Generalizing Neural Additive Models via Statistical Multimodal Analysis
Authors: Young Kyung Kim
Juan Matias Di Martino
Guillermo Sapiro
Venue: Accepted by TMLR ()

Abstract: Interpretable models are gaining increasing attention in the machine learning community, and significant
progress is being made to develop simple, interpretable, yet powerful deep learning approaches.
Generalized Additive Models (GAM) and Neural Additive Models (NAM) are prime examples. Despite these
methods' great potential and popularity in critical applications, e.g., medical applications, they fail to
generalize to distributions with more than one mode (multimodal\footnote{In this paper, multimodal refers to the context of distributions, wherein a distribution possesses more than one mode.}). The main reason behind this limitation is that these "all-fit-one"
models collapse multiple relationships by being forced to fit the data unimodally. We address this critical
limitation by proposing interpretable multimodal network frameworks capable of learning a Mixture of Neural
Additive Models (MNAM). The proposed MNAM learns relationships between input features and outputs
in a multimodal fashion and assigns a probability to each mode. The proposed method shares similarities with Mixture Density Networks (MDN) while keeping the interpretability that characterizes GAM and NAM. We
demonstrate how the proposed MNAM balances between rich representations and interpretability with
numerous empirical observations and pedagogical studies. We present and discuss different training alternatives
and provided extensive practical evaluation to assess the proposed framework. The code is available at \href{https://github.com/youngkyungkim93/MNAM}{https://github.com/youngkyungkim93/MNAM}.

Published in Transactions on Machine Learning Research (02/2024)
Generalizing Neural Additive Models via Statistical Multi- modal Analysis
Young Kyung Kim
yk206@duke.edu
Deportment of Electrical and Computer Engineering
Dake University
J. Matías Di Martino
matias. di mawtinoo@duuuk.cc. edu
Department of Electrical and Computer Engineering Duke University
Guillermo Sapiro
gulllermo..spiiroddaaeee. edu
Department of Electrical and Computer Engineering
Duke University
Reviewed on OpenReview: /ttps://openneee opw.reveew....eee.. net/ forum? id- algg81/EEba
Abstract
Interpretable models are gaining increasing attention in the machine learning community, and significant progress is being made to develop simple, interpretable, yet powerful deep learning approaches. Generalized Additive Models (GAM) and Neural Additive Models (NAM) are prime examples. Despite these methods' great potential and popularity in crit- ical applications, c.g., medical applications, they fail to generalize to distributions with more than one mode (multimodal)). The main reason behind this limitation is that these "all-fit-one" models collapse multiple relationships by being forced to the data uni- modally. We address this critical limitation by proposing interpretable multimodal net- work frameworks capable of learning Mixture of Neural Additive Models (MNAM). The proposed MNAM learns relationships between input features and outputs in multimodal fashion and assigns a probability to each mode. The proposed method shares similari- lies with Mixture Density Networks (MDN) while keeping the interpretability that char- acterizes GAM and NAM. We demonstrate how the proposed MNAM balances between rich representations and interpretability with numerous empirical observations and peda- gogical studies. We present and discuss different training alternatives and provided ex- tensive practical evaluation to assess the proposed framework. The code is available at MNAM
1
Introduction
Deep neural networks (DNN) achieve extraordinary results across several important applications such object detection (Redmon et al. 2016 Girshick et al. 2014 Ren et al. 2015), object classification (He et al. Krizhevsky et al. 2017 Dosovitskiy et al. 2020), and natural language processing (Mikolov et al, Devlin al.) 2018 Brown et al. 2020). Yet DNN's popularity is still low in critical applications where cdiss-llassificationn has high consequences or transparency is required for Decision-makinne e.g., to prevent unfairness toward certain groups; examples are medical-related risk estimation and machine learning (ML) based public policies. According to experts in these domains, one of the main factors limiting the adoption of based approaches is the lack of interpretability and trustworthiness associated with these algorithms
2016
2013
Shorten et al.) 2021 Amarasinghe et al. 2020 Li et al. 2022). Even though several techniques have been
In this paper, multimodal refers the of distributions, wherein distribution possesses than one mode.

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proposed to increase the understanding of DNN (Agarwal et al., 2021 Ribeiro et al. 2016 Pedapati et al, 2020), medical professionals and policymakers still prefer simple models for which they can understand directly the factors that lead to a particular prediction. On the opposite end of DNN are algorithms such as linear regression and its multiple variants (Montgomery et al.) 2021). which are simple and interpretable but lack the flexibility and high performance that DNNs have. Notably, linear models can not capture nonlinear relationships nor exploit numerous novel tools that efficiently optimize modern DNN approaches. A recent approach proposed by Agarwal et al., named Neural Additive Models (NAM), generalizes Additive Models (GAM) and achieves interesting balance between interpretability and learning power. Individual features undergo nonlinear transformations independently, and these transformed features are merged in a regression-like paradigm, allowing the user to understand the weight of each factor leading to the prediction. This enables the algorithm to learn non-trivial relationships between the fealures and the larget outcomes while leveraging powerful state-of-the-an optimization tools developed for deep learning.
Although most of the rescarch on GAM has focused on minimizing the trade-offs between accuracy and interpretability (Nori et. al. 2019 Zuur 2012 Agarwal et al. 2021), addressing the lack of power for GAM and NAM in capturing multimodal relationships between input and target variables has been rare nonexistent. This limitation is crucial especially when datasct has multiple distinctive relationships between features and outputs. For example, imaging in the context of medical application where we are predicting the glucose level y using electronic health records (EHR) as input variables I1, .... Ini let us assume there are two subpopulations identified by the variable d E {0, 1}, which can be observed or a latent feature. For both cases, NAM would fail to capture a relationship in which y is positively correlated with (æ1ld= = 0) but is uncorrelated with (dy = 1). This is due to NAM only learning one deterministic relationship between inputs and outputs. When d is latent variable, NAM will fail to diffcrentiate them and collapse two relationships into one average deterministic relationship. Even if d is an observed feature, NAM will fail to differentiate them as a DNN assigned for X1 does not take d as an input to have information on two subpopulations.
To address this while preserving the virtues of NAM, we propose a probabilistic Mixture of Neural Additive Models (MNAM). The main idea is apply mixture density networks (MDN), a neural network with mixture of le Gaussian distributions as an outcome, as a linking function for GAM to model the relationship between inputs and outcomes in a multimodal fashion and associating a probability to each mode. The probability of each mode enables the model to be flexible in representing multiple subpopulations as MNAM is able to activate accurate relationships for certain subpopulations by increasing their probability.
Figure I illustrates the power and flexibility of MNAM. These strengths are also illustrated in Section 3 through applying MNAM on real datasets. Such flexibility will be especially crucial in decision-making with high consequences. For example, for analyzing the side effects of medicine, 99% of participants might have steady glucose levels but 1% might have high and dangerous glucose levels after taking medicine. NAM will collapse both levels into one indicating no side effects average, but MNAM will accurately show, with probability, two glucose levels of different subpopulations.
It is important to highlight the interpretability of the model. Similar to NAM having one fixed relationship between input features and output variables, MNAM will have fixed multiple relationships, which makes the model interpretable. Only the probability of cach mode will change from the change in other features, which indicates changes in a subpopulation.
Our main contributions are: (i) we identify the overlooked limitation that GAM and NAM have when they are trained with datasct that has multiple subpopulations: (ii) we provide a practical alternative to solve the critical problem of one-fits--ll"" standard in interpretable DL approaches; (iii) we propose a model called MNAM that could learn multiple relationships among subpopulations for the solution; (iv) we propose a method to train MNAM, with objectives to learn multiple relationships and activate one or more matching relationships for a given subpopulation; and (v) we demonstrate MNAM is more expressive in accuracy and flexible in interpretability compared to NAM. We describe the proposed method in Section 2 Section 3 presents empirical evidence and pedagogical studies, showing strengths of MNAM. We discuss related work in Section and limitations in Section 5 Finally, we provide conclusions in Section 0
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Linear Regression
NAM
MNAM
10
Linear
W
Nonlinear
Bimodal Nonlinear
10
10
10
10
Figure Linear regression, NAM, and MNAM on linear, nonlinear, and bimodal data. The left column illustrates the input for three datasets. The columns illustrate the representations learned by linear regres- sion, NAM, and MNAM, respectively. As expected, linear regression fails to learn datasets with nonlinear relationships. NAM fails to learn datasets with relationships that have more than modality, and only MNAM is able to learn nonlinear and multimodal relationships.
2 Method
2.1 Architecture
MNAM produces outcome that is mixture of k Gaussian distributions. These distributions can be represented as "i). distribution with mean μ4 and standard deviation o,, while म; denotes the associated probability. The pur- pose of utilizing a mixture Gaussian distributions, a universal approximator for any density distribution, as the outcome for the model, to represent multiple relationships between inputs and outcomes. For rep- resentation, one more Gaussian distributions are assigned to each input-output relationship, and MNAM activates specific relationships for given subpopulations of the input by increasing the probability of the ap- propriate Gaussian distributions. This property is significant because it allows to successfully capture and represent modes for relationships within various subpopulations in the datasct. Importantly, this approach does not require prior knowledge of the number of modes, as long as the model has a sufficiently large value of relative to the number of modalities. In contrast, models like GAM and NAM are unable to accurately represent multimodal relationships since they only provide single estimated outcome per representation. In Section 3.3 we provide a demonstration of this property and also compare the representation of multimodal relationships between MNAM and NAM through a pedagogical example.
....,,, TR). Here, NN of) represents a standard Gaussian
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Similar to NAM, MNAM predictions are built from linear combination of embeddings Zi of each imput feature X; mapped through neural network. In contrast with NAM, MNAM embedding consists of pa- rameters for & Gaussian distributions and a latent variable for predicting the probability of the mixture of I Gaussian distribution models (NG,1(15.1, o2,))...NNAA(5,,,. 2, 2ÿ). 2.)) The left index j is a reference to one of the input features and the right index of the Gaussian distributions is a reference to the number of components for the mixture. As shown in Equation [ we compute the mean and variance of the Gaussian distributions for the MNAM outcome by linearly combining the mean and variance of matching components for Gaussian distributions of features' embedding.
Ni L.d) +... + NN,,lunn m
(1)
-NO
Piá Σ (..) -Ni(ue, o?
The linear property is crucial for the interpretability of additive models. In the case of GAM and NAM, where the output is a lincar combination of non-lincarly transformod features, the models are interpretable because one can observe how changes in a feature impact the outcome. Therefore, the mixture of Gaussians as the outcome has practical advantages. By leveraging the linear property of a mixture of Gaussians, MNAM can capture the magnitude of changes in the overall mean and uncertainty of predictions in response to variations in a feature.
Latent variables for predicting the probability of the mixture of & Gaussian distributions for all features' embeddings will be the input for a separate neural network that predicts the mode associated with the output. This neural network will learn to identify which subpopulation is being represented based on input from all features, and activate the correct relationships by assigning a high probability to the matching Gaussian distributions. The description of how MNAM computes predictions summarizod in Algorithm and the comparison of the architecture for NAM and MNAM is illustrated in Figure 2
Algorithm Mixture Neural Additive Models
Input: Data: (X1,,. Number of Features: n, Number of Gaussian Distributions: k Neural Networks for Feature Transformation: (fi..., Neural Network for Probability: 9
Output: Mixture of Gaussian Distributions: (5),,, ( Tl,..., for i= to n do (1 "ii,,,,,)).,,,........ N..,,,N, o2 = (X,,)
end for for
i= to k do
#4 > 1== Mjs
o, =²= ンドー 1 Jz
end for
#1, Th (Z7 ; ....y Z
2.2 Training and Optimization
As mentioned in Section 1 statc-of-the-art optimization tools for deep learning are applicable for training MNAM. For this work, we used Adam (Kingma & Ba 2014) with a learning rate decreasing by 0.5% for each epoch. The objective of the training and optimization of MNAM is to assign one more Gaussian distributions to each relationship in the dataset. Another objective is to learn to identify subpopulations from the given features to activate the correct relationship associated with the given sample. We devise hard-thresholding (HT) and soft-thresholding (ST) algorithms for the given objectives. The HT algorithm trains or updates a single mode or a Gaussian distribution with a minimum loss, while the ST algorithm updates all l modes with weights computed by the likelihood of each mode on the label. Our optimization objective shares similarities with the work proposed in the context of Mixture Density Networks (MDN) [Bishop 1994). Bishop discussed probabilistic formulations for deep neural networks that use mixture of
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NAM
MNAM
(a)
(c)
Zu (M ofg)
Zn (112. 0z2)
4
X,
Li (12))
(b)
(d)
(e)
X
4
X,
PLi
A
8
HAI
X
4
(12211
X2
P12
+
4
Map of
PiA
PLA
8
X,
X,
(gu o
(g)
M outcome
(A, R..A)
(A. (g.)
(8 (.
Figure 2: Illustrative schemes of NAM and MNAM network architectures. As shown in (a) and (b), NAM independently maps features into embedding through neural networks and then linearly combines embeddings for a prediction. Similar to NAM, MNAM independently maps features into embeddings through neural networks. The difference is that embedding consists of & Gaussian distributions and a latent variable for predicting probabilities for a mixture of the l Gaussian distributions, which illustrated in (c) and (d). (c) illustrates linear combinations of each component of the Gaussian distributions for all features' embeddings. (I) depicts the mapping of latent. variables for a mixture of & Gaussian distributions (ZT, TT,72,,., Z5) into probabilities for the mixture of & Gaussian distributions through neural network. (g) is example of the outcome for MNAM, which is the mixture of & Gaussian distributions.
Gaussians for outcomes and provided alternative optimization methods. MDN is optimized by maximizing the log- likelloooood of the data with respect to their mixture of Gaussian outputs,
log(
m x Ni(Y Mii of)).
(2)
In contrast to MDN, we optimize two terms, one that focuses on fitting one of the modes to a particular sample and the other that learns to predict the mode to which the instance belongs (this is described next in Section 3.2 Moreover, despite the similarities between our training objective and MDN log-likelihood, we observed that our proposed method is significantly more stable, easier to train in practice and solves some of MDN's s known and reported issues (Choi et. al. 2018 Makansi et al. 2019). We expand on this in the coming sections.
Among the three algorithms discussed next, we chose the HT algorithm since it is more numerically stable and performs better in likelihood metrics; this is demonstrated in the empirical experiment in Section 3.2
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2.2.1 Sonr--hhrssolding (HT) Algorithm
In order to address the two training objectives, the algorithm calculates two separate losses. For the objective of assigning one or more Gaussian distributions to each relationship, the algorithm computes the Gaussian negative log-likelihood (GNLL) loss for each Gaussian distribution of the outcome against a label. Among k losses, only the minimum factor will be used compute the total loss, which means only weights used to compute the minimum loss updated via the backpropagation. This enables the model to assign one more Gaussian distributions to learn each relationship. For the second objective, which involves learning to identify subpopulations, the algorithm computes the cross-cntropy loss between the probabilities of a mixture of Gaussian distributions for a prediction and the index number of the Gaussian distribution with the minimum loss. This loss measures how well MNAM activates the corresponding Gaussian distribution for the input and enables the model to learn to identify subpopulations for a given imput to increase the probability of the correct Gaussian distribution for representation. Algorithm 2 summarizes the proposed training algorithm. It is important to highlight that the proposed learning method is unsupervised, in the sense that the data subgroups do not need to be known or defined in advance.
Algorithm Hard-TThreshole (HT) Algorithm
Input: Data: Y), MNAM: f. GNLL loss: g, Cross-entropy loss function: h, Rate for cross-entropy
loss: À
Ni Nu(H,, ox), 11 #I,... Th = f(X)
min loss 0
for i= k do
gau loss (Ni oi), Y)
if min_loss gau loss then min mos_loss gau loss min__nnerr = end if end for
prob_loss loss h(( Tk), min mii_mmiidernr inder) total loss Min loss X prob loss
2.3 Soft- Thttshrreshlling Algorithm
Similar to the EM algorithm [Dempster et al. 1977 the ST algorithm has expectation and maximization steps for training. In the expectation step, we compute the posterior probability of subpopulations P(Z k|X, Y). As shown in Equation 3 we compute the posterior probability as,
k)P = P(X,Y) P(XY = k)P(Z = =k)
P(Z = ,X,,)
(3)
EL P(X, YI
where P(X,Y Y k) the likelihood of 4th Gaussian distribution for the given input, and P(Z k) is the prior probability of subpopulation predicted from MNAM. In the maximization step, update the weights of MNAM to maximize posterior probability of the subpopulations. First, compute GNLL losses for all Gaussian distributions, and then GNLL losses for all the Gaussian distributions are lincarly combined with weights matching their posterior probabilities from the expectation step. This ensures weights used to compute Gaussian distribution with a higher likelihood are updated more. Cross-entropy loss between the prior probability prodicted from MNAM and the posterior probability computed the expectation step is computed with a similar purpose in the HT algorithm. Algorithm summarizes the proposed training algorithm.

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Algorithm 3 Soft- Toft-Thresooling (ST) Algorithm
Input: Data (X, Y), MNAM f, GNLL loss Crossentropy loss function h, Rate for cross- entropy loss
oI). No(us ok), Ti..., "I., f(X)
for i= & do
gau loss; ((Ni(Hi, oi),,
gau like; p(Y ;44,
end for
mar prob = Σー= gou likey J
$1..., R2 gau likei T1 gau like Tik
mar prob ... mar prob
gau loss Σi=1 gav loss: 4 prob loss h((F1,..., Tk), (f1,,.., fx)) lotal loss you loss + À prob loss
2.4 Regularization
Similar to NAM, all regularization methods for deep learning be applied to MNAM, including weight decay, dropout, and output penalty. For this study, utilized weight decay and output penalty.
3
Experiments
3.1
Empirical Observations
3.1.1
Datasets
We evaluate six datasets: the California Housing (CA Housing) (Pace & Barry 1997). the Fair Isaac Cor- poration (FICO) (FICO) 2018), the New York Citi Bike (BIKE) (Vanschoren al. 2013), the Medical Information Mart for Intensive Care (MIMIC-III) JJohnson et al. 2016), the US Census data Income (ACS Income) for California (Ding et al. 2021), and the US Census data on Travel time (ACS Travel) for California Ding et al. 2021].
The CA Housing dataset has the task predicting housing prices and it consists of eight features.
FICO: The FICO datasct has the task of predicting crodit scores and it consists of 24 fcatures.
BIKE: The BIKE dataset has the task of predicting the duration of trips and it consists of four features.
Due to limited computation resources, we dropped data points that had more than 4000 seconds of duration for bike trip and sampled 25% of the remaining datasct for analysis.
MIMIC-III MIMIC-IIT dataset has the task of predicting the length of hospitalization for patients and it consists of various static and dynamic features. For comparing NAM and MNAM, we have used only static fcatures, which consist of seven fcatures.
ACS Income: The ACS Income dataset. for California has the task of predicting income and it consists of ten features.
ACS Travel: The ACS Travel dataset for California has the task of predicting travel time to work and it consists of 16 features.
3.1.2 Training and Evaluation
Similar to how the original paper trained NAM, we used Bayesian optimization (Močkus 1975) to finetune variables to train NAM and MNAM. Learning rate, weight decay, and output penalty are finetuned for NAM. Learning rate, weight decay, output penalty, number of Gaussian distributions, and lambda for the cross-entropy loss are finetuned for MNAM. For both models, utilized early stopping to reduce overfitting.
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Optimized parameters from Bayesian optimization can be found in the table from Appendix A We used 5-fold cross-validatio for CA Housing, FICO, and MIMIC-III datasets, and a 3-fold cross-validation for BIKE, ACS Income, and ACS Travel datasets. For evaluation, we trained 20 different models by randomly splitting the train set into train and validation sets for each fold. We ensembled 20 models to evaluate on the test set. We calculated confidence intervals for each model, based on the mean and variance from the n-fold cross-validation.
To compare between deterministic and probabilistic models, we use the mean absolute error (MAE). How- ever, the MAE fails to account for the uncertainty in predictions made by probabilistic models. Even if a probabilistic model accurately predicts true distribution for the label distribution, it may still receive the same MAE score as a deterministic model if it is correct in predicting the mean of the label distribution. To address this limitation, we transformed a deterministic model into a probabilistic one, fitting a global variance, which is determined by computing the mean of the variances between the actual labels and the predicted labels from the training set. We then utilized mean likelihood (LL) as a metric for comparision, to emphasize the importance of having multimodal compared to unimodal distribution have an extensive comparison between unimodal and multimodal models, also trained NAM with tuned variance, which we will refer to as probabilistic Neural Additive Models (pNAM). We trained pNAM by setting k = 1 within the MNAM framework. Theoretically, pNAM offers enhanced flexibility compared to standard NAM approaches utilizing a globally fitted variance, as it can effectively represent data, exhibiting heterogeneity of variances.
an outcome. To
3.1.3 Results
Table I displays the MAE and LL. scores of NAM, PNAM, and MNAM on datasets described above. MNAM consistently exhibited similar or superior MAE scores compared to other models across all six datasets. Moreover, MNAM showcased a significantly improved performance in terms of likelihood scores when com- pared other models for all datasets, except for the FICO dataset. Notably, the optimized number of Gaussian distributions for MNAM was 1, which means that pNAM and MNAM are identical models. This finding underscores MNAM's remarkable ability to effectively learn the output distribution, surpassing both NAM and PNAM in this aspect.
Differences in performance between MNAM and NAM models differ greatly by datasets. Specifically, the discrepancy in likelihood scores between NAM models and MNAM is much more pronounced for the CA Housing dataset compared to the ACS Income dataset. Several explanations could account for this obser- vation. Firstly, the CA Housing dataset might exhibit more intricate interaction relationships among its features, rendering it more challenging for NAM to accurately capture the underlying patterns without any interaction term learning. Conversely, MNAM, with its enhanced capability to model complex interactions, would demonstrate an improved likelihood score on such datasets. Secondly, the CA Housing dataset might possess modes that differ more significantly from one another, making it harder to fit using a single Gaussian distribution for NAM. In this scenario, MNAM would enhance the likelihood score by accommodating the complexity of interaction relationships and the differences among modes within the data.
--NAM
PNAM
MNAM
DATASET
MAE
LLt
MAE
LLt
MAE/
LLt
CA HOUSING
0.48+ 9e 005
0.43 19c -08
0.48 =3e 00
0.58 16e -04
U.46 + 4e 00
0.73 + 0.001
FICO
2.7 1 0.002
0.073 18c-U0
2.7 10.002
0.084 1 200
2.7 1 0.002
0.084 1 2c 20-00
MIMIC
1.5.t 0.0002
0.15 -0c-0
1.5 =0.0001
0.15
-3e-
1.5:: 0.0003
0.25 - 6c -0-66
BIKE
3.4 1 0.0005
0.064 -0
3.4 O. 00008
0.069 -0
3.4 0.0006
0.092 =1c 006
ACS INCOME
37.2 + 0.003
0.0049
:2c -0
35.7 + u.07
0.011 14 -44-07 007
35.7 + 0.02
0.013 + 4e 007
ACS TRAVEL
15.6. 0.0004
0.013 -10
15.5 10.0008
0.017 122-
15.5 1 0.002
0.036 1 2c -05
Table 1: MAF and L.L scores for NAM, PNAM, and MNAM on CA Housing, FICO, MIMIC BIKE, ACS Income, and ACS Travel datasets. We highlighted best and second-best. performances for each metric.
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3.1.4 Interpretability
In this section, visualize the relationships between features and labels, and how different relationships are activated from changes in subpopulations; we illustrate this for the CA Housing dataset. This illustrates the strength of the interpretability of MNAM. Relationships plots for other datasets can be found in Appendix C As illustrated in Figure 3 MNAM is able to learn and represent multiple relationships between features and labels, which NAM fails to do as collapses those relationships into a mean. Therefore, MNAM is more flexible in explaining and representing multiple relationships between features and labels by activating one or multiple of them. Another example of such visualization for CA Housing dataset is presented in Appendix B where each relationship is plotted in separate subplots to enhance readability.
First mode
Housing Price
6094
3.54
Second mode
429
150
3204
Median Income
Median House Ago
Average Number Rooms
Average Number Bedrooms
143g
1.5
Housing Price
120
3.54
-30
Population
Average Number Household
Latitude
Longitude
Figure 3: Two relationships between features and label learned by MNAM on CA Housing dataset. The solid lines represent the mean of these relationships, with highlighted regions indicating their uncertainties. By examining the solid lines and highlighted areas, users can analyze the changes in the relationship and the associated uncertainties between features and labels. Additionally, these subplots facilitate easy comparison, allowing users to discern whether the two relationships differ significantly.
Allowing multimodal data representations sheds light on non-trivial data relationships that are otherwise hidden average "one-fit-all" models. For example, as illustrated in Figure 4 we identified that the price of house could increase decrease the number of people in the neighborhood increases (the first column of Figure 4 illustrates the two modes recognized by MNAM). If we group the algorithm's output by median income (the first row of the second column represents the bottom one percent, and the first row of the third column represents the top one percent), we can recognize that one of the modalities is associated with higher income households and the other with lower income households. For example, the first row of the second column shows that the top mode is activated more frequently on this subgroup (darker blue represents higher frequency), suggesting that the larger the number the people in the neighborhood, the higher the house prices. The opposite can be recognized for the higher-income subgroups (see the first row of the third column). In other words, the output of the model suggests that for wealthier neighborhoods, the more people, the less expensive houses are, while the opposite occurs in poor communities. A similar story is illustrated when we group the algorithm's output by proximity to the beach (the second row of the second column represents inland, and the second row the third column represents the area near the beach). The output of the model suggests that for areas near beaches, the more people the more expensive houses are, while the opposite occurs inland. Notice how these rich data interpretations would have been missed using NAM, where "one-fit-all model optimized.
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The strength of MNAM in identifying unfairness is more evident when trained the MIMIC dataset. In Figure 5 the variance or the discrimination of the length of stay among different ethnicities significantly differ between the two modes recognized by MNAM (left graph of Figure 5). The first relationship, which is a red line, has more variance or discrimination among ethnicitics compared to the second relationship, which is a blue line, the length of stay. If we group the algorithm''s 's output by admission type (the middle graph represents common admission and the right graph represents urgent admission). we recognize the model activates more on relationships with less discrimination among ethnicities with urgent admission and vice versa with common admission. In other words, there is more variance and discrimination in the length of stay among ethnicities for common admission compared to urgent admission. Overall, these findings highlight the capabilities of MNAM compared to NAM in identifying potential unfairness of a model as NAM will provide the same relationship for all subpopulations.
Bottom One Percent
Top One Percent
Median Income
Median Income
First mode
1M
1M
Varlance of first mode
Housing Price
0.8M
0.8M
Second mode
0.6M
0.6M
Variance of second mode
0.4M
0.4M
0.2M
0.8
1M
0
Housing Price
0.8M
-0.2M
-0.2M
0.6M
-0.4M
-0.4M
0.4M
10K
20k
30k
10k
20k
30k
0.8
Population
Population
Inland
Near Beach
0.4M
1M
1M
10k
20k
30k
Housing Price
0.8M
0.8M
Population
0.6M
0.4M
0.4M
0.2
0.2M
0.2M
-0.2M
0.2M
10k
20k
30k
10k
20k
30k
Population
Population
Figure 4: Line graph and heatmaps on the relationship between housing price and population for CA Housing. The first column, a line graph, represents two modes recognized by MNAM between housing prices and populations. Second and third columns, heatmaps, represents changes in the activation two modes from changes features interest. The first row represents changes in median income and the second row represents changes in proximity to the beach. Except for each row's feature of interest, all other remaining features have been fixed to their mean values. The magnitude of the mode's activation is illustrated through the intensity of the color in the heatmaps. Darker blue represents higher activation of mode. The blue color bar represents the magnitude of mode's activation.
3.1.5 Training Efficiency
Table 2 shows the comparison of average training time, training time per epoch, and the number of epochs required to train NAM and MNAM on different datasets. As expected, MNAM takes longer to train per epoch than NAM because has an additional neural network for computing mode weights. However, interestingly, MNAM was faster than NAM training for half of the datascts, since MNAM required, several, fewer epochs than NAM. Our hypothesis is that MNAM's assignment of modes to subpopulations effectively shrinks the space for the model to explore, resulting in fewer epochs needed for training,
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First Relationship
Common Admission
Urgent Admission
Second Relationship
0.15
0.15
0.15
HH
Length of Stay
0.1
0.1
0.1
C.8
0.05
0.05
0.05
0
-0.05
-0.05
-0.05
-0.1
-0.1
-0.1
-0.15
-0.15
-0.15
asian
black
hispanic
multi
native
other
unknown
white
asian
black
hispanic
multi
native
other
unknown
white
asian
black
hispanic
multi
other native
unknown
white
Ethnicity
Ethnicity
Ethnicity
Figure Line graphs of the relationship between the length stay and the ethnicity for MIMIC with whisker representing a variance. The left graph represents two modes recognized by MNAM between the length of stay and the ethnicity. The middle and right graphs represent changes in the activation of modes from changes in admission type. Except. for admission type, all other remaining features have been fixed to mean values. The magnitude of the mode's activation is illustrated through the intensity of color and thickness of lines. The darker blue and thicker line represents higher activation of a mode. The blue color bar and different. thickness of lines on the right side of the color bar represents the magnitude of a mode's activation.
each mode only needs to represent one subpopulation. In contrast, NAM has only one outcome that must represent multiple subpopulations, causing it to oscillate among subpopulations for representation during training. Overall, MNAM may be more efficient than NAM in terms of training time, despite having more parameters to compute.
TRAINING TIME (SECONDS)L
TRAINING TIME
NUMBER EPOCHS
PER EPOCH (SECONDS)L
DATASET
NAM
MNAM
NAM
MNAM
NAM
MNAM
CA HOUSING
636.38
411.73
0.72
0.84
885.36
489.85
FICO
326.26
337.54
0.78
0.85
417.34
397.67
MIMIC
173.10
278.58
0.93
1.09
186.64
256.53
BIKE
410.42
477.62
1.37
1.85
298.76
258.36
ACS INCOME
1460.31
896.41
3.22
4.47
453.65
200.40
ACS TRAVEL
2411.51
946.55
3.86
5.75
624.43
164.58
Table 2: Comparisons of average training time in seconds, training time per epoch in seconds, and number of epochs between MNAM and NAM. We highlighted the best performance for each metric.
3.2 Comparison of Training Algorithms
In this section, evaluated HT, ST, and learning algorithm originally proposed by Bishop. This original algorithm optimizes the model to maximize the log- ligelioel of mixture of Gaussians distribution. We will refer to Bishop's algorithm as the original (ORG) algorithm. We asscssed their performance based on numerical stability (NS), number of epochs per training (NE), and accuracy. Employing the dataset from the pedagogic study, we conducted 20 rounds of training using various learning rates to evaluate these metrics. NS was determined by calculating the percentage successful training scssions without encountering gradient divergence. Accuracy was measured through MAE and LL computed on the test set. The evaluation results are presented in Table 3
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The HT algorithm exhibited superior performance in terms of NS and LL. One possible explanation for its better NS performance is that the HT algorithm exclusively considers the minimum GNLL loss term, whereas the ST and ORG algorithms incorporate all GNLL losses with associated weights for updates. These latter algorithms pass more gradients compared to the HT algorithm, introducing numerical instability during training, particularly the initial slages when the loss is higher due to random weight initialization. Thus, the HT algorithm demonstrated successful training across all learning rates, while the ST and ORG algorithms struggled to converge higher learning rates. This numerical instability in the ORG algorithm has been documented across various studies (Choi et al. 2018 Makansi et al. 2019). Furthermore, researchers have noted instances of mode collapse, which is also reflected in Table 3 While the ORG algorithm shows competence in predicting means, it sometimes struggles to effectively represent the multimodal distribution, potentially due to modes collapsing around the mean. In contrast, the HT algorithm altains the highest LL score, as is able to accurately assign modes. Given that MNAM operates as a probabilistic model and the HT algorithm offers superior numerical stability, we chose the HT algorithm over the others.
HT ALGORITHM
ST ALGORITHM
ORG ALGORITHM
LR
NSf
NET
MAET
LLt
NS1
NET
MAEL
LL
NS1
NEL
MAET
LL1
0.1
100%
191.1
681.26
0.024
0%
0%
0.05
100%
226.4
10.38
0.045
0%
0%
0.01
100%
343.0
3.58
0.66
0%
35%
230.8
3.02
0.31
0.005
100%
357.6
3.14
0.76
5%
145.0
3.74
0.015
95%
248.6
3.11
0.30
0.001
100%
410.6
3.18
0.87
100%
245.3
2.95
0.50
100%
243.7
3.08
0.36
0.0005
100%
503.4
3.19
0.89
100%
362.3
2.94
0.66
100%
247.7
3.07
0.36
0.0001
100%
418.3
3.12
0.76
100%
389.8
3.25
0.74
100%
252.9
3.10
0.35
5e" 005
100%
394.4
3.34
0.61
100%
342.8
3.36
0.57
100%
269.0
3.17
0.29
Table 3: Comparision of Hard-Thresholdingg Soft- Toft-hhresholdng, Original algorithm on data from pedagogic study
3.3 Pedagogical Example
For pedagogical value and further illustrate the differences between the original NAM and the proposed MNAM, we created synthetic dataset with different subpopulations, which differentiated by either observed or latent variables. NAM has limitations in accurately representing such dataset it collapses four relationships between X1 and Y into one deterministic relationship by averaging them. When X2 is an observed variable, NAM is not able to differentiate relationships, since a neural network assigned to X does not take X₂ as input. The neural network for X2 simply uses the average relationship for representation, which shown when X2 = 0 and X₂ 1. The representation differentiate subpopulations are latent variables, which is the case for X₂ = 2 and X₂ 3 in the synthetic datasct. NAM trics to represent multiple relationships with one relationship,
worsened for NAM when variables that
shown in the second column of Figure D MNAM overcomes such limitations as it is able to learn four relationships and activate the right relationships for each subpopulation. Another strength of MNAM is that as long as k is larger than the number of relationships in a dataset, MNAM will be able to represent the relationships accurately. In other words, tuning is not critical, as long as its value is higher than the expected number of modes. Furthermore, MNAM is able to learn the uncertainty of each relationship, which NAM is unable to do. Described limitations of NAM and strengths of MNAM are illustrated in Figure
Another advantage of MNAM is its ability to help discern whether subpopulation is differentiated by observed or latent variables. This differentiation is possible by examining MNAM's output probabilities for each Gaussian distribution. If only one of the Gaussian distributions is activated, it suggests that the subpopulation differentiation is driven by observed variables. Conversely, if multiple distributions are activated, this implies the influence of a latent variable. For instance, in Figure 6 we observe that the subpopulation is differentiated by observed variables when X₂ = and X₂ = 1. However, in scenarios where X₂ 2 and X₂ 3, the differentiation appears to be due a latent variable. By analyzing MNAM's latent-baseel based the
plots, users can determine the nature of subpopulation infferetiationnolseee
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count of activated Gaussian distributions. This insight is crucial for interpreting the model, it provides understanding of whether there is uncertainty in subpopulation identification and whether the collection of additional unobserved variables is necessary for more accurate characterization of these subpopulations.
MNAM k =
MNAM k = 10
Dataset
NAM
X2
X2=1
Y
10
X2 =
10
X2 =
5 10
10
5 10
10
X1
Figure 6: NAM versus MNAM datasel that has variables that identify subpopulations as observed and latent variables. The left column is scatter plot for dataset with different values of X2- The remaining columns represent predictions from training on the dataset for NAM, MNAM with k = 4, and MNAM with = 10. NAM clearly fails to represent the dataset as collapses multiple relationships into one relationship. On contrary, MNAM with and activates the right ones for different values of X2.
and
accurately represents the dataset as learns four relationships
3.4 Trade-offs between Accuracy and Interpretability
We evaluated Linear Regression (LR); NAM; MNAM; Explainable Boosting Machine (EBM) Nori et al. 2019), which form of Generalized Additive Models (GAM) with pairwise interaction terms; Gradient Boosting Trees (GBT) (Friedman, 2001 Pedregosa et al.) 2011 and Mixture Density Networks (MDN)
Bishop 1994p. We used grid search for LR, EBM, GBT, MDN to finetune hyperparameters for training. 4 shows the MAE and LL scores for these five models. The order of the columns, left to right, represents an increase in complexity and decrease in interpretability (here defined as simple one-to- one relationship between input and oulput). The lable is split into two, which are models with direct relationships and complex relationships (left and right respectively). LR, NAM, and MNAM are models with direct relationships because their feature and output relationships are fixed even from changes in other features. Meanwhile, EBM, GBT, and MDN are considered models with complex relationships as their feature and output relationships changes from change in other features due to their interaction terms.
Table
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The performance generally improves with an increase in the complexity of models, except for MDN. As mentioned above, numerical instability and mode collapse have been observed in various studies while training MDN (Choi et al. 2018 Makansi et al. 2019).
Excluding MDN, the MAE score improves from an increase in the complexity of models for most datasets (as expected). Interestingly differences performances among models fluctuate greatly by datasets. This could be result of datasets having different complexity. For example, models have similar performances on the MIMIC datasets. This could be due to the datasets being too simple to not even require nonlincarity interaction terms of models for representations. In contrast, for the ACS Income dataset, the performance increases with an increase in complexity. This could be duc to the dataset being more complex and requiring nonlinearity and more interaction terms with higher degrees for models to represent the dataset well. In contrast, the LL score did not exhibit improvement with an increase in complexity across all datasets, excluding MDN. For example, the BIKE and MIMIC datasets, MNAM outperforms other more complex models. This potentially arises from the inherent need of the data to obtain model with multimodal output to accurately represent its underlying distribution.
DIRECT INPUT
COMPLEX INPUT
OUTPUT
OUTPUT
RELATIONSHIPS
RELATIONSHIPS
DATASETS
LR
O-NAM
PNAM
MNAM
EBM
CBT
MDN
MAES
LL1
MAEІ
LLf
MAE
LL1
MAET
LL
MAE
LL
MAEJ
LLf
MAE
LL
CA HOUSING
0.54
0.41
0.48
0.43
0.48
0.58
0.46
0.73
0.34
0.69
0.31
0.78
1.15
0.27
FICO
3.4
0.066
2.7
0.073
2.7
0.084
2.7
0.084
2.5
0.095
2.4
0.11
8.9
0.034
MIMIC
1.5
0.15
1.5
0.15
1.5
0.15
1.5
0.25
1.5
0.15
1.5
0.16
1.5
0.20
BIKE
3.65
0.061
3.4
0.064
3.4
0.069
3.4
0.092
3.4
0.065
3.4
0.065
3.93
0.083
ACS INCOME
40.0
0.0046
37.2
0.0049
35.7
0.011
35.7
0.013
33.3
0.0052
31.8
0.018
55.2
0.0081
ACS TRAVEL
16.8
0.013
15.6
0.013
15.5
0.017
15.5
0.036
14.2
0.015
13.8
0.090
18.4
0.018
COMPLEXITY
INTERPOETABILITY
Table 4: MAE and LL scores for LR, NAM, MNAM, EBM, and GBT on CA Housing, FICO, MIMIC BIKE, ACS Income, and ACS Travel datasets. The complexity of models increases from left to right and the interpretability of models increases from right to left.
4 Related Works
For interpretable models, GAM (Hsstic 2017 has been widely used. GAM transforms each feature by a function and linearly combines the transformed features, which enables features have a fixed relationship with the output. For transforming each feature, various functions have been used such as boosted decision troes (Nori et al.) 2019) and piecewise linear functions (Zuur) 2012). NAM (Agarwal et al. 2021 uses neural networks while GAM uses boosted decision trees (Lou al. 2012 Guisan et al. 2002 to transform the features. Compared to those models, MNAM has multiple outputs with probability, instead of one single estimate. These multiple outputs enable the model to represent multiple subpopulations in the dataset. Furthermore, it is more flexible for interpretation as it is able to show multiple relationships between features and labels, and how different relationships are activated by changes a subpopulation.
To overcome the limitations of GAM in representing multiple subpopulations, the Generalized Additive Model with Pairwise Interactions (GA2M) [ouu et al.) 2013) was introduced, enhancing GAM by integrating interaction terms. The FAST algorithm was developed to efficiently rank all potential pairwise interactions, selecting only the top-k terms. This concopt has inspired the development of various algorithms for ranking pairwise interactions. Researchers have applied sparsily and heredity constraints to identify significant features and interaction terms (Yang et al., 2021 Greenwell et al.) 2023). The Constrained Generalized Additive Model (CGA2M+) (Watanabe al. 2021) further extends GA2M by incorporating higher-order interaction terms improve expressiveness. However, since higher-order terms are not interpretable, they assess the validity of the interpretability of model based the importance score of the higher-order
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interaction term. Despite ongoing research, the limitation of GA2M is that relationships between features and labels are not fixed due to its interaction terms, making the model less interpretable. The model requires users to read two graphs for interpretation. One is for a line graph on the relationship between label and features and another on the effect of the interaction terms.
Research on making ensemble methods interpretable has been progressing- In the realm of tree-based ensem- ble models, RuleFii (Friedman & Popescu 2008) improves interpretability by extracting splitting rules from each node and assigning weights to these rules in linear model. These rules also incorporate interaction terms. The Compressed Rule Ensemble (Nalenz & Augustin 2022) approach further enhances interpretabil- ity by clustering similar rules and combining them into onc. Another approach involves applying Bayesian model selection techniques to simplify tree ensembles, effectively reducing the nunber of decisions without losing the accuracy of the original model (Hara & Hayashi 2018). The Interpretable Mixture of Experts (IME) (Tsmail et al., (023)) model, a variant of the Mixture of Experts, extends these principles beyond tree-based ensembles. This model consists of multiple expert models and an assignment model, which as- signs a specific expert to given input. IME explores the trade-off between accuracy and interprelability by making the assignment model either interpretable or not while maintaining the interpretability of the expert models. Additionally, a hicrarchical model is proposed that combines deep neural network (not interpretable) with IME, thus providing interpretability to a subset of the dataset. Similar to one variant of IME, MNAM features multiple interpretable relationships and a non-interpretable assignment model, which is a neural network that predicts the weights of each relationship. The key difference is that IME uses one expert model for prediction instead of assigning weights to all expert models.
Furthermore, MNAM can show the relationship for minority subpopulations in the case when the subpopu- lation is differentiated by latent variables by assigning one of the k relationships to the subpopulation. For deterministic models like NAM and DNN, they will fail as they are limited to showing only one relationship. This is demonstrated in Section 3.3 as the synthetic dataset for pedagogical experiment contains multiple subpopulations diffcrentiated by latent variables. This can be critical, for example, in medical applications, where a drug might be effective a certain subgroup of the population, tools like MNAM, would allow identifying from data modes or outliers that might not fit the general expected therapeutic trend.
Our work shares similarities with the Mixture Density Networks (MDN) proposed by Bishop in 1994. Bishop describes practical alternatives to use a mixture of k Gaussian distributions to fit. the outcome of target variable using neural network. In this work, the network loss is designed to optimize the posterior log- likelihood of the data. A notable difference between this work and the previously described work is that all the imput features are processed to predict the target variable; hence, the method lacks the property GAM and NAM have to be able to map the output prediction to independent factors on the input features. Yet compared to NAM and GAM, MDN can handle multimodal distributions and does not collapse multimodal relationships into single "all-fit-one' rule. Our work can be interpreted combining those two paradigms, where we aim to preserve the good properties of the described methodologies. Integrating these two methods could prove beneficial for other machine learning models that rely on singlo-estimate approach. In this paper, however, our focus is solely on NAM and GAM, as they are among the most widely recognized models for interpretability. Applying the tools here developed to other models is the subject of future research.
In our proposed mothod, we linearly combine features that have been separately transformed with neu- ral networks to maintain independent relationships between features and output. while fitting multimodal distribution with probabilistic interpretations. This approach allows the model to effectively learn from subpopulations. In contrast with MDN, we not only model the output distribution as multimodal but also assume an underlying subpopulation structure in the data and aim to detect from the input variables to which modality a given sample belongs. In addition, despite the similarities between our objective and MDN, extensive experimental evaluation suggests that our training objective is significantly easier to train, leading to faster, more stable, and better overall performance.
5 Limitations
MNAM trade-offs between accuracy and interpretability of model. Increasing the number of k Gaussian distributions for MNAM will increase accuracy. Yet, if the number of k Gaussian distributions is large, then
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it will be harder to interpret since there are too many possible relationships between features and outputs. The larger the number of k Gaussian distributions in MNAM, the more the model will become similar to neural networks as it covers all separate relationships for all possible combinations of features. For our future work, we plan to explore mothods such as the clbow method or other clustering techniques to find an optimal number for to balance between accuracy and interprelability.
As the MN AM model becomes more complex with an increased number of k Gaussian distributions, a new challenge emerges: the model starts to treat outliers as additional mode. This can be advantageous in certain domains. For instance, in medicine, recognizing small subpopulation with significant side effects distinct mode could be pivotal for pinpointing critical issues. However, this approach may pose problems when outliers result from noise. In future work, we aim to develop more objective methods for outlier identification. A key strength of our model is that an outlier impacts only the Gaussian distributions assigned to it. Therefore, if we can accurately identify an outlier and its corresponding Gaussian in MNAM, we can exclude that Gaussian during the evaluation phase to mitigate the issuc.
One limitation of the MNAM lies in its neural network, which predicts probabilities for a mixture of & Gaus- sian distributions. This network's lack of interpretability makes it challenging to understand how changes in features affect the probabilities of the mixture. We can partially interpret the network by rigorously examining the model through various examples. Additionally, by modifying specific features, we can observe if there are variables that differentiate subpopulations, either observed or unobserved. Another approach involves examining the impact of each feature on the activation of probabilities for the mixture of Gaussian distributions. This can be done by dividing the data into bins of equal frequency and analyzing the average weight of each component across these bins. Examples of these examinations are provided in Section 3.3 and Appendix D However, there are still limitations in fully interpreting the network. For future work, we aim to explore more interpretable models for predicting probabilities in mixtures of k Gaussians, which would eliminate the need for such detailed examination.
MNAM learns means and variances of the outcome by minimizing the GNLL. However, this method is known to have limitations, as it may fail to effectively capture optimal means and variances of the outcome (Stirn let al.) 2023). One observed issue is that the model exhibits a bias towards learning in regions with low variance, since the variance serves as denominator for the gradient the mean. Consequently, in regions with high variance, the gradient becomes small, impeding updates to the weights used to compule the mean. To address these limitations, researchers have introduced a technique where the variance is multiplied by the gradient (Stirn et al. 2023). This approach aims to prevent the high variance from causing the gradient to diminish to zero In future works, we intend to apply this technique and explore other methods to further enhance the performance of the proposed model.
6 Conclusion
In this work, we introduced the Mixture Neural Additive Model (MNAM), interpretable model more flexible than GAM and NAM, capable of capturing multimodal relationships between the input and output variables and with predictions that can be interpreted probabilistically. We showed that our ideas also connect with the Mixture Density Networks (MDN) framework. At the same time, MNAM preserves the desirable properties of NAM, and empirical evaluation suggests it trains more robustly and efficiently than MDN. We showed that our proposed model can be applied in practice and evaluated across several real- world datasets and applications. Despite the recent progress and efforts in developing interpretable and robust deep learning alternatives, it is clear that there are still opportunities to improve the collection of tools that are available for practitioners working in critical applications such health care and data-driven policymaking. Morcover, and despite great work like the one presented by Bishop in 1994, gaps remain between the theory and practice of deep learning, and the stability and practical application and training of neural networks remain a very active research topic (Choi et al.) 2018 Makansi et al. 2019). Our work provides useful, robust, yet interpretable deep learning alternatives to help design and deploy data-driven solutions in critical applications. The code and pipelines to reproduce our work are open and can be found here: https///itthub.com htwpa//oooooooooogggggggn/////// MMNMM
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Acknowledgments
Work partially supported by ONR, NGA, NSF, and Simons Foundation. GS is also affiliated with Apple.
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acm. 11//10..114//6411990 264119
Akihisa Watanabe, Michiya Kuramata, Kaito Majima, Haruka Kiyohara, Kondo Kensho, and Kazuhide Nakata. Constrained generalized additive 2 model with consideration of high-order interactions. In 2021 International Conference on Electrical, Computer and Energy Technologies (ICECET), pp. 1-6. IEEE,
2021.
Zebin Yang, Aijun Zhang, and Agus Sudjianto. Gami-net: An explainable neural network based on gener- alized additive models with structured interactions. Pattern Recognition, 120:108192, 2021.
Alain F Zuur. beginner's 's guide to generalized additive models with R. Highland Statistics Limited New- burgh, 2012.
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Published in Transactions on Machine Learning Research (02/2024)
A Table of optimized parameters for MNAM
LEARNING
WEIGHT DECAY
OUTPUT
NUMBER OF
CROSS-ENTROPY
DATASET
RATE
PENALTY
GAUSSIAN DISTRIBUTION
LOSS
CA HOUSING
0.009896
3.8512E-05
0.03363
2
0.6118
FICO
0.05
1E-06
0.0166
0.7762
MIMIC
0.01805
7.0946E-05
0.01908
2
0.7214
BIKE
0.01172
9.10226-05
0.09256
6
0.3537
ACS INCOME
0.02873
9.13E-05
0.00167
0.494
ACS TRAVEL
0.01894
9.33777-05
0.0028
4
0.3634
Table 5: Optimized parameters for MNAM on six datasets
Different Relationships plots on Housing
First
Second mode
200
Median Income
Median Income
Media House Age
Media House Age
145
Average Number of Rocms
Average Number of Rooms
Average Number of Bedrooms
Average Number Bedrooms
13%
123
-004
-0.94
10c
30%
30
180
300
1988
Population
Population
Average Number Household
Average Number Household
1,00
Kon
0.00
Lattudo
Latiuudo
Longitude
Longitude
Figure 7: Learned relationships between features and labels for the MNAM on Housing datascts. Solid lines represent the mean of the relationships. while the highlighted regions represent their uncertainties. For enhanced readability, each subplot illustrates one distinct relationship.
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Published in Transactions on Machine Learning Research (02/2024)
C
Relationships plots on other datasets
C.1
FICO
First mode
Variance of first mode
FS00 sooe
FECO
PI00 store
RunThdoodeentaatti
FSD C0PE
Percrttintalladdee
Numinglant
NunninganmSHecciddes
F100 Mere
300
200
RedtractendrtaalaDoooo Burden
-27
+2.0
20
05
85
PaccesTrcceoWaalace
delnquett
Ingulry
Figure 8: Learned relationships between features and labels for the MNAM on FICO datasets. Solid lines represent the mean of the relationships and dotted lines represent their uncertainties.
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Published in Transactions on Machine Learning Research (02/2024)
C.2 MIMIC
First made
0.
Length Stay
Variance of first made
Second
0.20
Variance of second mode
-0.05
-2.12
Age
Gender
Ethnicity
Insurance
Length Stay
-02
-00
10000
Admission Type
Martial Status
Admission Location
Figure 9: Learned relationships between features and labels for the MNAM MIMIC datasets. Solid lines represent the mean of the relationshipe and dotted lines represent their uncertainties.
C.3
BIKE
Test first
Secend mode
Diraton
Third
Pickup Longitude
Pickup Latitude
The
Pickup Hour
Gender
Figure 10: Learned relationships between features and labels for the MNAM on BIKE datasets. Solid lines represent the mean of the relationships and dotted lines represent their uncertainties.
22

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Published in Transactions on Machine Learning Research (02/2024)
C.4 ACS Income
mode
Vaniacee first mode
Beoond made
Vamanoe
Incime
Variance ofieied
Pourth made
Vanance hourth mode
Age
Educational Adtai maatt
Trome
Occupation
spouss AfTERE
Hours Worked For Week
Home
Gondor
Figure 11: Learned relationships between features and labels for the MNAM on ACS Income datasets. Solid lines represent the mean of the relationships and dotted lines represent their uncertainties.
23

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Published in Transactions on Machine Learning Research (02/2024)
C.5
ACS Travel
mode
Vanance firtt mode
Beoond made
Vamance
Vartance ofieid
Pourth made
Vanance hourth mode
Nartal Stata
Gender
Dessiity
Moblity
Relggias Aftiaatoo
R200
Code
Oitzeneiip Statu
Ocoupetion
Transportation
Ploce of Work
3ooore Powerty Patio
Figure 12: Learned relationships between features and labels for the MNAM on ACS Travel datasets. Solid lines represent the mean of the relationships and dotted lines represent their uncertainties.
24

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Published in Transactions on Machine Learning Research (02/2024)
D Relationship between feature and probabilities for mixture of k Gaussian distributions
In Figure 13 we examine the relationship between features and the probabilities associated with a mixture of k Gaussian distributions. It is evident that the location variables have more significant impacts on altering probabilities compared to other variables, such population and average number of beds. This analysis allows us to determine which features are pivotal in influencing the outcome of the neural network, which outputs probabilities for mixture of & Gaussian distributions.
Median Income
Median House
Housing Price
60%
First mode
Second mode
20%
20%
096
(0.5, 1.91
11.5, 2.36)
12.35, 2.74]
12.74 2.34]
13.14 2.5
13.53, 3997
(3.97, 4,44)
(4.44 5.11]
16.11, 6.16)
16.1&, 15.0)
0%
[1.0, 13.0)
[12.0. 17.0]
[17.0. 20.01
(20.0, 25.0
123.0, 29 B)
(29.9, 33.01
13300 36.0 3..0
(26.0, 40.0)
(40 46.01
(46.0, $2.0)
Average Number of Rooms
Average Number of Bedrooms
Housing Price
60%
20%
0%e
to 3.79)
13.79, 4.271
14.27, 4411
[4.5 4333
14,03,
15.23, $25
(5.32, 9.88)
(5.28 (6.27, 98)
98. 141.1]
10.33, 0.91 0.971
10.97, 1.0)
[t.0, 1.013
11.01 1.03]
11.03, 1.88]
11.08 1.07)
11.07 1.09)
11.0% 1.13)
(1-12 1.7)
[1.17 24,77
5.231
6.27)
Population uoppennon
Average Household
Housing Price
60%
60%
40%
49%
12.0, 518.01
[s10.6, 219.0 o)
(110. farra 1007. 0)
f1007 f1166.0 1348,1
(1349.0, 1580.01 B)
fimee.0, too0. D)
Etnoa.0 2566.0)
(2066.0 35692.00
10.00, 2 2.00)
[2.08 2.34]
[2.34 2.51
1251. 2.471
12.87 2.823
12.22 2.08)
12.9 3.17)
BAx, 3.42)
12.42 3.89)
[3,59, 1243 33)
ass on
1166.0
Lattude
80%
Longibude
Housing Price
60%
40%
49%
20%
122 (22.54, 23,63)
(32.63, 23.97)
132 34,01
De 34.11
DM 26)
the 30 eej
120 so, 37
137 137 a8 AB)
138 41, (1.95)
[-124 35, -122 -222.00
1-122 29, -22 ma)
f-121.98 -121 37)
1-121.7 1189 911
1-119.91
f-118 40, -11
1-118 -31.133)
E-18 13, -137.89
[-117 20, -117.99)
(-117.25, (11331
37.80
-120 491
The relationship between features and probabilities in mixture of k Gaussian distributions. Each point on the x-axis represents bin for the feature, which has been divided to ensure equal frequency. Fach bar depicts the average probabilities for the mixture of & Gaussian distributions across each bin.
Figure 13:
25


Reviews:
[{"review_id": "6DY5SmNxj6", "reviewer": "Reviewer_MoRB", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The motivation behind adding the mixture model is compelling.\n- For the tested datasets, MNAM fits the data better than NAM.  \n\nWeaknesses:\n- The interpretability evaluation is based on a few worked examples.  Further, these examples don't really seem to discuss $g$ (the function that determines how to weight each of the gaussian components for a particular point) which seems essential to interpreting MNAM.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This works introduces MNAM, which extends NAM in two ways.  First, rather than having each of the per-feature neural networks predict a single value, MNAM has the each of those neural networks predict the mean and standard deviation of a gaussian distribution.  Second, rather than using a single gaussian for each feature, MNAM uses a mixture of gaussians. By learning when to use each gaussian component, MNAM can model feature interactions that NAM cannot (eg, if $X_1 = 0$, $X_2$ has a positive slope and, if $X_1=1$, $X_2$ has a negative slope).", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "NAMs are generally considered interpretable because their structure makes simple for a user to 1) estimate the model's prediction for a new point and 2) estimate how the model's prediction would change if the point being explained was changed.  However, it's not as clear that this is possible for MNAM because of the $g$ function (ie, one can construct situations where all of MNAM's modeling power comes from $g$).  Please add discussion and/or experiments related to interpreting $g$. \n\nA few potential ways to do this that came to mind are: \n- How can the user tell if the subpopulations are differentiated by an observed or unobserved set of features?\n- What do the plots in Figure 4/5 look like if you use a point drawn randomly from the dataset?  What if you first bin the dataset based on the feature on the x-axis and then show the weight of each component averaged across those bins?", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "YYd6FIQB6T", "reviewer": "Reviewer_qZmq", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The idea of extending NAM to multi-modal area is very interesting.\n- The author explained in detail how the model works and how the training procedure is conducted.\n- Multiple experiments are conducted to explore different properties of the proposed model.\n- The authors describe the connection of the proposed model to other related baselines and how the model is placed in terms of the accuracy vs interpretability tradeoff, as well as the limitation of the proposed model.\n\nWeaknesses/Questions:\n\nOverall, the proposed model is well explained in the paper. However, I have a few questions about the model.\n- The model requires a parameter $k$ that describes the number of Gaussian distributions in the mixture. Are there any guidelines on how to choose it? Particularly in the case where we do not really know how much modality there is in our datasets.\n- One of the downsides of a more complex model is that it may overfit the data. How does the model behave in the case where there are outliers? Does the model treat the outliers as another source of modality?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors present a method that extends the Neural Additive Models (NAM) to handle multi-modal data distributions. Instead of producing a single output like NAM, the proposed model, Mixture of Neural Additive Models (MNAM), outputs the means, variances, and mixture membership probabilities of a mixture of k Gaussian distributions. The authors then presented two different ways to train the proposed model: the soft and hard thresholding algorithms. Finally, the authors demonstrate the benefits of the proposed model against several baselines, highlighting that the model provides a good tradeoff between accuracy and interpretability in datasets with multiple modalities.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I would be grateful if the authors put some discussions about the question I mentioned above in the revision.", "broader_impact_concerns": "I do not have any broader impact concerns for this paper.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "brUpWpyJBH", "reviewer": "Reviewer_9cDv", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n\nCompared to the previous version, there are several improvements:\n\n- The comparison to MDNs and the discussion of training differences is helpful.\n- The inclusion of a NAM with global variance parameter, and a pNAM with a single Gaussian mode are helpful. This makes it easier to see that 1) fitting a variance parameter and 2) fitting multiple modes are both helpful additions to the existing NAM.\n- The inclusion of the log-likelihood metric, compared to the earlier usage of metrics designed for point predictions, is an improvement.\n\n## Weaknesses\n\nThe premise of the paper is that NAMs and GAMs fail to represent multimodal relationships between predictors and response variables. This is somewhat confusing, because this problem exists for all mainstream regression models, whether they are interpretable or not: linear regression, random forests, gradient boosting and neural networks all output point predictions. So it's a bit confusing to focus on this issue in the special context of GAMs/NAMs. \n\nCan the authors describe how they obtained confidence intervals for the results in Table 1? Ideally these would have been generated using multiple independent runs of each algorithm, but the confidence intervals are remarkably small in most cases, so that seems unlikely. \n\nThe utility of this extension to NAMs from a predictive point of view seems somewhat clear. In terms of the MAE and LL performance metrics, it seems like the MNAM provides a more flexible and therefore more accurate prediction. However, the utility from an interpretation perspective seems limited. The plots in Figure 3 are not simple to read, and this is for a case with only two modes and a couple features - it could become far more difficult with more modes. With this in mind, I have limited optimism about the use of mixture of Gaussian predictions and the HT training algorithm in this context, and I wonder if it would be more useful with generic DNNs rather than NAMs. \n\nThe proposed extension to classification problems using LIME still doesn't make sense to me. Unlike regression models, classification models automatically output distributional predictions because they assign a probability to each class. So it's unclear what a mixture would add here (a mixture of categorical variables is still a categorical), and the usage of LIME, a post-hoc explanation method, doesn't seem to make sense.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes an extension to neural additive models (NAMs) to accommodate data that requires multi-modal, distributional predictions. The method involves predicting a mixture of Gaussians distribution, where each Gaussian is additive between features, and the mixture components are weighted according to a network that depends on all features. Ultimately, this is intended to provide a model (MNAM) that balances rich nonlinear predictions with the interpretability of NAMs.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "No", "audience_explanation": "", "requested_changes": "Several small issues:\n\n- Typo in the abstract: GAN -> GAM.\n- Citations throughout the paper lack parentheses where they should be used. The authors are perhaps using the \\cite{} command rather than \\citep{}. \n- The authors use \"multimodal\" to refer to response variable distributions that are not well approximated by point predictions. It would be helpful to distinguish this from multimodality in the sense of using multiple training data types, e.g., text and images, perhaps in a footnote. This usage of term is popular today and could lead to confusion for some readers.\n- Missing section link (\"Section ??\") on page 2.\n- \"log\" should not be italicized in equation 2, and there should be more description leading up to this equation - it currently looks isolated.\n- I agree that there could be advantages to the hard-thresholding training algorithm over the direct log-likelihood optimization, but I don't understand this sentence: \"In contrast, MDN assumes the data is multimodal by nature but does not assume there is information in the input to learn the mode associated with a given sample.\" Training using the log-likelihood does provide a way to learn the mode associated with each prediction, because it would increase the weight on the mode that most accurately predicts the label. Unless I'm missing something, I think the authors' point here is incorrect.", "broader_impact_concerns": "No broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: Reviewers agree that the extension of the original NAM with mixture density network is interesting and useful in practice. Specific concerns about experiments are addressed by the author feedback, and the reviewers recommend acceptance for this submission. 

In camera ready I suggest the authors to include all the provided edits in the current revised version. In the final recommendation, a reviewer further mentioned some related work -- pairwise GAMs (eg, https://www.microsoft.com/en-us/research/wp-content/uploads/2017/06/kdd13.pdf and subsequent work) and other interpretable ensemble models (eg, https://openreview.net/pdf?id=DdZoPUPm0a), for which the authors should consider adding at least a discussion regarding these relevant works.
Decision: Accept (poster)