NODE-GAM: Neural Generalized Additive Model for Interpretable Deep Learning
Authors: Chun-Hao Chang
Rich Caruana
Anna Goldenberg
Venue: ICLR 2022 Spotlight (2022)
Keywords: Generalized Additive Model
Deep Learning Architecture
Interpretability

Abstract: Deployment of machine learning models in real high-risk settings (e.g. healthcare) often depends not only on the model's accuracy but also on its fairness, robustness, and interpretability. Generalized Additive Models (GAMs) are a class of interpretable models with a long history of use in these high-risk domains, but they lack desirable features of deep learning such as differentiability and scalability. In this work, we propose a neural GAM (NODE-GAM) and neural GA$^2$M (NODE-GA$^2$M) that scale well and perform better than other GAMs on large datasets, while remaining interpretable compared to other ensemble and deep learning models. We demonstrate that our models find interesting patterns in the data. Lastly, we show that we are able to improve model accuracy via self-supervised pre-training, an improvement that is not possible for non-differentiable GAMs.

Published as a conference paper at ICLR 2022
NODE-GAM:
NEURAL GENERALIZED ADDITIVE MODEL FOR INTERPRETABLE DEEP LEARNING
3 Chun-Hao Chang, 2 Rich Caruana, 1,2,3 Anna Goldenberg
l University of Toronto, 2 Vector Institute, Hospital of Sickkids, Miicosooft Research k iingll syscey. w.r.cetooootttt................. toronto. edu, rea ruana @mi crosoft com, anna gol denber wttobberrrooooooooooo
ABSTRACT
Deployment of machine learning models in real high-risk settings (e.g. healthcare) often depends not only on the model's accuracy but also its fairness, robust- ness, and interpretability. Generalized Additive Models (GAMs) are a class interpretable models with long history of use in these high-risk domains, but they lack desirable features of deep learning such as diffcrentiability and scalability. In this work, we propose neural GAM (NODE-GAM and neural GA 2 (NODE- GA MM) that scale well and perform better than other GAMs on large datasets, while remaining interpretable compared to other ensemble and deep learning models. We demonstrate that our models find interesting patterns in the data. Lastly, we show that we improve model accuracy via celf-superiied pre-training, improvement that is not possible for non-differentiable GAMs.
INTRODUCTION
As machine learning models become increasingly adopted in everyday life, we begin to require models to not just be accurate, but also satisfy other constraints such as fairness, bias discovery, and robustness under distribution shifts for high-stakes decisions (e.g., in healthcare, finance and criminal justice). These needs call for an easier ability to inspect and understand a model's s predictions.
Generalized Additive Models (GAMs) (Hastie & Tibshirani 1990) have long history of being used to detect and understand tabular data patterns in a variety of fields including medicine (Hastie & Tibshirani 1995 [zadi] 2020), business (Sapra (013) and ecology (Pedersen et al., 2019). Recently
proposed tree-based GAMs and GA Ms models (Lou et al. 2013) further improve on original GAMs (Spline) having higher accuracy and better ability to discover data patterns (Caruana al. 2015).
These models are increasingly used to detect dataset bias (Chang et al., 2021) or audit black- models (Tan et al.
2018a b). As powerful class of models, they still lack some desirable features of deep learning that made these models popular and effective, such as differentiability and scalability.
In this work, propose deep learning version of GAM and GA that enjoy the benefits of both worlds. Our models are comparable to other deep learning approaches in performance on tabular data while remaining interpretable. Compared other GAMs, our models can be optimized using GPUs and mini-batch training allowing for higher accuracy and more effective scaling on larger datasets. We also show that our models improve performance when labeled data is limited by self-supervised pretraining and finetuning, where other non non-differentiable GAMs cannot be applied.
Several works have focused on building interpretable deep learning models that are effective for tabular data. TabNet (Arik& & Prister, 2020) achieves state-of-the-as art performance on tabular data while also providing feature importance per example by its attention mechanism. Although attention seems to be correlated with input importance (Xu et al. 2015). in the worst case they might not correlate well (Wiegreffe & Pinter, 2019). Yoon et al. (2020) proposes to use self-supervised learning on tabular data and achieves ofate-of-the-ar performance but does not address interpretability. NIT (Tsang jet al. 2018) focuses on building a neural network that produces at most K-order interactions and thus include GAM and GA? M. However, NIT requires two-stage iterative training process that requires longer computations. And their performance is slightly lower to DNNs while ours are overall on par with it. They also do not perform purification that makes 2 graphs unique when showing them.

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Published as a conference paper at ICLR 2022
The most relevant approaches our work are NODE (Popov al. 2019) and NAM Agarwal et al. 2020). Popov et al. (2019) developed NODE that mimics an ensemble of decision trees but permits differentiability and achieves ofute-o-thhe-ar performance on tabular data. Unfortunately, NODE suffers from a lack of interpretability similarly to other ensemble and deep learning models. On the other hand, Neural Additive Model (NAM) whose deep learning architecture is GAM, similar to our proposal, thus assuring interpretability. However, NAM can not model the pairwise interactions and thus do not allow GA Also, because NAM builds small feedforward net per feature, in ligessimensional datasets NAM may require large memory and computation. Finally, NAM requires training of 10s to 100s of models and ensemble them which incurs large computations and memory, while ours only trains once; our model is also better than NAM without the ensemble (Supp.
To make our deep GAM scalable and effective, we modify NODE architecture (Popov et al. 2019) to be a GAM and M, since NODE achieves state-of-the-ar performance on tabular data, and its tree-like nature allows GAM learn quick, non-linear jumps that better match patterns seen real data (Chang et al. 2021
We thus call our models NODE-GAM and NODE-GA"M "M respectively.
One of our key contributions that we design several novel gating mechanisms that gradually reduce higher-order feature interactions learned in the representation. This also enables our NODE-GAM and NODE-GA M to automatically perform feature selection via back- pack-propagation for both marginal and pairwise features. This is a substantial improvement tree-based GA that requires additional algorithm to select which set of pairwise feature interactions to learn (Lou et al. 2013).
Overall, our contributions be summarized follows:
Novel architectures for neural GAM and GA²M thus creating interpretable deep learning models.
Compared to mate-of-the-arl GAM methods, our HODE-GAM and NODE-GA 2 achieve similar performance on medium-sized datasets while outperforming other GAMs larger datasets.
We demonstrate that NODE-GAM and NODE-GA discover interesting data patterns.
Lastly, we show that NODE-GAM benefits from self-supervised learning that improves performance when labeled data is limited, and performs better than other GAMs.
We foresee novel deep learning formulation of the GAMs to be very useful in high-risk domains, such as healthcare, where GAMs have already proved to be useful but stopped short from being applied to new large data collections due to scalability access to labeled data is limited. Our novel approach also benefits the deep learning community by adding high accuracy interpretable models to the deep learning repertoire.
accuracy issues, well settings where
BACKGROUND
GAM and GA
GAMs and GA²Ms are interpretable by design because of their functional
forms. Given an input & E RD,, a label a link function (e.g. is log r-p in binary classification), main effects fj for each feature j. and feature interactions siy, GAM and GA²M are expressed as:
D GAM: ((y) = fo E GA g(v) fo E +
D
D ΣΣ for (2,, 3=13'3
1=1
J=1
Unlike full complexity models (e.g. DNNs) that have y f(.,,-.., TT)), GAMs and GA interpretable because the impact of each feature fÿ and each feature interaction Siy can be visualized as graph (i.e. for f,. x-axis shows z, and y-axis shows fz,)) Humans can easily simulate how they work by reading fjs and fix off different features from the graph and adding them together.
GAM baselines: We compare with Explainable Boosting Machine (EBM) Nori al. 2019 that implements tree- GAM and GA?M. We also compare with splines proposed in the 80s (Hastie Tibshirani 1990) using Cubic splines in pygam package (Servén & Brummitt 2018).
Neural Oblivious Decision Trees (NODEs): We describe NODEs for completeness and refer the readers to PPppov et al. (2019) for more details. NODE consists of L layers where each layer has differentiable oblivious decision trees (ODT) of equal depth C. Below we describe single ODT.
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Differentiable Oblivious Decision Trees: An ODT works like traditional decision tree except for all nodes in the same depth share the same imput features and thresholds, which allows parallel computation and makes it suitable for deep learning. Specifically, an ODT of depth C' compares C chosen input feature to C' thresholds, and returns one of the 2 possible responses. Mathmatically, for feature functions Fe which choose what features split, splitting thresholds de, and response vector R E R the tree output (a) is defined as:
h(2) = R (HE - 64) b4) &
(x) S 62) [(F 62) @
I(F x S I[FC 2 80
(I)
Here II is the indicator function, @ is the outer product, and is the inner product. Both feature functions F and Il prevent differentiabilityy To make them differentiable, Popov et al. (2019) replace (x) as a weighted sum of features:
(x) Σ eFttaxx((FF)) .enemaax (F").
Here F E RD are the logits for which features choose, and entmax, (Peters et al. 2019) is the entmax transformation which works like a sparse version of softmax such that the sum of the output equals to 1. They also replace the I with entmoid which works like sparse sigmoid that has output values between 0 and 1. Since all operations are differentiable (entmax, entmoid, outer and inner products), the ODT is differentiable.
Stacking trees into deep layers: Popov al. (2019) follow the design similar to DenseNet where all tree outputs h (z) from previous layers (each layer consists of total I trees) become the inputs to the next layer. For input features æ, the imputs æ' to each layer / becomes:
x
='= x'
|z, h (x ).... (1-1) (x"1)) for 1 1.
(3)
And the final output of the model (xx) is the average of all tree outputs hi, hr all layers:
((x) LI
hi (x:)
(4)
3
OUR MODEL DESIGN
GAM design: See Supp. for complete pseudo code. To make NODE a GAM, we make three
key changes to avoid any feature interactions the architecture (Fig. [ First, instead of letting [T) be a weighted sum of features (Eq. 2p. we make it only pick T feature. We introduce temperature anncaling parameter T that linearly decreases from to 0 for the first S learning steps to make entmax. /T') gradually become one-hot:
(((x) x entmax, (F"
stops
(5)
Second, within each tree, we make the logits F' the same across depth C' i.e. =l.... PC F to avoid any feature interaction within tree. Third, we avoid the DenseNet connection between two trees that focus on different features j.j', since they create feature interactions between features j and j' if two trees connect. Thus introduce gate that only allows connections between trees that take the same features. Let G, entmax,, (F,/T) of the tree i. For tree i in layer I and another tree i in layer for i the gating weight It:: and the feature function Fys for tree become:
1-1
Ia: G Gi, Fa( G.
ΣΣ
(6)
ΣI Iis [=1
Since G becomes gradually one-hot by Eq. after S steps 9;: would only become 1 when G; = G; and ( otherwise. This enforces feature interaction between tree connections.
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Input
F.
WL
Layer
Layer
The NODE-GAM architecture. Here we show 4 features with 4 different colors. Each layer consists of differentiable oblivious decision trees that outputs hy...rr, where each h; only depends on 1 feature. We only connect trees between layers if two trees depend on the same features. And concatenate all outputs from all layers as inputs to the last linear layer Wい to produce outputs.
Figure 1:
Attention-based GAMs (AB-GAMs): To make the above GAM more expressive, we add an attention weight age the feature function (a) to decide which previous tree to focus on:
1-1
Fi(()) Σ G. + ΣΣ hg(z)9aam where ΣΣ Iu 1.
To achieve this, we introduce attention logits Au for each tree i that after entmax it produces aga
0z = entmaxx (log(g:) + Au);-
(8)
This forces the attention of a tree that >; am 1 for all that It.- = 1 and ag when Ii 0.
The attention logits A requires large matrix size [/, (1 1)/] for each layer / which explodes memory. We instead make A as the inner product of two smaller matrices such that A BC' where of size [1, E] and C is of size [E, (I 1)4]. where a hyperparameter for the embedding dimension of the attention.
Last Linear layer: Lastly, instead of averaging the outputs of all trees as the output of the model (Eq. 4), we add the last linear layer be weighted sum of all outputs:
((x) = ΣΣ
bu(x)))w
==
Note that self-supervised learning, wi has multiple output heads to predict multiple tasks.
Regularization: We also include other changes that improves performance. First, we add Dropout (rate pl) on the outputs of trees hx(2z), and Dropout (rate p2) on the final weights wy, Also, to increase diversity of trees, each tree can only model a random subset of features (7)), an idea similar to Random Forest. We also add an & penalization (A) on h4 (xz). In binary classification task where labels y are imbalanced between class 0 and 1, we constant log I-P(y) that added the final output of the model such that after sigmoid it becomes the p(y) if the output of the model O. We find it's crucial for f penalization to work since & induces the model to output 0.
p(y)
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NODE-GA'Ms extending NODE-GAMs to two-way interactions: To allow two-way in- teractions, for each tree we introduce two logits F' and F instead of just one, and let F = F(c-1) mod 2+1 for > 2: this allows at most features to interact within each tree (Fig. 7 Besides temperature annealing (Eq. 5). we make the gating weights Pi, 1 only if the combination of F*, FF the same between tree and (i.e. both trees i and focus on the same features). We set Ij as:
I = min( G x (G: G + (G) G.) x (G G 1).
(10)
We cap the value at 1 to avoid uneven amplifications as 9a: = 2 when G G. G G
Data Preprocessing and Hyperparameters: We follow Popov al. (0019 to do target encoding for categorical features, and do quantile transform for all features to Gaussian distribution (we find Gaussian works better than Uniform). We use random search to search the architecture space for NODE, NODE-GAM and NODE-GA GA MM. We use QHAdam (Ma & Yarats 2018) and average the most recent 5 checkpoints (Izmailov al., 2018). addition, we adopt learning rate warmup (Goyal 2017). and do early stopping and learning rate decay on the plateau. More details in Supp. G
et al.
Extracting shape graphs from GAMs: We follow Chang et al. (021) 1) to implement a function that extracts main effects f, from any GAM model including NODE G-MM, Spline and EBM. The main idea is to take the difference between the model's outputs of two examples (x x²) that have the same values except for feature j. Since the intercept and other main effects are canceled out when taking the difference, the difference f(x²) f(z') is equal to fz]) fz(z)). If we query all the unique values of ø,, we get all values of f, relative to fz())). Then we center the graph of fy by
setting the average of (zy) across the dataset as 0 and add the average to the intercept term fo-
Extracting shape graphs from Gs: Designing a black box function to extract from any GA is non-trivial, as each changed feature 2, would change not just main effect term fy but also every interactions Vy lay' that involve feature j. Instead, since we know which features each tree takes, we can aggregate the output of trees into corresponding main fj and interaction terms fjz--
Note that GA M can have many representations that result in the same function. For example, for a prediction value associated with æ2, we can move to the main effect fa(x2) = u. or the interaction effect (z2, = v that involves z2. To solve this ambiguity, we adopt "purification" (Lengerich al. 2020) that pushes interaction effects into main effects if possible. See Supp. for details.
RESULTS
We first show the accuracy of our models in Sec. 4.1 Then we show the interpretability of our models on Bikeshare and MIMIC2 datasets in Sec. 4.2 In Sec. 4.3] we show that NODE-GAM benefits from self-supervised pre-training and outperforms other GAMs when labels are limited. In Supp. A we show our model outperforms NAM without ensembles. In Supp. B we provide a strong default hyperparameter that still outperforms EBM without hyperparameter tuning.
4.1 ARE NODE-GAM AND NODE-GA 2 ACCURATE?
We compare our performance on popular binary classification datasets (Churn, Support2, MIMIC2, MIMIC3, Income, and Credit) and 2 regression datasets (Wine and Bikeshare). These datasets are medium-sizee with 6k-300k samples and 6-57 features (Table 6). We use 5-fold cross validation to derive the mean and standard deviation for each model. We use 80-20 splits for training and val set. To compare models across datasets, we calculate summary metries: (1) Rank: we rank the performance on each dataset, and then compute the average rank across all 9 datasets (the lower the rank the better). (2) Normalized Score (NS): for each dataset, we set the worst performance for that dataset and the best 1, and scale all other scores linearly between 0 and 1.
In Table I we show the performance of all GAMs, GA and full complexity models. First, we compare 4 GAMs (here NODE-GA G (M-main is the purified main effect from NODE-GA MM) We find all 4 GAMs perform similarly and the best GAM different datasets varies, with Spline as the best in Rank and NODE-GAM in NS. But the differences are often smaller than the standard deviation.
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Table I: The performance for 8 medium-sized datasets. The first 6 datasets are binary classification (ordered by samples) and shown the AUC (%). The last 2 are regression datasets and shown the Root Mean Squared Error (RMSE). We show the standard deviation of 5-fold cross validation results. We calculate average rank (Rank, lower the better) and average Normalized Score (NS, higher the better).
GAM
GA
Full Complexity
NODE
NODE GAM
NODE GA
EBM GA
GA Main
EBM
Spline
NODE
XGB
RF
Chur
84.9 us
84.9 099
85.0 0.7
85.1 0.
85.0+ 0.8
85.0
84
84.7+ 09
82.9 as
Support2
1.3
-1
81.5+ L.
81.5
82.7 0.7
82.6+
82.7+10
82.3+ 10
82.1+
Mimic2
83.2+
83.4=1.3
83.5+
82.5
84.6
84.8+
84.3+
84.4+
85.4 1.3
Mimic3
81.4
81.0=a6
80.9
81.2=
82.2
82.1
82.8
81.9+
79.5
Income
92.7= 0.3
91.8=a5
92.7=
91.8=
92.3 0.3
92.8
91.9
92.8+
90.8
Credit
1.
98.4=
98.2= 97.4=
98.6
98.21
98.1
97.81 0.
94.61 1.8
Wine
0.71 0.03
0.70 0.02
0.70 0.02
0.72+ 0.02
0.67 0.02
0.66+ 0.0
0.64 0.01
0.75# 0.03
0.61 0.01
Bikeshare
100.7 1.6
100.7 1.4
100.0 1.4
99.8= 144
49.8 0.8
50.1
36.2
49.2: 0.9
42.2
Rank 5.8 0.533
6.2 0.471
5.9 0.503
5.3
3.2 0.808
3.5
4.5
3.9
6.6 0.301
NS
0.464
0.812
0.737
0.808
Table 2: The performance for 6 large datasets used in NODE paper. The first datasets (Click, Epsilon and Higgs) classification datasets and shown the Error Rate. The last 3 (Microsoft, Yahoo and Year) are shown Mean Squared Error (MSE). We show the relative improvement (Rel Imp) of model NODE-GAM to EBM and find it consistently outperforms EBM up 7%.
GAM
GA
Full Complexity
NODE GAM
Rel Imp
NODE GAPM
EBM
Rel Imp
EBM
Spline
NODE
XGB
RF
GA
0.3342
0.3328
0.3369
0.3307
0.3297
0.3312
0.3334
0.3473
Click
-0.4%
-0.2%
+ 0.0001
= 0.0001
0.0002
+ 0.0001
= 0.0001
0.0002
0.0002
+ 0.0001
0.1040
0.1050
0.1034
0.1112
0.2398
Epsilon
+ 0.0003
+. 0.0002
0.0003
0.0006
+ 0.0008
0.2970
0.3006
0.2566
0.2767
0.2101
0.2328
0.2406
Higgs
1.2%
7.3%
+ 0.0001
+ 0.0002
+ 0.0003
+ 0.0004
0.0005
0.0003
1 0.0001
0.5821
0.5890
0.5618
0.5780
0.5570
0.5544
0.5706
Microsoft
1.2%
2.8%
+ 0.0004
+ 0.0006
+ 0.0003
+ 0.0001
+ 0.0002
+ 0.0001
+ 0.0006
0.6101
0.6082
0.5807
0.6032
0.5692
0.5420
0.5598
Yahoo
-0.3%
3.7%
= 0.0006
= 0.0011
+ 0.0004
= 0.0005
0.0002
0.0004
1 0.0003
85.09
85.81
79.57
83.16
76.21
78.53
86.61
Year
0.8%
4.3%
0.01
+ 0..11
0.12
0.01
0.12
0.09
0.06
Average
0.5%
3.6%
Next, both NODE-GA'M and EBM-GA²M perform similarly, with NODE-GA M better in Rank and EBM-GA better NS. Lastly, within all full complexity methods, XGB performs the best with not much difference from NODE and RF performs the worst. In summary, all GAMs perform similarly. NODE-GA2M M is similar EBM-GA M, and slightly outperforms foll-complexity models.
In Table 2 we test our methods on 6 large datasets (all have samples > 500K) used in the NODE paper, and we use the same train-test split to comparable. Since these only provide test split we report standard deviation across multiple random seeds. First, on a cluster with 32 CPU and 120GB memory, Spline goes out of memory on 5 out of 6 datasets and EBM also can not be run on dataset Epsilon with 2k features, showing their lack of ability to scale to large datasets. For 5 datasets that EBM can run, our NODE-GADAEE GAMS runs slightly better than EBM. But when considering G NODE-GA M outperforms EBM-GA M up to 7.3% in Higgs and average relative improvement of 3.6%. NODE outperforms all GAMs and GA Ms substantially on Higgs and Year, suggesting both datasets might have important higher-order feature interactions.

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(a) Hour
(b) Temperature
(c) Month
(d) Week day
Rental Counts
149
socce SUM care
309
Figure 2: The shape plots of 4 (out of 11) features of 4 models (NODE-GA' NODE-GAM, EBM, and Spline) trained on the Bikeshare dataset.
(a) Hr x Working day
(b) Hr x Week day
(c) Hr X Temperature
(d) Hr Humidity
Rontal Counts
Week day
Tomperratur
Slumialty
wo Manninggg Say wo Kingday
24
Hour
Hour
Hour
Hour
Figure 3:
The shape plots of interactions of NODE-GA trained on the Bikeshare dataset.
4.2 SHAPE GRAPHS OF NODE-GAN AND NODE-GA M: BIKESHARE AND MIMIC2
In this section, we highlight our key findings, and show the rest of the plots in Supp. M
Bikeshare dataset: Here we interpret the Bikeshare dataset. contains the hourly count of rental bikes between the years 2011 and 2012 in Capital bikeshare system located in Washington, D.C. Note that all 4 GAMs trained on Bikeshare are equally accurate with < 0.1% error difference (Table D.
In Fig. 2 we show the shape plots of 4 features: Hour, Temperature, Month, and Week day. First, Hour (Fig. 2) is the strongest feature with two peaks around 9 AM and PM, representing the time that people commute, and all 4 models agree. Then we show Temperature in Fig. 2b. Here temperature is normalized between 0 and 1, where O means -8 and means 39'C. When the weather is hot (Temp 0.8, around 30CC), all models agree rental counts decrease which makes sense. Interestingly, when it's getting colder (Temp < 0.4, around 1 C)) there a steady rise shown by NODE-GAM, Spline and EBM but not NODE-GA (blue). Since it's quite unlikely people rent more bikes when it's getting colder especially below O'C, the pattern shown by GA seems more plausible. Similarly, in feature Month (Fig. 2 NODE-GA2M shows a rise in summer (month 8) while others indicate a strong decline of rental counts. Since we might expect more people to rent bikes during summer since it's warmer, NODE- M might be more plausible, although we might explain it due to summer vacation fewer students ride bikes to school. Lastly, for Weekday (Fig. ! all models agree with each other that the lowest number of rentals happen at the start of the week (Sunday and Monday) and slowly increase with Saturday as the highest number.
Fig. 3 we show the 4 feature interactions (out of 67) from our NODE-GA The strongest effect happens in Hr Working day (Fig. I[))): this makes sense since in working day (orange), people usually rent bikes around 9AM and 5PM to commute. Otherwise, if the working day is 0 (bluc), the number peaks from 10AM to 3PM which shows people going out more often in daytime. In Hr x Weekday (Fig. b)), we can see more granularly that this commute effect happens strongly on Monday Thursday, but on Friday people commute bit later, around 10 AM, and return earlier, around 3 or 4 PM. In Hr x Temperature (Fig. c)). it shows that in the morning rental count is high when it's cold, while in the afternoon the rental count high when it's hot. We also find in Hr Humidity (Fig. d)) that when humidity is high from 3-6 PM, people ride bikes less. Overall these interpretable graphs enable to know how the model predicts and find interesting patterns.
MIMIC2: MIMIC2 is the hospital ICU mortality prediction task (Johnson et al. 20) We extract 17 features within the first 24 hour measurements, and we use mean imputation for missingness.
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(a) Age
(b) PFratio
(c) Bilirubin
(d) GCS
Log oddd
NODE GAZM
NODE GAM EBM
Spline
44
Figure 4: The shape plots of 4 GAMs trained on MIMIC-II dataset (4 of the 17 features are shown).
(a) Age x Bilirubin
(b) GCS x Bilirubin
(c) Age GCS
(d) GCS x PFratio
Biliroiin
Biliruhin
20
109
Age
GCS
Age
GCS
Figure 5: The shape plots of 4 interactions of NODE-GA2M trained on the MIMIC2 dataset.
We show the shape plots (Fig. 4 of features: Age, PFratio, Bilirubin and GCS. In feature Age (Fig.. 4a)). we see the all 4 models agree the risk increases from age 20 to 90. Overall NODE- GAM/GA M are pretty similar to EBM in that they all have small jumps in similar place age 55, 80 and 85; spline (red) is as expected very smooth. Interestingly, we see NODE-GAMIGA'S shows risk increases a bit when age 20. We think the risk is higher in younger people because this is generally a healthier age in the population, so their presence in ICU indicates higher risk conditions.
In Fig. 4(b), we show PFratio: a measure of how well patients oxygenate the blood. Interestingly, NODE-GAMIGA'M and EBM capture sharp drop at 332. It turns out that PFratio is usually not measured for healthier patients, and the missing values have been imputed by the population mean 332, thus placing group of low-risk patients right at the mean value of the feature. However, Spline (red) is unable to capture this and instead have a dip around 300-600. Another drop captured by NODE-GAM/GAPM from 400- 500 matches clinical guidelines that >> 400 is healthy.
Bilirubin shape plot is shown in Fig. 4c). Bilirubin is yellowish pigment made during the normal breakdown of red blood cells. High bilirubin indicates liver or bile duct problems. Indeed, we can see risk quickly goes up as Bilirubin is > 2, and all 4 models roughly agree with each other except for Spline which has much lower risk when Billirubin is 80, which is likely caused by Spline's 's smooth inductive bias and unlikely to be true. Lastly, in Fig. d) we show Glasgow Coma Scale (GCS); a bedside measurement for how conscious the patient is with 1 in a coma and conscious. Indeed, we find the risk is higher for patients with GCS=1 than 5, and all 4 models agree.
In Fig. 5, show the 4 of 154 feature interactions learned in the NODE-GA2M. First, in Age Bilirubin (Fig. 5ja)), when Billirubin is high (>2), we see an increase risk (blue) in people with age 18 18-70. 70. Risk decreases (red) when age > 80. Combined with the shape plots of Age (Fig [a) and Bilirubin (Fig. (c)), we find this interaction works as a correction effect: if patients have Bilirubin > 2 (high risk) but are young (low risk), they should have higher risk than what their main (univariate) effects suggest. On the other hand, if patients have age > 80 (high risk) and Bilirubin 2 (high risk), they already get very high risk from main effects, and in fact the interaction effect is negative to correct for the already high main effects. It suggests that Billirubin=2 is an important threshold that should affect risk adjustments.
Also in GCS Bilirubin plot b()), we find similar effects: if Bilirubin >2, the risk of GCS is correctly lower for GCS= and higher for 3-5. In Fig, ((c) we find patients with GCS=1-3 (high risk) and age>80 (high risk), surprisingly, have even higher risk (blue) for these patients; it shows models think these patients are more in danger than their main effects suggest. Finally, in Fig. d(d) show interaction effect GCS PFratio. We find PFratio also has a similar threshold effect: if PFratio > 400 (low risk), and G,S= (high risk), model assigns higher risk for these patients while decreasing risks for patients with GC5=3,4.5.
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Churn
Support2
MIMIC2
Relative Imp (%)
55
EBM
Spline
SS EAM
FBM
Spline
Labeled data ratio (%)
Labeled data ratio (%)
Labeled data ratio %
MIMIC3
Income
Credit
Relative Imp (%)
SS
SS
EBM
EBM
EBM Spline
Spline
Spline
20
05
300
155
353
Labeled data ratio(%) (%)
Labeled data ratio (%)
Labeled data ratio(%)
Figure 6: The relative improvement (%) over NODE-GAM without self-supervision (No-SS) datasets with various labeled data ratio. The higher number is better.
4.3 SELF-SUPERVISED PRE-TRAINING
By training GAMs with neural networks, it enables self-supervised learning that learns representations from unlabeled data which improves accuracy in limited labeled data scenarios. We first learn NODE-GAM that reconstructs input features under randomly masked inputs (we use 15% masks). Then remove and re-initialize the last linear weight and fine-tune it on the original targets under limited labeled data. For fine-tuning, we freeze the embedding and only train the last linear weight for the first 500 steps; this helps stabilize the training. We also search smaller learning rates [5e-5, 1e-4, 3c-4, 5e-4] and choose the best model by validation set. We compare our self-supervised model (SS) with 3 other baselines: (1) NODE-GAM without self-supervision (No-SS), (2) EBM and (3) Spline. We randomly search 15 attention based AB-GAM architectures for both SS and No-SS.
In Figure 6 we show the relative improvement over the AUC of No-SS under variously labeled data ratio 0.5%, 1%, 2%. 5%, and 10% (except Credit which 0.5% has too few positive samples and thus crashes), And run 3 different train-test split folds derive mean and standard deviation. Here the relative improvement means improvement over No-SS baselines. First, we find NODE-GAM with self-supervision (SS, blue) outperforms No-SS in 6 of 7 datasets (except Income) with MIMIC2 having the most improvement (6%). This shows our NODE-GAM benefits from self-supervised pre-training. SS also outperforms EBM in 3 out of datasets (Churn, MIMIC2 and Credit) up to 10% improvement in Churn, demonstrating the superiority of SS when labeled data is limited.
5 LIMITATIONS, DISCUSSIONS AND CONCLUSIONS
Although we interpret and explain the shape graphs in this paper, we want emphasize that the shown patterns should be treated as an association not causation. Any claim based on the graphs requires proper causal study to validate it.
In this paper, we assumed that the class of GAMs is interpretable and proposed new deep learning model in this class, so our NODE-GANE is as interpretable to other GAMss. But readers might wonder if GAMs are interpretable to human. Hegselmann et al. (2020) show GAM is interpretable for doctors a clinical user study; Tan et al. (019) find GAM is more interpretable than the decision tree helping users discover more patterns and understand feature importance better; Kaur et al. (2020) compare GAM to post-hoe explanations (SHAP) and find that GAM significantly makes users answer questions more accurately, have higher confidence in explanations, and reduce cognitive load.
In this paper we propose a deep-learning version GAM and G that automatically learn which main and pairwise interactions to focus without any two-stage training and model ensemble. Our GAM is also more accurate than traditional GAMs in both large datasets and limited-labeled settings. We hope this work further inspire other interpretable design in the deep learning models.

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ACKNOWLEDGEMENT
We thank Alex Adam to provide valuable feedbacks and improve the writing of this paper. Resources used in preparing this research were provided, in part, by the Province of Ontario, the Govermment of Canada through CIFAR, and companies sponsoring the Vector Institute www. vectorinstittuee ai/#part.ners
ETHICS STATEMENT
A potential misuse of this tool is to claim causal relationships of what GAM models learned. For example, one might wrongfully claim that having asthma is good for pneumonia patients (Caruana et al. 2015). This is obviously wrong conclusion predicated by the bias in the dataset.
The interpretable models are great tool discover potential biases hiding in the data. Especially in high-dimensional datasets when it's hard to pinpoint where the bias is, GAM provides easy visualization that allows users to confirm known biases and examine hidden biases. We believe that such models foster ethical adoption of machine learning in high-stakes settings. By providing deep learning version of GAMs, our work enables the use of GAM models on larger datasets thus increasing GAM adoption in real-world settings.
REPRODUCIBILITY STATEMENT
We released our code in https::///// /ttp::://gthubbe http://ggittb.oo....... wwwpo//awcee00o /odeeqaa with instructions and hyperparameters to reproduce our final results. Our datasets can be automatically downloaded to the directory with train and test fold using included code. We report the details of our datasets in Supp. F) and hyperparameters for both our model and baselines in Supp. G We list the hyperparame- ters corresponding to the best performance in Supp.
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Alistair EW Johnson, Tom J Pollard, Lu Shen, Lehman Li-wei, Mengling Feng, Mohammad Ghassemi, Benjamin Moody, Peter Szolovits, Leo Anthony Celi, and Roger G Mark. Mimic-iii, freely accessible critical care database. Scientific data, 3: 3:160035, 2016b.
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www. www.wclwwbbb.. org wttp://rg//.tttoooooeee1------
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neurips Paper---ppp... pdf
A COMPARISON TO NAM (AGARWAL ET AL., 2020) AND THE UNIVARIATE NETWORK IN NID (SANG ET AL. 2017)
First, we compare with NAM's (Agarwal et al. 2020) performance without and with the ensemble. Since NAM requires an extensive hyperparameter search and be fair with NAM, we focus on 3 of their 5 datasets (MIMIC2, COMPAS, Credit) and use their reported best hyperparameters to compare. We find that NODE-GAM is better than NAM without ensemble consistently across 3 datasets.
Table 3: Comparison to NAM NODE
Agarwal et al.
2020) with and without the ensemble.
NAM
NAM (Ensembled)
EBM
Spline
GAM
(no ensemble)
COMPAS MIMIC-II
74.2 (0.9)
73.8 (1.0)
73.7 (1.0)
74.3 (0.9)
74.1 (0.9)
83.2 (1.1)
82.4 (1.0)
83.0 (0.8)
83.5 (1.1)
82.5 (1.1)
Credit
98.1 (1.1)
97.5 (0.8)
98.0 (0.2)
97.4 (0.9)
98.2 (1.1)
As the reviewer points out, in the NID paper (Tanng al. 2017) they also have similar idea to NAM that trains a univariate network alongside an MLP to model the main effect. But there are two key differences between the univariate network in NID and NAM: (1) NAM proposes new jumpy activation function ExU- - that can model quick, non-linear changes of the inputs as part of their hyperparameter selection, and (2) NAM uses multiple networks ensemble. To be thorough, we compare with NAM that only uses normal units to resemble the univariate network in NID. We also considered whether removing the ensemble would disproportionately impact ExU activations since ExU activations are more prone to overfitting.
We show the performance Table We show the results in 2 of their datasets (MIMIC-II, Credit) that find ExU perform better than normal units (in other 3 datasets normal units perform better). First, we find that after ensemble normal units perform quite similar to ExU in MIMIC2 but worse in Credit. And given that in 3 other datasets NAM already finds normal units to perform better, we think normal units and ExU probably have similar accuracy. Besides, ensemble helps improve performance much more for ExU units but not so much for normal units, since ExU is more low-bias high-variance unit that benefits more from ensembles. In either case, their performance without ensemble still inferior to NOME-GAM.
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Table 4: Comparison NAM with normal units ExU units, and with and without ensemble. NAM-normal NAM-normal (Ensembled)
NAM-ExU
NAM-ExU (Ensembled)
MIMIC-II
82.7 (0.8)
82.9
(1.0)
83.0 (0.8)
Credit
97.3 (0.8)
97.4
(0.8)
(0.2)
COMPAS
73.8 (1.0)
73.7 (1.0)
Table 5: The default performance for 6 large datasets. The NODE-GA GA2M-Defaull is the model with default hyperparameter, and the Rel Diff the relative difference of performance between default and tuned NODE-GA "M. The first 3 datasets (Click, Epsilon and Higgs) classification datasets and shown the Error Rate. The last (Microsoft, Yahoo and Year) are shown in Mean Squared Error (MSE).
NODE GA Default
NODE GA
EBM GA
Rel Diff
0.3332
0.3307
0.3297
Click
-0.4%
+ 0.0001
= 0.0001
+ 0.0001
0.1063
0.1050
Epsilon
-0.8%
+ 0.0001
= 0.0002
0.2656
0.2566
0.2767
Higgs
-3.7%
+ 0.0003
+ 0.0003
1 0.0004
0.5670
0.5617
0.5780
Microsoft
-0.9%
+ 0.0003
0.0003
+ 0.0001
0.6002
0.5807
0.6032
Yahoo
-3.4%
+ 0.0004
= 0.0004
+ 0.0005
80.56
79.57
83.16
Year
-1.1%
+ 0.22
= 0.12
+ 0.01
B THE PERFORMANCE oF NODE-GA²M AND TABNET WITH DEFAULT HYPERPARAMETERS
To increase the ease of use, provide strong default hyperparameter of the NODE-GA 2. In Table 5) compared to the tuned NODE-GA the default hyperparmeter increases the error by but still consistently outperforms EBM-GA²M.
C PSEUDO-CODE FOR NODE-GAM
Here we provide the pseudo codes for our model in Alg. We highlight our key changes that make NODE as a GAM in red, and the new architectures or regularization in blue. We show single GAM decision tree in Alg. single GA M tree in Alg. model algorithm in Alg. 3 and the model update in Alg.
D PURIFICATION OF GA
Note that GA can have many representations that result in the same function. For example, for prediction value v associated with 22, we can move v to the main effect fz(22) = U, or the interaction effect (22,
that involves z2. To solve this ambiguity, et al. 2020 that pushes interaction effects into main effects if possible.
adopt "purification" (Lengerich
To purify an interaction fiy, we first bin continuous feature xj into most K quantile bins with K unique values 2) a.. and for nz well. Then for every aty move the average az of interactions faz' to main effects fy:
K'
1 -
Σ Ssr
= (x),,
f,45)) = -
NK
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Algorithm I A GAM differentiable oblivious decision tree (ODT)
Input: Input X E RR, Temperature T (I' + 0), Previous layers' outputs Xp E R²,, Previous layers' feature selection Gp. & IRPxD Attention Matrix A E IRP Hyperparameters: Tree Depth C, Column subsample ratio !J Trainable Parameters; Feature Selection Logits F "R, Split Thresholds b E RC, Split slope
S E RR, Node weights W E R
v then
maxx(Dn, 1)
Number of subset features Randomly choose features to exclude Exclude features
i shufflo(range(())fiittt)1
F[i] inf
end
G EntMax FF/F F/)
Go through EntMax to generate soft one-hot vector (Eq. 5) >> Pick feature softly
K = X G
if X None then
> Calculate gating weights g (Eq. g' g/ Σ g is None else g g-entmaxa (log(g) + A) > Attention-basee GAM (Eq
GpG Rº
K = K g'
b Add previous outputs with g normalized to
end
H EntMoid
(K bb/S) gC
Generate soft binary value
H
(1 - 11,,))
(H²)
E R20
② .... 8
D Go through the decision tree Select one weight value softly as the output
(1 H
e W
Return: h,
Return tree response and feature selection G
Algorithm 2 A GA differentiable oblivious decision (ODT)
Input: Input X TRD, Temperature T (T - 0), Previous layers' outputs X, E R.P. Previous layers' feature selection G',, G E PxD Attention Matrix A E IR.
Hyperparameters: Tree Depth C, Column subsample ratio !]
Trainable Parameters: Feature Selection Logits F*, F E RRP Split Thresholds E RC, Split
slope RC, Node weights W 2
if 1 < 1 and first time running then max(Dr) 1) i = shuffle (nuffle(range(D))mm]
Number of subset features Randomly exclude features Exclude features
F' F' inf, F inf
end if
G EntMax FL (T) G2 EntMax FF
Get soft one-hot vector (Fq. 5) D Pick feature softly
K X.G G K2 = G2
if Xx is not None then
g mån G x . (G' G%) (G² G)), Gating weights (Eq. 10
g' = g g/gg/// g if A None else g (lomaa,.loo(( + A) & AAtention based GAM (Eq
K = K X g'
> Add previous outputs with g normalized to
1² = K2 X, g''
K = =K', 22 Kl ..... K2 E
D Alternating between K and R²
end ((
H EntMoid (K- - b)/S)
E R
Generate soft binary value
H
X .... &
(1 H10))
(HO)
> Go through the decision tree Select one weight value softly as the output Return tree response and feature selection [G' e
220
(1 H
W
Return: h, |G G2)
This is one step to purify fiz' to fy- Then we purify fiz' to fyr, and so on until all at and at are close to O.
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Algorithm 3 The NODE-GAMI GOAGGGAMMNNNNNNN NODE-GA'M algorithm
1: Input: Input X IRD
2: Hyperparameters:: number of layers I, number of trees per layer, tree depth C, current optimization step s, temperature annealing step S, Attention Embedding E
3: Trainable Parameters: the decision trees Mi in each layer / (either GAM trees (Alg. I or ), the final output weights W2 E (LL)) and bias up
4: GA trees (Alg.
5: if E > 0 then
Use attention- based GAM
6:
Initialize B E (1-11)IE and C' E EXI for = 2...L
7: end if
8: T = 10 10=2)s if < S else 0
> Slowly decrease temperature to 0
9: X, None, G,- None
b Initialize previous trees' outputs Xp and feature selections Gp
10: for = 1 to I do A' None
0 or 1 else B'C h. G: MX(X, .X.T T, X p: Gp. A')
Calculate attention matrix
12:
> Run total I trees in Mj by Alg, 1 or
13:
h1 = Dropout(hz)
b Dropout rate p
14:
X, h1 if X, is None else [X p: hi]
Concatenate outputs h
15:
G5 = Gi if G2 is None else Gp. Gil
> Concatenate feature selection G
16: end for
17: WL = Dropout W1)
Dropout rate pa
18: R= X, WL + wh
Go through last linear layer
19: Return: R, Xp
D Return model response and all trees' outputs Xp
Algorithm 4 Our model's update
J: Input: An input X E RR", target Node-GAM model M
R, Xp M(X)
Run Node-GAM (Alg.
4: I BCELoss(y, (y. R) if binary classification else MSELoss(y, R)
5: L I.+ (XX
Add the lg on the output of trees
6: Optimize L by Adam optimizer
E NODE-GA FIGURES
Here we show the architecutre of NODE-GAAM in Figure 7
F DATASET DESCRIPTIONS
Here describe all datasets and we summarize them in Table
Churn: this is to predict which user is potential subscription churner for telecom company. http:///eeeeoo-cu....oeeee--a-ae biostat. vanderbis wtt.te.dd/..ataaeee.e
Attps://www http:/aaagg...coo//aatttaaaaaaaa
Support2: this is to predict mortality in the hospital by several lab values. http://
MIMIC-II and MIMIC-III dataset (Johnson al. 2016b): this ICU patient datasets predict mortality of patients in tertiary academic medical center in Boston, MA, USA.
Income: UCI Dua & Graff (017)). This is a dataset from census collected in 1994, and the goal is to predict who has income >50//year. htttps:/aarchvee ics. uci edu mll/ataaeets/ s/tul///////lll
this is to predict which transaction is fraud. The features provided are the coeffi- cient of PCA components to protect privacy. lhtps:: //www/ www.wwwagll..cm wwt.:/commmmo/meccee.e Icreditcardfraut
Credit:
Bikeshare (Dua & Graff 2017): this is the hourly bikeshare rental counts in Wash-
ington D.C., USA. https: pettps//aarchieee /ttpr//aa..ee..e. ics. uci. wttpm///aaaattss///////
sharing+datasst
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Input
WL
Layer
Layer
Figure 7: The NODE-GA architecture. Here we show 4 features with 4 different colors. Each layer consists of I differentiable oblivious decision trees that outputs hı...,, where each h; depends on at most 2 features. We only connect trees between layers if two trees depend the same two features. And we concatenate all outputs from all layers as inputs to the last linear layer WL to produce outputs.
Wine (Dua & Grail 30177: this is to predict the wine quality based on variety of lab values. attps://archive ics uci http:///a.aaaa///aaaaaaa/a////aa
For 6 datasets used in NODE, we use the scripts from NODE paper (https: /tt:::/gttttbb com/
Qwi Owicen/node /node) which directly downloads the dataset, Here we still cite and list their sources:
Click:
track2
Higgs: UCI
Dua & Graff 2017
http:///rchlveeccsuu..ddddd
datasets
//IGGGs///IGGS HIGGS
Microsoft: malr/
Yahoo: datatype=c
Year (Dua & Graff https://archive.tt http::///aaeeieee........./////.
rearpredictionnss.
F.I PREPROCESSING
For NODE and NODE-GAN AODE-GAM/GAMM we follow Popov et al. (0019) to do target encoding for categorical features, and do quantile transform" with 2000 bins for all features to Gaussian distribution (we find Gaussian performs better than Uniform). We find adding small gaussian noise (c.g. le-5) when fitting quantile transformation (but no noise in transformation stage) crucial have mean and standard deviation close to 1 after transformation.
sxttemmpprrooceessngquuiineeeee transffrrmm
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Table 6: All dataset statistics and descriptions.
Domain # Samples # Features Positive rate
Description
Churn
Retail 7,043
19
26.54%
Subscription churner
Support2
Healthcare
9.105
29
25.92%
Hospital mortality
MIMIC-II
Hcalthcare
24,508
17
12.25%
ICU mortality
MIMIC-III
Healthcare
27,348
57
9.84%
ICU mortality
Income
Finance
32,561
14
24.08%
Income prediction
Credit
Retail
284,807
30
0.17%
Fraud detection
Bikeshare
Retail
17,389
16
Bikeshare rental counts
Wine
Nature
4,898
12
Wine quality
Click
Ads
1M
11
50%
2012 KDD Cup
Higgs
Nature
11M
28
53%
Higgs bosons prediction
Epsilon
500K
2k
50%
PASCAL Challenge 2008
Microsoft
Ads
964K
136
MSLR-WEB 10K
Yahoo
Ads
709K
699
Yahoo LETOR dataset
Year
Music
515K
90
Million Song Dataset
G HYPERPARAMETERS SELECTION
In order to tune the hyperparameters, we performed random stratified split of full training data into train set (80%) and validation set (20%) for all datasets. For datasets we compile of medium-sioe sized (Income, Churn, Credit, Mimic2, Mimic3, Support2. Bikeshare), we do 5-fold cross validation for 5 different test splits. For datasets in NODE paper (Click, Epsilon, Higgs, Microsoft, Yahoo, Year), train/val/test split provided by the NODE paper author. Since they only provide test split, we report standard deviation by different random seeds on these datasets. For medium-sized datasets, we only tune hyperparameters on the first train-val-test fold split, and fix the hyperparameters to run the rest of folds. This means that we do not search hyperparameters per fold to avoid computational overheads. All NODE, MODE-GAM/CEE M are run with 1 TITAN Xp GPU, 4 CPU and 8GB memory. For EBM and Spline, they are run with machine with 32 CPUs and 120GB memory.
Below describe the hyperparameters we use for each method:
G.1 EBM
For EBM, we set inner bags=100 100 and outer bags=100 and set the maximum rounds as 20k to make sure converges; we find EBM performs very stable out of this choice probably because we total bagging as 10k that makes it stable; other parameters have little effect on final performance.
For EBM GA M, we search the number of interactions for 16, 32, 64, 128 and choose the best one on validation set. On large datasets we set the number of iterations as 64 as we find it performs quite well on medium-sizee datasets.
G.2 SPLINE
We use the cubic spline in PyGAM package (Servén & Brummitt) 2018) that we follow Chang et al. (202T) to set the number of knots per feature to a large number (we find setting it larger would crash the model), and search the best lambda penalty between 1e-3 to le3 for 15 times and return the best model.
G.3 NODE, NODE-GAPM AND NODE
We follow NODE QHAdam (Ma & Yarats 2018) and average the most recent 5 checkpoints. In addition, we adopt learning rate warmup at first 500 steps. And we early stop our training for no improvement for 11 k steps and decay learning rate 1/5 if no improvement happens in 5k steps.
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Published as a conference paper at ICLR 2022
Here we list the hyperparameters we find works quite well and we do not do random search these hyperparameters:
optimizer: QHAdam (Ma & Yarats 2018) (same as NODE paper)
Ir wwrrmp_ssepps: 500
num checkpoints_avgedd 5
cattrratture_ammaaiiiee stepssssstpssseeeee (S): 4k
min tem__mmmeertrre: 0.01 (0.01 is small enough for making one-hot vector. And after S steps we set the function to produce one-hot vector exactly.)
batch_size 2048, or the max batch size that fits in GPU memory with minimum 128.
Maximum training time: 20 hours. This is just avoid model training for too long.
We random search to find the best hyperparameters which we list the range in below. We list the random search range for NODE:
num layer:: 3,,3,4. 4, Default: 3.
total tree counts (= num_trees x mum_layers): {500, 1000, 2000, 4000). Default: 2000.
depth: (2, 4, 6). Default: 4.
tree dim_ (0, 1). Default: 0.
output_dropon (pl): (0, 0.1, 0.2). Default:
colsample_bytree (1, 0.5, 0.1, le-5). Default: 0.1.
Ir: 10.01, 0.005). Default: 0.01.
12 lambda: {O., 1c-7, le-6, le-5). Default: 1e-5.
add_Jast_ke Jast_ineur (to add last linear weight not): (0, 1). Default: 1.
last_ last_dropout (p2, only if hod_last_linerre (0, 0.1, 0.2, 0.3). Default: 0.5.
seed: uniform distribution [1. 100].
For NODE-GAM and NODE-GA we have additional parameters:
arch: (GAM, AB-GAMJ. Default: AB-GAM.
dim att (dimension attention embedding E): (8, 16, 32). Default: 16.
We show the best hyperparameters for each dataset Section H And we show the performance of default hyperparameter in Suppl. B
G.4 XGBooST
For large datasets in NODE, directly report the performance from the original NODE paper. For medium-sized data, we set the depth of xghoost as 3, and learning rate as 0.1 with r_estimators=50k and set early stopping for 50 rounds to make sure it converges.
G.5 RANDOM FOREST (RF)
We use the default hyperparameters from sklearn and set the number of trees to large number 1000.
H BEST HYPERPARAMETERS FOUND IN EACH DATASET
Here report the best hyperparameters find for 9 medium-sized sized datasets in Table (NODE. GAM). Table 8 (NODE-GA: GA M)), and Table 9 (NODE). We report the best hyperparameters for large datasets in Table 10 (NODE-GAM) and Table 11 M
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Published as a conference paper at ICLR 2022
Table 7:
The best hyperparameters for NODE-GAM architecture.
Dataset
Compas
Churn
Support2 Mimic2
Mimic3 Adult
Credit
Bikeshare Wine
batch
2048
2048
2048
2048
512
2048
2048
2048
2048
size
num
5
3
4
4
3
3
5
layers
800
166
125
500
1333
666
400
250
800
trees
depth
4
4
2
4
6
4
2
2
addi
Iree
2
2
1
1
0
1
2
dim
output dropout
0.3
0.1
0.1
0
0.2
0.1
0.2
0.2
colsample
0.5
0.5
le-5
0.5
1e-5
0.5
0.1
0.5
0.5
bytree
Ir
0.01
0.005
0.01
0.01
0.005
0.01
0.01
0.005
0.005
12
le-5
1e-5
1e-6
1e-7
1e-7
0
0
1e-7
1e-5
lambda
add
last
1
1
0
1
1
linear
last
0
0
0
0
0
0
0.3
0.1
dropout
seed
67
48
43
99
97
46
87
55
31
arch
AB-GAM AB-GAM GAM
AB-GAM GAM GAM AB-GAM
GAM
GAM
dim
16
8
32
8
att
I COMPLETE SHAPE GRAPHS IN BIKESHARE AND MIMIC2
We list all main effects of Bikeshare in Fig. g and top 16 interactions effects in Fig. We list all main effects of MIMIC2 in Fig. 10 and top 16 interactions effects in Fig.
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Published as a conference paper at ICLR 2022
Table 8: The best hyperparameters for NODE-GA'M architecture.
Dataset
Compas
Churn
Support2
Mimic2
Mimic3
Adult
Credit
Bikeshare
Wine
batch
2048
2048
256
256
512
256
512
2048
512
size
num
4
2
2
4
4
layers
num trees
1000
333
2000
2000
1000
2000
1000
125
1000
depth
2
6
6
6
6
6
6
6
addi
tree
2
2
0
2
0
1
dim
output dropout
0.2
0.1
0
0.2
0.2
0.2
colsample
0.2
0.5
0.2
0.5
0.2
0.5
0.5
bytree
Ir
0.005
0.005
0.01
0.005
0.01
0.01
0.01
0.01
0.01
12
0
0
le-5
0
0
0
lambda
add
last
1
0
0
0
1
0
linear
last dropout
0.2
0.2
0
0
0
0.3
0
seed
32
31
33
10
87
33
38
83
87
arch
GAM AB-GAM AB-GAM AB-GAM AB-GAM GAM AB-GAM
GAM
AB-GAM
dim
32
32
8
16
32
16
att
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Published as a conference paper at ICLR 2022
Table 9: The best hyperparameters for NODE architecture.
Dataset
Compas Churn Support2 Mimic2 Mimic3 Adult Credit Bikeshare Wine
batch
2048
2048
2048
2048
2048
2048
512
2048
2048
size
num
5
4
2
3
2
3
3
2
layers
num
100
125
1000
166
1000
1000
1333
333
500
trees
depth
2
2
6
4
6
4
4
addi
tree
1
0
0
0
0
1
1
1
dim
output
0
0.2
0.2
0.2
0.2
0.2
0.1
dropout
colsample! bytree
0.2
0.5
0.2
0.2
0.2
0.2
0.2
0.5
Ir
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.01
12
1e-5
1e-7
1e-6
1e-7
1e-7
1e-6
1e-5
lambda
add
last
0
0
0
linear
last
0
0
0
0
0
0.3
dropout
seed
3
26
93
17
93
93
82
49
73
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Published as a conference paper at ICLR 2022
Table 10: The best hyperparameters for NODE-GAM architecture for 6 large datasets.
Dataset
Click
Epsilon
Higgs
Microsoft
Yahoo
Year
batch
2048
2048
2048
2048
2048
2048
size
num 5
5
4
4
2
layers
num
800
400
200
125
500
500
trees
depth
4
4
4
6
4
2
addi
tree
2
2
2
0
dim
output dropout
0
0.1
0.1
0.2
0.1
colsample
le-5
0.1
0.5
0.1
0.1
0.5
bytree
Ir
5e-3
le-2
5e-3
5e-3
5e-3
le-2
12
lc-7
0
Le-5
0
le-6
le-6
lambda
add
last
0
0
0
linear
last dropout
0
0.1
0.1
0.2
0.1
seed
97
31
67
67
14
58
arch
AB-GAM AB-GAM AB-GAM AB-GAM AB-GAM AB-GAM
dim
32
16
8
8
16
att
22

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Published as a conference paper at ICLR 2022
Table 11: The best hyperparameters for NODE-GA architecture for 6 large datasets.
Dataset
Click
Epsilon
Higgs
Microsoft
Yahoo
Year
batch
2048
2048
2048
1024
2048
512
size
num
3
4
5
layers
nur trees
1333
2000
1000
500
800
800
depth
4
2
6
4
6
addi
tree
2
2
0
0
0
dim
output dropout
0.2
0.2
0.1
0.2
0.2
colsample
0.5
0.5
0.5
bytree
Ir
0.005
0.01
0.01
0.005
0.005
0.005
12
Ie-6
le-6
le-6
0
lc-6
lambda
add
last
1
1
linear
last dropout
0.15
0.3
0.15
0
0
seed
36
95
69
25
78
arch
AB-GAM AB-GAM AB-GAM AB-GAM AB-GAM AB-GAM
dim
32
32
8
32
16
att
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Published as a conference paper at ICLR 2022
-22
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Figure 8: importance (Imp).
The shape plots of all features (main effects) in Bikeshare. We also show the feature
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The shape plots of top 16 interactions in Bikeshare. We also show the feature importance (Imp).
Figure 9:
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Published as a conference paper at ICLR 2022
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Figure 10:
The shape plots of all features (main effects) in MIMIC2, We also show the feature importance (Imp).
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Published as a conference paper at ICLR 2022
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The shape plots of top 16 interactions in MIMIC2, We also show the feature importance (Imp).
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Reviews:
[{"review_id": "wWtFnQwf3w", "reviewer": "Reviewer_qoW7", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "Pros:\n- I believe that there is well backed motivation for work based off of the plentiful literature review.\n- There is novel integration of 2 methods previously not combined before\n- A variety of datasets seem to show that the proposed method is useful.\n- Code is available for reproducibility.\n\nCons: The only issues I have with this are things that could improve clarity.\n- If it is possible, I would love to see a full algorithm of NODE-GAM for clarity and understanding of each novel step\n- I would rather the the argument for the interpretability be moved to the introduction / literature review section rather than in the conclusion for flow. Then, the authors could summarize their work in the conclusion instead.", "summary_of_the_paper": "Generalized Additive Models (GAMs) are a class of interpretable models with a long history of use in these high-risk domains, but they lack desirable features of deep learning such as differentiability and scalability. \n\nThe authors propose a neural GAM (NODE-GAM) and neural GA2M (NODE-GA2M) that scale well and perform better than other GAMs on large datasets, while remaining interpretable.\n\nPopov et al. (2019) developed NODE that mimics an ensemble of decision trees, but suffers from lack of interpretability similarly to other ensemble and deep learning models. Agarwal et al. (2020) developed a Neural Additive Model (NAM) whose deep learning architecture is a GAM, but does not scale well.\n\nThe authors combine the above mentioned 2 approaches via a 3 step process:\n1. Instead of letting the ODT feature function be a weighted sum of features, they make it pick only one feature.\n2. Within each tree, they make a logit layer the same across all depths to remove interactions between features.\n3. Finally, they avoid the DenseNet connection between two trees that focus on different features.\n\nThe authors then output performance on 6 popular binary classification datasets (Churn, Support2, MIMIC2, MIMIC3, Income, and Credit) and 2 regression datasets (Wine and Bikeshare). Looking at the experiments, their NODE GA^2M version seems to perform similar to the compared state of the art.\n\nThe authors additionally perform an ablation experiment by testing their method's semi supervised learning capabilities, show that it is able to learn on mask data and be fine tuned on a limited labeled dataset.\n\nFinally, the conclusion has arguments to support the interpretability of GAMs and to defend their model's interpretability", "summary_of_the_review": "I would tend to accept this paper as it is novel enough and supported by empirical experiments.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "iRhwOqDznyb", "reviewer": "Reviewer_5LN4", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "For the design of the new architecture, the authors made three major changes to the NODE architectures. However, it was not very clear to me the considerations behind, e.g. why avoiding features interactions in a tree. It will be great if the authors can provide evidence, either theoretical or empirical, to support the suggested designs.", "summary_of_the_paper": "The authors proposed novel architectures for neural GAM and GA2M, which preserves the interpretability of GAM and leveraging the deep learning architectures for performance gains. The new method was assessed on 14 different dataset which covers a wide variety of prediction tasks. The performance of methods (NODE-GAM and NODE-GA2M) were comparable with the other GAM methods. Compared to other similar methods, the proposed method has a better scale ability, and better performance on larger datasets.", "summary_of_the_review": "The paper proposed an interesting architecture for interpretable deep learning models. However, based on the empirical results, the method does not seem to be superior or only marginally better than traditional methods such as the spline model.\n", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "GH2iXEne2h-", "reviewer": "Reviewer_L13g", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "The authors added a comparison to NAM without bagging and provided more context for the advantages of their approach relative to other works. As a result, I am increasing my score.\n\n----\n\nStrengths\n- The authors put a lot of thought into designing the architectures of NODE-GAM and NODE-GA$^2$M. I appreciated the clever tricks adopted, like the gating mechanism and the annealing to constrain to 1 feature, as well as the small extensions to NODE like adding attention and regularization.\n- The authors similarly put careful thought into extracting the interactions. I especially appreciate the application of the \"purification\" technique (Lengerich et al. 2020), which addresses an ambiguity problem that has bothered me about approaches like these in the past.\n\nWeaknesses\n- The biggest weakness for me is primarily some missing baselines that I would be interested in seeing. In reading this paper, I felt there was an \"elephant in the room\" question lurking in my mind about why a much simpler neural-network-based architecture, such as the \"univariate\" networks present in the \"Neural Interaction Detection\" (NID) paper for modeling main effects (Fig 2 in https://arxiv.org/pdf/1705.04977.pdf; not cited) was not applied. I believe this is similar to the approach taken in the NAM paper as well (https://arxiv.org/pdf/2004.13912.pdf; cited). Regarding the NAM paper, the authors write \"NAM requires training on tens to hundreds of neural networks and requires an extensive hyperparameter search...in addition, NAM's shape graphs are too smooth after bagging\". As a reader, having seen the \"univariate networks\" used in the NID paper (which, to my knowledge, were not trained with tens to hundreds of networks), I'm left wondering \"what happens if you use fewer neural networks and don't use bagging\"? I'm sure other difficulties arise in this case - perhaps the model used in the NID paper is also not good at modeling abrupt changes - but it would be helpful to the reader to showcase to what extent this is an issue in practice. I get the sense that the authors are familiar with a lot of not-well-documented limitations of other approaches that led them to develop the NODE-GAM architecture, and it would be helpful if these limitations were made explicit.\n- Still on the subject of prior work/baselines, I am interested to hear how this approach compares to the \"Neural Interaction Transparency\" (NIT) approach (https://proceedings.neurips.cc/paper/2018/hash/74378afe5e8b20910cf1f939e57f0480-Abstract.html) - it does not seem as though NIT is cited in this work. To quote from the abstract of NIT: \"NIT is also flexible and efficient; it can learn generalized additive models with maximum $K$-order interactions by training only $O(1)$ models\" - so it seems quite topical.\n\nMinor\n- There are a few very noticeable typos scattered around (e.g. \"Mathamatically\" at the bottom of page 2, \"distirbution\" at the top of page 5).\n- In equation 7, it looks like the attention weights $a_{\\hat{i}i}$ are not indexed by the layer $l$ - is this just missing notation? Presumably the attention weights vary depending on the layer?\n- I was a bit confused by the sentence \"To allow two-way interactions, for each tree we introduce two logits $\\boldsymbol{F}^1$ and $\\boldsymbol{F}^2$ instead of just one, and let $\\boldsymbol{F}^c = \\boldsymbol{F}^{\\lfloor c/2 \\rfloor }$ for $c > 2$\". Maybe the authors meant to write $\\boldsymbol{F}^c = \\boldsymbol{F}^{(c \\mod 2) + 1}$ so that $\\boldsymbol{F}^c$ alternates between $\\boldsymbol{F}^1$ and $\\boldsymbol{F}^2$? $\\lfloor c/2 \\rfloor$ can take on values beyond 1 and 2.\n- In the section on regularization, the authors mention that adding a constant corresponding to the log of the class imbalance is needed for the $l_2$ penalty to work because \"$l_2$ induces the model output to 0\". Wouldn't it be easier to solve this by adding an (unregularized) learnable bias term to the output?\n", "summary_of_the_paper": "\n- The authors develop the NODE-GAM and NODE-GA$^2$M architectures by modifying the previously-developed NODE architecture with constraints and a gating mechanism to ensure that, both within each tree and across layers, the model is only allowed to learn feature interactions of order 1 (for NODE-GAM) or 2 for (NODE-GA$^2$M).\n- They show that their proposed NODE-GAM and NODE-GA$^2$M architectures perform comparably to the baselines of Explainable Boosting Machines, tree-based GAM/GA$^2$M and the traditional spline-based GAMs on medium-sized datasets while outperforming them on larger datasets. They also show that NODE-GAM can benefit from self-supervised learning by pretraining a model to reconstruct the original input from a masked input.\n- They apply NODE-GAM and NODE-GA$^2$M to real-world datasets and showcase the patterns uncovered.", "summary_of_the_review": "Overall, I think a lot of careful thought went into this paper and the design of the NODE-GAM and NODE-GA$^2$M architectures is clever enough to be inspiring in its own right. However, the missing comparisons, particularly to the \"univariate networks\" for modeling main effects from the \"Neural Interaction Detection\" paper (which is the first thing that tends to come to mind when I think of a \"neural network based GAM\"), leave me hesitant to recommend the paper outright. I hence currently rate it as only marginally below the acceptance threshold, but expect that can easily change on revision.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper proposes two new generalized additive models (GAM) based on neural networks and referred to as NODE-GAM and NODE-GA2M. An empirical analysis shows that the proposed and carefully designed architectures perform comparably to several baselines on medium-sized datasets while outperforming them on larger datasets. Moreover, it is shown that the differentiability of the proposed models allows them to benefit from self-supervised learning. 

Reviewers agreed on the technical significance and novelty of the proposed models and valued the clever design of the new architectures. Most concerns and open questions could be answered in the rebuttal and by changes in the revised manuscript. Based one the suggestions of one reviewer new experiments comparing the proposed models to NAM were added, which improved the paper further.  The paper should be accepted.
Decision: Accept (Spotlight)