EB²M implementation, but only for regression and binary classification tasks. We use the interpretml library [44].
EBM [36, 37].
XGBoost [13]. EXtreme Gradient Boosted trees (XGBoost) are another non-interpretable black-box model that learn high-order feature interactions. We use the xgboost library [13].

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Table 2: Number of parameters (#par.) and throughput as imputs per second (æ/ssc).. Here NAM and
2 M refers to our optimized implementation; Trefers to sparse NBM optimization.
CA Housing
FICO
CoverType
Newsgroups
iNat. Birds
Model
#par.
ø/sec
#par.
ø/sec
#par.
ø/sec
#par.
3/sec
#par.
ø/sec
NAM NBM
54K
0.5M
262K
123K
363K
80K
984M
23
2.3M
15K
65K
3.4Mx68
68K
821Kx67
70K
530K X6.6
18M
19Kx30
x91
0.5M
74Kx49
N
243K
119K
5.3M
6K
10M
3K
320M
NB
161K
641Kx54
0.3M
30Kx50
0.5M
15K
66M
374x38
4.3
Implementation details
NBM. We use the following architecture for NBMs and NB: MLP containing hidden layers with 256, 128, and 128 units, ReLU [23]. B = 100 basis outputs for NBMs and B = 200 for NB3Ms. Source code is available at github. com/ wwwebookresearccc bsamm......................mmam.
Training details. Linear, NAM, NBM, and MLP models are trained using the Adam with decoupled weight decay (AdamW) optimizer [35], on 8x V100 GPU machines with 32 GB memory, and batch size of at most 1024 per GPU (divided by every time a batch cannot fit in the memory). We train for 1,000, 500, 100, or, 50 epochs, depending on the size and feature dimensionality of the dataset. The learning rate is decayed with cosine annealing [34] from the starting value until zero. For NBMs on all datasets, we tune the starting learning rate in the continuous interval [le-5, 1.0), weight decay in the interval [1e-10, 1.0), output penalty coefficient in the interval [1e- 7, 100), dropout and basis dropout coefficients in the discrete set {0, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5. 0.6, 0.7, 0.8.0.8.... 0.9). We find optimal search is performed for Linear, NAM, and MI.P baselines. Finally, for EBMs and XGBoost, CPU machines are used, with typer-paramete search using the guidelines in the original works. See Appendix Section A.2 for more training and hyper-pann ppper-parameter search details.
using validation set and random search. Similar hyper-
Evaluation details. Performance metrics: (i) for regression we report mean-squares error (MSE) or root MSE (RMSE); (ii) for binary classification we report area under the ROC curve (AUROC) error rate (Error): (iii) for multi-class classification in tabular and image domains we report accuracy @ (acc@) I); and, (iv) for object detection in image domain
average precision (mAP) averaged over IoU thresholds from 0.5 to 0.9 as per MS-COCO [33] definitions. We report average performance and standard deviation over 10 runs with different random seeds.
report
4.4 Comparison with baselines
In this section we compare NBM, as well as the pairwise interactions N counterpart, with popular and widely used GAMs and non-interpretable black-box models.
First, we perform extensive comparison the number of parameters and throughput against the most similar architecture, i.e. Neural Additive Model (NAM) [2]. Both NAMs and NBMs utilize deep networks and are able run on GPUs, which helps to scale the models large datasets. Results on representative datasets are presented in Table 2. The throughput is measured as the number of input instances that we can process per second (ar sec) on one 32 GB V100 GPU, in inference mode. For each model, we take the largest batch size (up to 8, 8,192) that fits on the GPU and calculate the average time over 100 runs to process that batch. With that, we calculate the number of input instances processed per second. Throughput can vary depending on the implementation, library used, etc. Hence, in order to be as fair as possible, we compare both models with our own optimized implementation in the same deep learning library, i.e., PyTorch [48]. The only dataset where NAM narrowly beats NBM in the number of parameters is CA Housing, which has the smallest number of input features D = 8. On other datasets, NBM has around 5x-50x fewer parameters than NAM, while having 7 x smaller runtime. Finally, only the sparse version of NBM model can efficiently run on Newsgroups and Common Objects (with pairwise interactions) datasets, where standard NAMs (and other GAMs) have throughput too low for any practical application. Note that these comparisons with our optimized implementation of NAM.
7

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Table 3: Performance comparison with baselines. L: lower is better; f: higher is better.
CA
iNat. Birds
Common
FICO
CoverType
News.
CUB
Model
Housing
Objects
RMSE +
AUROC T
acc
acc
acc
acc
mAP 1
Linear
0.7354
0.7909
0.7254
0.8238 1:0.0005
0.7451
0.3932 1100000
0.1917
1:000001
11000001
1:000000
2:0.0000
1:00001
0.5586
0.7985
0.7392
EBM
=0.0002
#0.0001
0004
NAM
0,5721
0.7993
0,7359
0,7632
0.4194
0.2056
2:000054
20,0004
2:000003
12000000
22000007
1:0 1.000005
NBM
0.5638
0.8048
0,7369
0.8446
0.7683
0.4227
0.2168
200013
200000
2000002
4:00011
2000000
#:0.0007
2000006
EB
0.4919
0.7998
1000004
1:0.0005
NA M
0.4921
0.79 0.7992
0.8872
0.7713
0.4591
1:000008
110.0003
0:000006
11.0.011
11000000
NB
0.4779
0.8029
0,8908
0.7770
0.4684
0.2378
20,0000
20.0003
200800
220.0006
20,0009
1:0.0006
XGBoost
0.4428
0.7925
0,8860
0.7677
0.7186
4-0,0009
0.5014
0.7936
0.9694
0.8494
0.7684
0.4584
0.2376
MLP
4:0,0011
=0000007
::0.000
:5000007
Next, we compare performance with GAM baselines, and non-interpretable black-box models in Table 3. We notice that NBMs achieve the best performance among all GAMs, even outperforming full complexity MLPs on several datasets. The difference in performance is more pronounced on larger datasets, such as iNaturalist and Common Objects, where NBMs generalize better. EBMs have a big downside as they do not support pairwise interactions on multi-class tasks, and do not scale at all to very large datasets. Finally, datasets with large number of features, i.e. Newsgroups and Common Objects (with pairwise interactions), NBMs are the only GAMs that scale, as we did not manage complete successful training with NAMs EBMs.
We finally observe that pairwise interactions are enough to match or beat black-box models on out of 7 datasets. The only exception is CoverType, where MLP has 0.9694 acc@] 1 NB with 0.8908. However, we scale NBMs further and train NB (i.e., including triplet interactions) that gets 0.9634 acc @1 1. Even though this is interesting result that helps understand the data behavior, argue that triplet feature interactions are very hard to visualize and hence not interpretable.
4.5 Comparison with state of the art
We finally compare NBM, as well as the pairwise interactions NB²M counterpart, with neural-based state-of-the-art GAMs. Namely, we compare with Neural Additive Model (NAM) [2] and Neural Oblivious Decision Trees GAM (NODE-GAM) [12], and their pairwise interactions versions NA²M and NODE-GA MM. Results presented in Table 4.
We compute NAM and NBM results using our codebase, while NODE-GAM results are reproduced from [12]. For fair comparison on these datasets, we use training, validation, and, testing splits from NODE-GAM [12], and perform the same feature normalization. see Section 4.1 for details. Without any additional model selection (architecture tuning, stochastic weight averaging, etc.) NBMs outperform NODE-GAMs on out of 8 presented datasets (albeit marginally), and outperform NAM on all datasets. Note that, the idea of NBM is perpendicular to that of NODE-GAM and it is possible that bigger gains can achieved by combining them, especially for Epsilon and Yahoo datasets, where 2B² Ms do not scale due to high dimensionality and non-sparse nature of features.

---

Table Performance comparison with state-of-the-an GAMs. NODE-GAM results are reproduced from [12]. J: lower is better; f: higher is better. State-of-ther of-the-arrr art performance shown in bold.
MIMIC-11
Credit
Click
Epsilon
Higgs
Microsoft
Yahoo
Year
Model
AUROC 1 AUROC1 1
Error
Error
Error 4
MSE 4
MSE
MSE
NAM
0.8539
0,9766
0.3317
0.1079
0.2972
0,5824
0.6093
85.25
1:00004
+000027
100005
2000002
2000001
2000002
2000003
20001
NODE
0.8320
0,9810
0.3342
0.1040
0.2970
0.5821
0.6101
85.09
GAM
1:00110
1:001100
10,000
1:00000
1000001
4:000004
1:000006
1:001
0.8549
0.9829
0,3312
0.1038
0.2969
0.5817
0,6084
85.10
NBM
1:00004
4:00014
4200002
4:00000
100001
1:00001
200001
2001
NA M
0.8639
0.9824
0.3290
0.2555
0.5622
79.80
1:0,001
1:0.0032
1:00005
1:0.0003
1:00003
110.05
NODE
0.8460
0.9860
0.3307
0.1050
0.2566
0.5618
0.5807
79.57
GA
1:0.0110
1:0.0100
1000001
1:000002
3 1.00000
1:000003
1000000
110.12
NBAM
0.8690
0,9856
0.3286
0.2545
0.5618
79.01
1:00000
1:00017
0:02
REIT Age
Figure 3: NAM (upper row) and NBM (bottom row) shape functions fi for the CA Housing dataset.
4.6 Stability and interpretability
The interpretability of GAMs comes from the fact that the learned shape functions can be easily visualized. In the same manner the other GAM approaches, each feature's importance in an NBM can be represented by a unique shape function that exactly describes how the NBM computes a prediction. We demonstrate this the CA Housing dataset because has small number of input features (D 8) and the full model can conveniently be visualized in a single row, see Figure 3.
Towards this purpose, we train an ensemble of 100 models by running different random seeds with optimal ppprrpparaeeter, in order to analyze when the models learn the same shape and when they diverge. Following Agarwal et al. [2], set the average score for each shape function to be zero by subtracting the respective mean feature score. Next, we plot each shape function as fi(x;) vs. ær; for each model in the ensemble using semi-transparee line, and an average ensemble shape function using a thick line. Finally, the z-axis divided by bars depicting the normalized data density, i.e., darker areas contain more data points. Figure 3 depicts ensemble of NAMs [2] (upper row) and NBMs (bottom row). Although the shape functions are correlated for most features, we observe that NBMs diverge much less in the cases where there only few data points (light white areas in the graphs). This is due to the fact that the basis network in NBM is trained jointly with only linear composing weights being trained for each feature separately. In contrast, each feature in NAM has its own network, which becomes unstable and diverges in cases of few training data points.
Finally, to quantify stability, we compute the standard deviation for each visualized shape function over 100 models, and report the mean standard deviation over all features. For NAMs, this mean standard deviation is 0.9921, while for NBMs it is 0.1987.

---

NBM, CoverType
NBAM, Coverlype NB"M, CUB
Table 5: Ablation on the number subnets S.
NBM, Newagroups
0.9
CoverType
0.88
Model
S
0.86
#param
ø/sec
acc 1
acc
0.82
NAM
363K
80K
0.7359
0.8
NBM
1
70K
530K
0.7369
0.78
0.76
NAM
5
1.8M
16K
0.7417
0.74
NBM
5
350K
88K
0.7435
0.72
5
10
25
50
100
200
500
1000
B
NB M
1
463K
15K
0.8908
Figure 4: Ablation on the number of bases B.
4.7 Ablation study
Number of basis functions. We evaluate the robustness of NBMs W.E. the choice of the number of basis functions B. Results are presented in Figure 4 for Newsgroups, CoverType, and, CUB. We observe that NBMs are not overly sensitive to the choice of B. Rather than tuning this hyperparameter, we recommend setting B = 100 for NBMs and B = 200 for 2B as it performs well across large variety of datasets we experimented with. Although, e.g., for 2 on CoverType, using a larger number of basis functions leads to some performance gains, it comes with throughput trade-off, as computing the linear combination of bases starts becoming the bottleneck. Thus, we suggest using the recommended values as great trade-off between accuracy and throughput.
Multiple subnets. Agarwal et al. [2] propose to extend NAMs to multi-task/ multi-class setups by associating each feature with multiple subnets. This method is directly applicable on NBMs as well, by using S different networks to learn S sets of basis functions. We compare few options on the multi-class CoverType predictions with the number of subnets S=1 and S=5 in Table 5. Both NAMs and NBMs have similarly small relative accuracy improvement when using 5 subnets over the subnet, however that comes at 5x increase in the number of parameters and 5x lower throughput. With the eecuracy-complee datasets, keep S across all other experiments. Interestingly, 2 with S= has comparable throughput to NAM with S=5, while achieving an impressive accuracy gain, see Table 5.
trade-off in mind, we did not see much benefit of using = on
Conclusion and future work
This work describes novel Neural Basis Models (NBMs), which belong to subfamily of Generalized Additive Models (GAMs), and utilize deep learning techniques to scale to large datasets and high-dimenniionl features. Our approach addresses several scalability and performance issues associated with GAMs, while still preserving their interpretability compared to black-box deep neural networks (DNNs). We show that our NBMs and NB'Ms Ms achieve state-of-the oofte-----hhhrrr performance a large variety of datasets and tasks, while being much smaller and faster than other neural- based GAMs. As result, they can be used as drop-in replacements for the black- DNNs.
We do recognize that our approach, though highly scalable, has limitations w.r.f. number of features. Beyond 10,000 dense (or million sparse) features, we would need to apply some form of feature selection [12, 37, 58] to scale further. Scalability issue is even more pronounced when modeling pairwise interactions in 2B However, NBMs still handle an order of magnitude more than what can be handled by NAM or other GAM approaches that do not perform feature selection.
There are many future directions for this line of work. First, for the computer vision domain, we assume an intermediate, interpretable concept layer [29] on which NBMs and in general GAMs can be applied. It would be interesting to explore visual interpretability by either directly going to the pixel space with NBMs or learning visual features that can recognition with lower order interactions in NBM framework (for example, Ms) Finally, the idea of using shared basis is generic. Although, we used neural networks to learn these basis, we can enhance the model interpretability further for higher order interactions, by using more interpretable learning functions such as polynomials.
10

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13


Reviews:
[{"review_id": "nPpQWqST3eP", "reviewer": "Reviewer_9iGQ", "summary": "# After Rebuttal\n\nI appreciate the authors including multiple datasets in rebuttal, but the performance improvement is still not as big. \nThe idea is still somewhat incremental in my idea, but the evaluation seems complete and the speed-up looks good compared to NAM. I believe this paper will have more impact if it releases a good codebase, and also shows more GAM graphs in the image datasets. \n\n# Original Review\n\nThis paper proposes to improve the NAM model by learning a shared basis function for each feature function i.e. for each MLP of each feature, the first few layers are the same across features. The proposed method, called NBM, allows a strong reduction of the # of parameters without sacrificing the accuracy. It also has higher speed and scales to higher number of features. It also models the interaction term, called NB2M, by modeling each pairwise interaction term through another MLP. The performance of NBM is slightly better than NAM in several tabular and image datasets although consistently. ", "questions": "1. **Dataset picking**: Why are there several datasets in NAM not compared? Also, image datasets are a very poor use of GAM - no one would want to visualize the GAM plot of a single pixel. I get that the shared basis functions of NBM should work better with images since each feature is homogenous, but again no one would use GAM on images. If the idea is to test NB2M in high-dimensional features, there are multiple high-dimensional tabular datasets used in the NodeGAM paper that can be compared to. Also, why do you mostly include multi-class datasets while NAM uses binary ones? \n\n2. Doesn't the NB2M still have the quadratic growth of the parameters? In Eq. (5), the parameter $b_{i,j,k}$ still grows quadratic.\n\n3. IMHO, the feature selection method in NodeGAM by using attention and back-propogation is not complex, just a type of regular deep learning architecture.\n\n4. It would be nice to show if NBM is better than NodeGAM since it's also a neural GAM and should have fewer #params and faster throughput than NAM already. It also supports multi-class classification unlike EB2M. But I understand it can be perpendicular to the paper's main point to improve from NAM.\n\n\n[1] Chang, Chun-Hao, Rich Caruana, and Anna Goldenberg. \"Node-gam: Neural generalized additive model for interpretable deep learning.\" arXiv preprint arXiv:2106.01613 (2021).", "limitations": "One limitation I can think of is that the hard sharing could lead to negative transfer among feature networks and deteriorates the performance. ", "rating": 5, "confidence": 5, "soundness": 1, "presentation": 4, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "# Strengths\n- Clear writing.\n\n# Weaknesses\n- The datasets, IMO, are poorly chosen in this paper. Several image datasets are used, but no one would use GAM on pixels to claim any interpretability. Also, several original datasets in NAM are not compared. In contrast, some rare datasets like CoverType are used. It makes me wonder if there are dataset picking and the reported performance is not representative.\n\n- Low originality: I believe the idea is not new as sharing basis functions is common, e.g. CNN v.s. MLP and multi-task learning. The basis sharing might work better but the hard sharing in NBM can incur negative transfer that lowers the performance, especially in tabular datasets when the features are very heterogenous. Therefore, I think it's more important the author shows more comprehensive evidences.\n\n- Even with the selected datasets, the performance benefit does not seems to be big. ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "YDkxWSBGXFa", "reviewer": "Reviewer_3MrK", "summary": "This paper proposes a new GAM-like model, denoted as Neural Basis Model (NBM) to analyze the learned feature in machine learning models.  Comparing with the traditional GAM, NBM learns a set of shape functions for each feature using a single neural network. Comparing with NAM, NBM requires less parameters when the dimensionality is greater than ten. The experiment shows NBM can achieve similar or slightly higher results to NAM on multiple datasets using less parameters. The visualized shape functions show that NBM is more stable than NAM, a lower standard deviation is achieved on 100 runs.", "questions": "Generally, this draft is well- written and organized, the only question I have is about the \"sparse architecture\"(line 120). It would be even better to add one more column about the average sparseness of those tabular datasets to Table 1, which could be another factor behind the acceleration shown in Table 2. ", "limitations": "yes", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The number of parameters of NBM is less than NAM when the dimensionality is high, the trick behind is the convolution-like shared shape function (with a fixed small number of B parameters, Eq.(6)), and this NBM can be extend to the bi-variate situation based on Eq(5). This simple yet effective design outperforms the previous state-of-the-art NAM.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ["I don’t know"]}, "extra_text": {}}, {"review_id": "RNTBuHyYZO", "reviewer": "Reviewer_uzMk", "summary": "The paper proposes a Neural Basis Model (NBM) that utilizes basis decomposition of shape function for regression, binary classification and multi-class classification tasks in the context of achieving state-of-the-art accuracy, model size, and throughput. The central idea of the proposed approach is to dissociate the input feature and individually feed it to a set of shared bases that allows for efficient in inference and compact model size.\n", "questions": "\n1) Have you try the proposed method on the ill-posed tasks? such as image pinpointing and radiance field reconstruction? How does it perform?\n2) Is there any similarity between the learned basis and the basis extracted from PCA-like approaches?\n3) Any further advantages of interpretable modeling apart from the visualization?", "limitations": "The paper lacks a limitations analysis.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "What's good:\n1) The paper organization and presentation are clear mostly.\n2) The idea of decomposing each feature’s shape function into a small set of basis functions seems novel and efficient.\n\nTo be improved:\n1) Seems the main motivation of the paper is interpretability (as addressed in the title). If so, I would like to see a detailed analysis of the shape of the learned basis and a visualization of how these basis represent the inputs. \n2) The comparison approaches are related old I think, how does NBM perform compared with recent implicit functions?\n3) The idea of basis learning is also related to the radial basis functions, I would recommend also including some related discussion.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "CBh_j0TVmlE", "reviewer": "Reviewer_pw3W", "summary": "This paper proposes a new transparent model called NBM. NBM improves upon NAM, which learns a NN (neural network) for each feature and the final output is a learned weighted-sum of the outputs of the neural networks. Instead of training independent NNs for each feature like NAM, NBM trains a set of NNs as basis and used them across all features. \nNBM improves the scalability issue of NAM by reducing the number of independent NNs. NBM is more stable than NAM because of the use of shared NNs. \nExperimental results show NBM and its variant NB^2M can get better results than NAM and NA^2M.", "questions": "- Do you do any data normalization before training NBM? I didn't remember any related discussions in the paper or supplimentary PDF. If no, when the features' magnitudes vary, does NBM essentially work the same way as NAM since each NN is supposed to work for a particular range of inputs? I believe it's worth to discuss this issue a little bit more.\n- Why does NBM achieve better experimental results than NAM? NAM seems to be more powerful in its representative power. Is it the case where NAM achieves better training accurracy/loss but NBM gets better validation/test results? If that's case, does the shared NN work as some implicit regularizations? Will there be some regularizations that make NAM achieve similar results as NBM?", "limitations": "- It would be great if the authors can discuss more about what kind of high dimension data can/can't be handled by NBM.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The proposed model achieved very good experimental results.\n- The proposed model is easy to follow.\n\nWeeknesses\n- I had to switch back and forth multiple times while reading the paper. For example, the NAM model, an extension of GAM and the predecessor of the proposed NBM, is not mentioned in Section 3.1 Background but in Section 3.3 Dicussion after NBM is introduced. I feel it's more natural to introduce GAM, NAM, then NBM, which helps readers understand the contribution of this paper.\n- The discussion regarding the limitation of NBM is quite limited and can be improved. See my questions below.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ["I don’t know"]}, "extra_text": {}}]

Author rebuttal:

Decision: Accept