Curve Your Enthusiasm: Concurvity Regularization in Differentiable Generalized Additive Models
Authors: Julien Niklas Siems
Konstantin Ditschuneit
Winfried Ripken
Alma Lindborg
Maximilian Schambach
Johannes Otterbach
Martin Genzel
Venue: NeurIPS 2023 poster (2023)
Keywords: Interpretable Machine Learning
Generalized Additive Models
Concurvity
Multicollinearity
Regularization
Time-Series Forecasting
Interpretability
TL;DR: We address concurvity, the non-linear equivalent of multicollinearity, in Generalized Additive Models (GAMs) by proposing a novel regularizer that effectively reduces the concurvity, enhancing interpretability without compromising prediction quality.

Abstract: Generalized Additive Models (GAMs) have recently experienced a resurgence in popularity due to their interpretability, which arises from expressing the target value as a sum of non-linear transformations of the features. Despite the current enthusiasm for GAMs, their susceptibility to concurvity — i.e., (possibly non-linear) dependencies between the features — has hitherto been largely overlooked. Here, we demonstrate how concurvity can severly impair the interpretability of GAMs and propose a remedy: a conceptually simple, yet effective regularizer which penalizes pairwise correlations of the non-linearly transformed feature variables. This procedure is applicable to any differentiable additive model, such as Neural Additive Models or NeuralProphet, and enhances interpretability by eliminating ambiguities due to self-canceling feature contributions. 
We validate the effectiveness of our regularizer in experiments on synthetic as well as real-world datasets for time-series and tabular data. Our experiments show that concurvity in GAMs can be reduced without significantly compromising prediction quality, improving interpretability and reducing variance in the feature importances.

Curve Your Enthusiasm: Concurvity Regularization in Differentiable Generalized Additive Models
Julien Siems
Konstantin Ditschuneit*
University of Freiburg
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Winfried Ripken"
Alma Lindborg"
Merantix Momentum winfried. www.dd.iieennnnnem www.eenmeernntiii..c.mmmmm com
Merantix Momentum
alma. lindborgdmerantixx com
Maximilian Schambach Merantix Momentum maximilian schambachheerrattii. maximml
Johannes S. Otterbach nyonic j Joannes@nyonic. ai
com
Martin Genzel
Merantix Momentum
martin.. pertin.ggnzeer wwwggeeeeeeeeaeene 1mweeeeiieaaaiccoomo com
Abstract
Generalized Additive Models (GAMs) have recently experienced a resurgence in popularity due to their interpretability, which arises from expressing the target value of non-linear transformations of the features. Despite the current enthusiasm for GAMs, their susceptibility to concurvity - i.c.. (possibly non- linear) dependencies between the features has hitherto been largely overlooked. Here, we demonstrate how concurvity can severely impair the interpretability GAMs and propose remedy: conceptually simple, yet effective regularizer which penalizes pairwise correlations of the non-linearly transformed feature variables. This procedure is applicable any differentiable additive model, such as Neural Additive Models or NeuralProphet, and enhances interpretability by eliminating ambiguities due self-canceling feature contributions. We validate the effectiveness of our regularizer in experiments on synthetic as well as real-world datasets for time series and tabular data. Our experiments show that concurvity GAMs can be reduced without significantly compromising their prediction quality, improving interpretability and reducing variance the feature importances.
Introduction
Interpretability has emerged as a critical requirement machine learning models in safety-critical decision-maainn processes and applications subject to regulatory scrutiny, Its importance is particu- larly underscored by the need to ensure fairness, unbiasedness, accountability and transparency in
"Equal contribution
Corresponding author
Work done while Merantix Momentum
Code: https://githul: //te////rr/////////aa////////rrr httpa//www.....ececeacettataaaaa
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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Task: Fit periodic timeseries with daily and weekly scasonality
Few Fourier terms (NouralProphet dofaults)
Many Fourier terms
Many Fourier terms
w/ concurvity regularization
Daily Compunent Component
+
Wookdy
Prediction
X RMSE = 3-85-10
vf RMSE 4.68 10"
RMSE = 4.37-10" 100
V Confuwekly, daily) 6.0-10-11 10-10 X Carr(wedly,, daly) - 0.64
V Corn(weekly, daily) 10
Figure 1: Concurvity in NeuralProphet model: Fitting time series composed of daily and weekly seasonalities, each represented by Fourier terms. (left) Using few Fourier terms results in uncorrelated components but a poor fit. (middle) A more complex model improves the fit but sacrifices interpretability due to self-canceling with our regularizer, achieves both good predictive performance and interpretable (decorrelated) components. See Section 4.2 for more details.
feghffreyeeccy termss (right) The same complex model, but
many applications and domains 66 42 such as medical diagnoses 10 loan approvals [4], and hiring practices [16]. In such cases, model interpretability can be favored over prediction accuracy 31
popular model class for interpretable machine learning is Generalized Additive Models (GAMs) [24] in which the target variable expressed as combine the interpretability of (generalized) linear models with the flexibility of capturing non-linear dependencies between the features and the target. GAMs have recently seen resurgence in interest with prominent examples being Neural Additive Models (NAMs) 2 and its variants [1] 19
of non-linearly transformed features. GAMs
20 39
50 for tabular data, well Prophet 445 and NeuralProphet 46 for time-series forecasting. Both domains will be further explored in our experiments.
A significant obstacle to the interpretability of additive models is the phenomenon of concurvity 9. As non-linear analog to multicollinearity. concurvity refers the presence of strong correlations among the non-linead non-linearly transformed feature variables. Similarly multicollincarity, concurvity can impair interpretability because parameter estimates become unstable when features are correlated [40], resulting in highly disparate interpretations of the data depending the model initialization. Although this issue is known and addressed by various techniques such as variable selection (28 in traditional GAMs, it has been overlooked in more recent works. Unlike the prevalent GAM package mgev [47]
we are not aware of any differentiable GAM implementations that include concurvity metrics.
In this work, we propose novel regularizer for reducing concurvity in GAMs by penalizing pairwise correlations of the non-linearly transformed features. Reducing concurvity improves interpretability because it promotes the isolation of feature contributions to the target by eliminating potentially correlated or self-canceling transformed feature combinations. As a result, the model becomes casier to inspect by clearly separating individual feature contributions. In addition, encourages the model to learn more consistent feature importances across model initializations, which increases interpretability. The trade-off between increased interpretability and prediction quality will be further explored in Section 4
regularizer
Figure I provides a first intuition of the practical effects of concurvity and how they can be addressed by our regularizer. We use the additive time-series model NeuralProphet [46] this example, restricted to daily and weekly seasonality components. Here, each component is modeled via periodic functions with adjustable complexity. For more details on the experiment, see Section 4.2 We find that while the default NeuralProphet parameters effectively mitigate concurvity by producing very

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simple model, they provide worse fit to the data than the more complex models. However, if left unregularized, more complex model is subject to strong correlation between the seasonalities, an effect visually apparent in self-canceling periodic components in the middle column of Figure [ In contrast, when using our regularization, the seasonalities are less correlated, resulting in a clearer separation between the components. While the predictive performance of the two complex models is comparable, the regularized model is more interpretable because daily and weekly effects are clearly separated.
This idea forms the basis of our argument: Comprehending a standard GAM typically requires answers to two questions: How do the features relate to my output? How do the features interact, possibly negating their contributions? By applying concurvity regularization, we simplify the interpretation of a GAM by eliminating the need for the second question.
Our main contributions be summarized as follows:
1. We showcase the susceptibility of modern GAMs to concurvity and present a revised formal definition of the concept.
2. We propose a concurvity regularizer applicable to any differentiable GAM.
3. We validate our approach experimentally on both synthetic and real-world data, investi- gating the trade-off between concurvity and prediction quality, as well as the impact of regularization on interpretability.
2 Background
2.1 Generalized Additive Models
Generalized Additive Models (GAMs) 24 form a class of statistical models that extends Generalized Linear Models (36) by incorporating non-linear transformations of each feature. Following 24 GAMs can be expressed as:
(B[yIx]) 3
E.
i=1 S((X.).
(GAM)
where Y = (yI,..., .........NN E RN is a vector of N observed values of a target (random) variable, X = [X,, 1XI...,,Xp) Xp] E NX assembles the observed feature variables X; (Zi,,,...,33 Zi,N) R-, and ʃ; R - R are univariate, continuous shape functions modeling the individual feature transformations Furthermore, 8 E R is a learnable global offset and g IR + that relates the (expected) target value to the feature variables, e.g., the logit function in binary classification or the identity function in linear regression. The shape functions f/ precisely describe the contribution of each individual feature variable in GAMs, and can be visualized and interpreted similarly to coefficients in a linear model. This allows practitioners to fully understand the learned prediction rule and gain further insights into the underlying data.
the link function
While early GAMs primarily used splines 24 or boosted decision trees 10
34
351
to model fs.
more recent GAMs such as Neural Additive Models (NAMs) [ use multilayer perceptrons (MLPs) to fit the functions fú, benefitting from the universal approximation capacity of neural networks [[5] as well as the support of automatic differentiation frameworks [ 38 and hardware acceleration.
As result, one can now solve the GAM fitting problem
GAM-Fit:
min B.(f.....D)E
E B[L + Σ f((x)))
by common deep learning optimization techniques such as mini- stochastic gradient descent (SGD). Here, I.: - IR is loss function and H C ((Si,..., Sp) fa R- [k] any function class with differentiable parameters, e.g., ML.Ps or periodic functions like in NeuralProphet [46].
2.2 Multicollinearity and Concurvity
Multicollinearity refers to situation in which two more feature variables within a linear statistical model are strongly correlated. Formally, this reads as follows
1 As usual, when fi or g are applied to a vector, their effect is understood elementwise. 'Our definition is based on fixed (deterministic) feature design, but one could also formulate a probabilistic version, also Appendix A.2 l).
3

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Definition 2.1 (Multccoliineartty). Let XX.....Xp E R be set of feature variables where X, = (fi,1...... N..N) E EN represents N observed values. We say that XI..... Xp are (perfectly) multicollinear if there exist co. Cp.......p R, not all zero, such that co Di= C;X1; 0.
According to the above definition, every suitable linear combination of features that fits target variable, say Y z do Ei= d,X,, can modified by adding trivial linear combination co + Ei=,c GX;; 0. i.e., Y s (co + do) + Ei=1 (c: + d,, So there exist other (possibly infinitely many) equivalent solutions with different coefficients. This can make individual effects of the features on a target variable difficult disambiguate, impairing the interpretability of the fitted model. However, even in the absence of perfect multicollinearity difficulties may arise 6 For example, if two features are strongly correlated, estimating their individual contributions becomes challenging and highly sensitive to external noise. This typically results in inflated variance estimates for the linear regression coefficients 40 among other problems [17].
The notion of concurvity was originally introduced in the context of GAMs to extend multicollinearit to non-linear feature transformations ( In analogy with our definition of multicollinearity, we propose the following definition of concurvity:
Definition 2.2 (Concurvity). Let XI...,,
RN be a set of feature variables and let
H C {(1,.... Sp) fs R RJ be a class of functions. We have (perfect) concurvity w.r.t XI...., X2 and H if there exist (o.....99p)) E H and cu E R such that co + - 9(,)) (x with co. 91 (X.)..... ..n.),........X) (N(NN) not all zero.
Technically, concurvity simply amounts to the collinearity of the transformed feature variables, and Definition 2.1 is recovered when H is restricted to affine linear functions. Concurvity poses analogous challenges to multicollinearity: Any non-trivial zero-combination of features can be added solution of GAM-Fit in Equation (I. rendering the fitted model less interpretable as each feature's contribution to the target is not immediately apparent. Finally we note that, although concurvity can be considered as a generalization of multicollineaarty, neither implies the other in general, see Appendix 2) for more details.
3 Concurvity Regularizer
Concurvity easily arises highly expressive GAMs, such as NAMs, since the mutual relationships between the functions fi are not constrained while solving GAM-Fit in Equation (T). This results in a large, degenerate search space with possibly infinitely many equivalent solutions. To remedy this problem, it appears natural to constrain the function space H of (I such that the shape functions fi do not exhibit spurious mutual dependencies. Here, our key insight that pairwise uncorrelatedness is sufficient to rule out concurvity. Indeed, using H from Definition 2.2 let us consider the subclass
H-. = ((f,..... fp) E H (ox,)
= j} C.
where Corr(.,)) is the Pearson correlation coefficient. It is not hard to see that concurvity w.r.t XI,...,X, Xp and Hi is impossible, regardless of the choice of H (see Aopendix/A.IJaae proof). From a geometric perspective, H., imposes an orthogonality constraint on the feature vectors. The absence of concurvity follows from the fact that an orthogonal system of vectors is also linearly independent. However, it is not immediately apparent how to efficiently constrain the optimization domain of Equation I to H.. Therefore, we rephrase the above idea an unconstrained optimization problem:
GAM-Fit
min S.JE
B[ 3 Ei fi(x.))] + X R, (ff.). (X,))
(2)
where Ri 7 x Nxp 10, denotes our proposed concurvity regularizer:
R. (Xi)),)(i))) 117-11/2
Corr (Jor) [xx))
(=1 =11
Using the proposed regularizer, GAM-Fit simultaneously minimizes the loss function and the distance to the decorrelation space H. In situations where high accuracy and elimination of
'Informally, non non-perfect multicollinearity describes situations where Σie, CxX; R 0. However, a formal detinition would also require an appropriate correlation distance metric.

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concurvity cannot be achieved simultaneously, trade-off between the two objectives occurs, with the regularization parameter À 2 determining the relative importance of each objective. An empirical evaluation of this objective trade-off is presented in the subsequent experimental section.
Since R. is differentiable almost everywhere, GAM-Fit, in Equation (2) can be optimized with gra- dient descent and automatic differentiation. Additional computational costs arise from the quadratic scaling of R in the number of additive components, although this can be efficiently addressed by parallelization (see Appendix E.3.2 for runtime analysis). A notable difference to traditional regularizers like f1- or fz-penalties is the dependency of R_ on the data {X:}:- As a consequence. the regularizer is also affected by the batch size and hence becomes more accurate with larger batches. It is also worth noting that our regularizer does not necessarily diminish the quality of the data fit, for instance, when the input feature variables are stochastically independent, see Appendix (32
Common aaglieehased concurvity metrics proposed in the literature (28 47 are not directly applicable to other differentiable models such as NAMs or NeuralProphet. Thus, similarly to [40]. we decided
to report the average R1 (1 (X.).) E [0, 1] as model-agnostic measure of concurvity.
4 Experimental Evaluation
In order to investigate the effectiveness of our proposed regularizer, we will conduct evaluations using both synthetic and real-world datasets, with a particular focus on the ubiquitous applications of GAMs: tabular and time-series data. For the experiments involving tabular data, we have chosen to use Neural Additive Models (NAMs), as they are differentiable and hence amenable to our regularizer. For time-series data, we investigate NeuralProphet models which contain an additive component modeling seasonality. Further elaboration on our experimental setup, including detailed specifications and parameters, can be found in Appendix
4.1 Toy Examples
In the following, we design and investigate two instructive toy examples to facilitate a deeper comprehension of the proposed regularizer as well as the relationship between the regularization strength X and the corresponding model accuracy. Results for an additional toy example from (28) are presented in Appendix [E.2
Toy Example 1:
Concurvity regularization with and without multicollinearity To compare the effect of concurvity regularization on model training in the presence of multicollinearrty. we generate synthetic data according to the linear model Y = X; -- X2, where X1 and X₂ are each drawn from a uniform distribution. We consider three different settings of input feature correlation: (1) the stochastically independent case where X1 and X2 are independently sampled, (2) the strongly correlated case where Corr(X,. X2) = 0.9, and (3) the perfectly correlated case with X; = X2-
We first investigate the effect of concurvity regularization on the contribution of each feature to the target by measuring the correlation of the transformed features fi (X1). f2(X2) with the target variable Y. The results are shown in Figure 2a In the stochastically independent case, the NAM accurately captures the relationship between the input features and target variable regardless of the regularization setting, as observed by the high correlation for (X, and zero correlation for fx(X;) with the target. This result emphasizes the minimal impact of the regularizer when features are independent (see Appendix 2 for details). In the perfectly correlated case, the NAM trained without concurvity regularization displays a high correlation for both Si (Xi) and (x(2) with the target, thus using both features for its predictions. In contrast, when concurvity regularization is applied, the NAM is pushed towards approximately orthogonal ft. which encourages feature selection, as indicated by high correlation of either (x or (x(2) with the target as there is natural preference. Interestingly, the situation is slightly different in the strongly correlated case, as X is more likely to be selected here, which means that the model has correctly identified the true predictive feature. This nicely illustrates the impact of the proposed regularization on correlated feature contributions. A more detailed visualization of the impact of regularization under perfectly correlated features can be found in Figure & in Appendix E.
Secondly, we examine the trade-off between the validation RMSE and concurvity Rú in the perfectly correlated case in Figure 2b Our findings suggest that with moderate regularization strengths we can effectively eradicate concurvity without compromising the accuracy of the model. Only when
5

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regularization
regullriiatiin
Cont X2)=1 CorfX X2)= Con(X, x2)=0
Con()s(Xa),.) (on( V) ( V7
Seed averrago
16a
10
Val
0.4
10
0.2
0.0
2000
10
X)
Val. RMSE
(a)
(b)
Figure 2: Results for Toy Example 1. (a) Effect of concurvity regularization on uncorrelated and correlated features. For each of the six settings, 40 random initializations were evaluated. (b) Trade- off curve between model accuracy (validation RMSE) and measured concurvity (validation Rt).
Results averaged over 10 random initializations per regularization strength A.
W/o regularization
W/ rugulatization (Corr 0.03)
(Corr 0.92)
10'
600
0.0000 0,0005 0,0010 0.0015
(x(1
0.01
f:(X1)
0.01
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0.8
10
(a)
(b)
Figure 3: Results for Toy Example 2. (a) Comparison of transformed feature correlation with and without concurvity regularization. (b) Trade-off curve between model accuracy (validation RMSE) and measured concurvity (R
the regularization strength is considerably high, the RMSE is adversely affected, without any further reduction in the measured concurvity.
Toy Example 2: Concurvity regularization in the case of non-linearly dependent features Next, we evaluate our regularizer in the case of non-linear relationships between features, a setting to which it is equally applicable. To this end, we design an experiment with feature variables that are uncorrelated, but not stochastically independent due to a non-linear relationship. We choose X1 = Z and X₂ = |Z| where 7 is standard Gaussian, and let Y = X2 be the target variable. In this case, there is no multicollincarity by design, but the model may still learn perfectly correlated feature transformations. For example, a NAM could learn f1 | and f2 id, then fi(X)) = (X2). which are fully correlated, yet providing a perfect fit. For our experiment, we use the same NAM model configuration in the previous toy example.
The transformed features for the NAM fitted with and without regularization are visualized in Figure 3a We find that the regularizer has effectively led the NAM to learn decorrelated feature transformations fi and /2, reflected in the striking difference feature correlation (Corr = 0.03 for the regularized NAM compared to Corr 0.92 for the unregularized model). Moreover, these results suggest that the regularized model seems to have learned the relationship between the features, where f2(X seems to approximate (X))).

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Finally, a trade-off curve of the validation RMSE and R1 is shown in Figure 3bb illustrating that even in the case of non-linearly dependent features, our proposed regularizer effectively mitigates the measured concurvity Rı with minimal impact on the model's accuracy measured by the RMSE.
4.2 Time-Series Data
In this section, we provide more context and additional results for the motivational example in Fn time-series forecasting using NeuralProphet [46], which decomposes a time-series into various additive components such as seasonality or trend. In NeuralProphet, each seasonality Sp is modeled using periodic functions
Sp(t) = E, cos (2xjt/p) + by sin (2mjt/p)
where & denotes the number of Fourier terms, p the periodicity, and az. by are the trainable parameters of the model. In the synthetic example of Figure [ we restrict the NeuralProphet model to two components, namely a weekly and daily seasonality. Our overall reduced model is therefore given by dt = S24n(t) + Sza(t).
If k is sufficiently large, it can cause the frequency ranges of S2t, and S7a to overlap, leading to con- curvity in the model. The default values of k provided by NeuralProphet for S2n. are intentionally kept low to avert such frequency overlap, as demonstrated in Figure (left.. However, this "safety measure" comes at the cost of prediction accuracy due reduced model complexity.
Individual soods Seed average
10"
and S7a = 3) 0.15
0.10
10"
10"
0.00
0.25
0.50
0.75
1:00
1.50
Analogously to the above toy examples, we present trade-off curve between RMSE and concurvity, av- eraging over 10 random initialization seeds per reg- ularization strength A. In this experiment, we choose k = 400 components for both daily and weekly seasonality, to allow concurvity to occur and fit the data almost exactly. Our findings are identical the toy examples, demonstrating a steep decline in concurvity when increasing À with only small increase in RMSE.
Val.
RMSE
Figure 4: Trade-off curve for NeuralProphet model trained on step-function data.
Finally, we note that concurvity can often be identified by visual inspection for additive univariate time-series models as each component is a function the same variable, see Figure In contrast, on multivariate tabular data, concurvity may go unnoticed if left unmeasured and hence lead to false conclusions, as we investigate next.
4.3 Tabular Data
In our final series of experiments, we investigate the benefits of the proposed regularizer when applied to NAMs trained on real-world tabular datasets domain often tackled with conventional machine learning methods such as random forests or gradient boosting. We concentrate our analysis
on six well-studied datasets: Boston Housing 223 California Housing (37). Adult [18], MIMIC-
29 MIMIC-III 27 and Support2 [14] These datasets were selected with the aim of covering
different dataset sizes (ranging between N = 506 for Boston Housing and N = 48, 842 for Adult) as well as target variables (binary classification for Adult, MIMIC-II & -III, and Support2, and regression for California and Boston Housing). NAMs are used throughout the evaluation, subject to a distinct hyperparameter optimization for each dataset; more details are presented in Appendix [.]]] Additionally, provide traditional, spline- based GAM using pyGAM 43) for comparison; details on the HPO for pyGAM can be found in Appendix B.2
First, we explore the trade-off between the concurvity measure R_ and validation fit quality when employing the proposed concurvity regularization. FigureBddisplass the results for the tabular datasets, including the PyGAM baseline. It is clear that the concurvity regularizer effectively reduces the concurvity measure R. without significantly compromising the model fit quality across considered datasets, in particular in the case of small to moderate regularization strengths. For example, on the California Housing dataset, we are able to reduce R, by almost an order of magnitude from around 0.2 to 0.05, while the validation RMSE increases by about 10% from 0.59 to 0.66. Additionally,
7

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Boston Housing
Califoria Housing
Adult
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Figure 5: Trade-off curves between model fit quality and measured concurvity R. for 50 levels of concurvity regularization strength X. Each regularization strength is evaluated over 10 initialization seeds to account for training variability, particularly noticeable in smaller datasets. The results of conventional GAM are shown for comparison.
observe the variation in the scale of R& across the datasets, exemplified by the Adult dataset where the transformed features are nearly decorrelated even in the unregularized case, potentially implying a reduced necessity for regularization. In practice, trade-off curves between concurvity and model accuracy can serve as valuable tool for identifying the optimal level of regularization strength. As expected, PyGAM obtains similar levels of concurvity as the unregularized NAM at equally similar levels of model performance. We refer to Appendix E.3.I E..31I for more verbose trade-off curves.
Case study: California Housing Our preceding experiments demonstrated that concurvity reduc- tion be achieved when training NAMs tabular data. However, the practical significance of this observation in relation to interpretability remains unclear so far. To address this, we perform a more detailed analysis of NAMs trained on the California Housing dataset (see Appendix E.3.3 for similarly detailed results on the other datasets). In the following analysis, we compare NAMs trained with and without concurvity regularization. More specifically, we evaluate based on Figure 5 and À = 0.0 both for 60 random weight initializations.
0.1 (determined
First, assess the effect of the regularizer on the model fit, finding that regularization increases the mean test RMSE by about 10% from about 0.58 to 0.64 and slightly decreases the spread between the seeds, shown in Figure 6d Note that the result in the non-regularized case is par with the original NAM evaluation serving as a sanity check of our experimental setup.
Second, we juxtapose the feature correlations of non-linearly transformed features for models trained with and without regularization. The results, as displayed in Figure 6aa (upper right triangular matrices), are contrasted with the raw input feature correlations (lower left triangular matrices). It is evident that without regularization, high input correlations tend to result in correlated transformed features,
in the left correlation matrix. Conversely, the right correlation matrix reveals that concurvity regularization effectively reduces the correlation transformed features. This effect is especially pronounced for previously highly correlated features such Longitude and Latitude, or Total Bedrooms and Households.
Third, we investigate how concurvity impacts the estimation of the individual feature importances, which of key interest for interpretable models such as NAMs. Following [2], we measure the where Te denotes the average of the shape function f;
importance of feature as N Eymls((())))) over the training datapoints. We visualize the distribution of feature importances over our regularized and unregularized ensembles of NAMs in Figure 6b| It is apparent that feature importances tend to have a larger variance in the unregularized case compared to the regularized case, pattern which is
We emphasize that the main purpose of Figure 6b is not a direct quantitative comparison of feature importances between different regularization levels but the discovery of qualitative insights.

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regularization W/ regularization
Householdk
Longitude
Lattrade
Housing modiám age
Housing
Latirade
Total halrooms
0.0
Longitado
Population
Househalds
-0.5
Modian
Mulan income
1.0
Ocean proximity
Longtine
Housingg medien ago
Total roonts
Towi butrromms Populatto
Houscholi
Median incommo
Lompinn
L
Housing mediaa
Tonal poom
Total bacroon
Popeiin
Houssbalds
Medius incoan
Population
Reg. str.
Total badrooms
0.0
Total
Importance
(a) Average feature correlations the features (lower (b) Combined box and strip plot the models' feature left) and non-linearly transformed features (upper importances. right).
Contribution [100k $]
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(c) Average and individual shape functions of 3 out of the selected features. A kernel density estimate of the training data distribution is depicted on top. The results for pyGAM are shown for comparison, including 95% confidence intervals.
(d) Test RMSES.
Figure 6: Results for the California Housing dataset. The considered NAMs were trained with and without concurvity regularization using 60 model initialization seeds each.
particularly clear for the strongly correlated features which we identified in Figure 6a Such variance in feature importance can detrimentally impair the interpretability the models, due to potential inconsistencies arising in absolute importance orders. However, our proposed concurvity regularizer effectively counteracts this issue, resulting in more consistent and compact feature importances across different random seeds. With regards to the varying effect of regularization features, two observations are particularly interesting: (1) Features that are mostly uncorrelated
the respective
remain unaffected by the regularization an effect we have previously seen in Toy Example which can, for example, be observed in the case of the Median income feature. (2) Input correlations lead bi-modal distribution in the corresponding feature importance as, for example, observable in the case of the Longitude and Latitude or Total bedrooms and Households features. Similarly to Toy Example 1, we see that the regularizer encourages feature selection for correlated feature pairs.
Finally, to visualize the impact of the regularization on model interpretability in more detail, the shape functions of three features shown in Figure 6c Here, the features Households and Population are strongly negatively correlated (see Figure 6a) which leads to their feature contributions largely canceling each other out. This problem is effectively mitigated by the proposed regularization, revealing naturally low contributions for both features. For comparison, the contribution of the mostly non-correlated Median income feature remains virtually unchanged by the regularization. Due to the concurvity in the data, the shape functions of the PyGAM model exhibit large variance which impedes their interpretability [40]. similar behavior can be observed for the remaining feature contributions, which are depicted in Figure [ in Appendix E.3.3
compare concurvity regularization to classical L1 regularization on the California Housing dataset. We find that L1 more strongly prunes features and removes finer details of each feature's contribution, which more gently preserved for concurvity regularization.
In supplementary experiment in Appendix E.3.4
In summary, our case study on the California Housing dataset establishes that concurvity regulariza- tion significantly enhances interpretability and consistency GAM in terms of shape functions and feature importance, whilst maintaining high model accuracy.

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5
Related Work
Classical works on concurvity in GAMs The term concurvity was first introduced in 9 for well-written introduction to concurvity, we refer the reader to [40]. Numerous subsequent works have developed techniques to address concurvity, such as improving numerical stability in spline-based GAM fitting [25] 48 and adapting Lasso regularization for GAMs [.] Partial GAMs 21 were proposed to address concurvity through sequential maximization of Mutual Information between response variables and covariates. More recently, [28] compared several feature selection algorithms for GAMs and found algorithms selecting a larger feature set to be more susceptible to concurvity,
property first noticed in (2). In addition, [28] proposes novel feature selection algorithm that chooses a minimal subset to deal with concurvity. We refer to [28] for a thorough comparison of different concurvity measures. In contrast, our proposed regularizer adds no additional constraints
the feature set size and does not explicitly enforce feature sparsity.
Recent advancements in neural approaches to GAMs, such as Neural Additive Models (NAMs) and NeuralProphet [46), have provided more flexible and powerful alternatives to classical methods [24). These have spurred interest in the subject leading to
Modern neural approaches to GAMs
several extensions of NAMs 11 19 39]. Our approach is compatible with the existing methodologies and can be readily integrated if they are implemented in automatic differentiation framework.
Regularization via decorrelation Similar types of decorrelation regularizers have previously been proposed in the machine learning literature but in different contexts. [33] found that regularizing the pross-covariance of hidden activations significantly increases generalization performance and proposed DeCov. OrthoReg [4] was proposed to regularize negatively correlated features to increase generalization performance by reducing redundancy in the network. Similarly, [49] proposes to add a regularizer enforcing orthonormal columns in weight matrices. More recent approaches, such as Barlow Twins [5I), leverage decorrelation as a self-supervised learning technique to learn representations that are invariant to different transformations of the input data.
Conclusion
In this paper, have introduced differentiable concurvity regularizer, designed mitigate the often overlooked issue of concurvity in differentiable Generalized Additive Models (GAMs). Through comprehensive empirical evaluations, we demonstrated that regularizer effectively reduces concurvity in differentiable GAMs such as Neural Additive Models and NeuralProphet. This in turn significantly enhances the interpretability and reliability of the learned feature functions, a vital attribute in various safety-critical and strongly regulated applications. Importantly, our regularizer achieves these improvements while maintaining high prediction quality, provided it is carefully applied. We underscore that while the interpetabbilty-accure trade-off is an inherent aspect of concurvity regularization, the benefits of increased interpretability and consistent feature importances across model initializations are substantial, particularly in real-world decision -making scenarios.
Nonetheless, our study is not without its limitations. The validation of our approach, while diverse, was limited to three real-world and three synthetic datasets. As such, we acknowledge that our findings may not fully generalize across all possible datasets and use cases.
An intriguing avenue for future work could be to examine the impact of our regularizer on fairness in GAMs. While prior work [12) suggests that GAMs with high feature sparsity can miss patterns in the data and be unfair to minorities, our concurvity regularizer does not directly enforce feature sparsity. Thus, comparison between sparsity regularizers and our concurvity regularizer in unbalanced datasets would be of high interest. addition, future work could explore how the joint optimization of concurvity and model fit could be improved by framing it as multi-objective problem. Finally, our concurvity regularizer could be adapted to differentiable GAMs that incorporate pairwise interactions. Specifically, it would be interesting contrast such an extension with the ANOVA decomposition proposed in 30 in terms of single and pairwise interactions.
We conclude by encouraging researchers and practitioners to "curve your enthusiasm" - that is, to seriously consider concurvity GAM modeling workflows. We believe this will lead to more inter- pretable models and hence more reliable and robust analyses, potentially avoiding false conclusions.
10

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Acknowledgements
This work has received funding from the German Federal Ministry for Economic Affairs and Climate Action as part of the ResKriVer project under grant no. 01MK21006H. We would like to thank Roland Zimmermann, Fabio Haenel, Aaron Klein, Christian Leibig, Felix Pieper. Sebastian Schulze, Ziyad Sheebaelhamd, and Thomas Wollmann for their constructive feedback in the preparation of this paper.
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A Additional Remarks and Theoretical Results for the Proposed Regularizer
A.1 The Decorrelation Space Hi Rules Out Concurvity
In this section, we formalize our claim from Section 3 that the space H. provably rules out concurvity. In the following, (+-) and || 222 denote the Euclidean scalar product and norm, respectively. For v (vi....,20 uN) ) E R we set 8 == N Σ 2i and denote the all-one vector by = (I...., 1) R The standard (empirical) Pearson's 's correlation coefficient is then given by
(o- 11.-ww.00 1833 v- 222 w-
o and w are non-constant, otherwise,
Corr(e, w) =
U, t E RN.
00
Lemma A.1. Let XI..... Xp E TRN be a set of feature variables with p > 1 and let H C (UI....;fp) fa IR + R) be class of functions. Consider the following subclass of:
H (Vi,..., H Corr(J((X.);
0 for all Hj}CH.
Then we do not have concurvity w.r.t X1,..., Xp and H..
Proof. Towards a contradiction, assume that there are 9p)) E My and co E R such that col + E (Xx, O.
We set U. = g:(X,) and trivially add the mean vectors to the linear combination:
P
p
0 col Σ v. co
Σ
04
U2 0,1).
Using that Corr(u;; 13)) for + j. we then obtain
P
0 (0, Vj 1 (co + Σ B. (1. 0,1)
Σ (u -
p
(co
U (No, No,) + +10, 0, ||0 6,11
We conclude that vy Fy 1, i... By is constant vector. But this contradicts the definition of H. because we would have Corr(U,, 1y) = oo with any + j.
A.2 Additional Remarks on Concurvity and our Regularizer
(1) The definitions of multicollinearity (Definition 2.1 and concurvity (Definition 2.2 are based on a fixed (deterninistic) feature design, but one could also formulate probabilistic versions, cf. [44). The latter typically facilitates theoretical analysis, which, however, is not the focus of our work. Moreover, a probabilistic definition would not cover an important practical source of multicollinearity namely underdetermined systems where N < p.
(2) Although closely related, multicollinearity does not necessarily imply concurvity and vice versa. Indeed, one can easily come up with setups where perfectly correlated features become decorrelated after a non-linear transform. Similarly, uncorrelated input features can be made perfectly correlated with non-linearity Two simple (toy) examples are presented in Section 4.1
(3) Our concurvity regularizer R; does automatically affect the predictive performance of GAM. For example, assuming that the input features are drawn from stochastically independent random variables, since non non-linedd linear transforms of independent random variables remain independent. Consequently, we have that R1 (. (X)).) 0, so that no (in this case undesirable) regularization takes effect.
can conclude that Corr(f((X,).
(X₃)) for large enough sample size N,
"The special case of constant vectors could be treated differently, e.g., by setting the correlation to 0. The version we use here is most convenient for our purposes as it excludes the treatment of additional special cases in Lemma
14

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B Hyperparameter Optimization
B.1 Tabular Datasets
For hyperparameter optimization we use Tree-structured Parzen Estimator (TPE) 7 as implemented in Optuna [3]. We run the optimization for budget of 500 function evaluations and optimize w.r.t. validation RMSE for Boston Housing and California Housing or validation binary cross entropy loss for Adult, MIMIC-2, MIMIC-3, and Support 2.
The hyperparameter space and default parameters are shown in Table and the hyperparameters per dataset are shown in Figure 7
Hyperparameter
Value / Range
Scaling
Learning Rate
[lc-4, le-1]
log
Weight Decay
[1e-6, 1]
log
Activation
[ELU, GELU, ReLU]
cat.
# of neurons per layer
[2, 256]
linear
# of hidden layers
[1, 6]
linear
Num. Epochs
[10, 500]
linear
Table Hyperparameter Search Space
Hyperpeaasker
Vilu:
Hyperpraneter
Vilue
Ryperaraneee
Vilue
Reperpennnte
Ville
Leartng
Ic-3
Leanting
7
Leanus
D-4e-3
Leasise
2
Weigh Decay
0.0
Daay
Activation
GRI
Actossion
B1.
Actioation
ReLl
Actisatinn
GBLD
ecances per layer
178
#dd reanons per layer of hiskn lagen
of nosees per layer
72
of aranm per boyer
204
oo hidees loes
hakden bpen.
hitden Layers
Epocls
Epocts
Epocss
Buach Soz
128
Bach
128
Baoh
512
Boii
Contiiion Devomitator:
16-12
Corclation Dasomicator:
1e:12
Condatos Decomiato
1e-12
Conolition Desonitatoe
lell2
Stort Con.. Reg. after x of stgs
Stat Coo Reg x of stps
Stan Core. Reg after x5 stgs
Stat Cono. Reg aft x sugs
(a) Toy Example 1&2
(b) Boston Housing
(c) California Housing
(d) Adult
Peperpeannter
Vilue
Hyperpaannter
Value
Hyporpannnterr
Value
Lousing
Woigh
1
Weigtt
1,06-22
Adtivtioa
GELU
Activaion
GELU
Activation
GBLU
of acorous per lye of bizskn layrn
190
of aearon per loyer
a acuicce pei layer
127
hideen byer
hadera byers
Batch Saz
312
Baich Size
512
Baah Siz
S12
Genslation Desoriinttrr
le-12
Consatio Denominater
Io-12
Condaiion Desomisarr
Ie-12
Sun Conn Bay
aeps
Stant Crne. Rep, after x alpps
Start Cane. Rez, alter aleee
(e) MIMIC-II
(I) MIMIC-III
(g) Support2
Figure 7: Hyperparameters per dataset for NAM experiments.
B.2 PYGAM
We optimized the second-order derivative penalties on the spline functions in PyGAM 443 using random search. We used budget of 300 function evaluations within a search range of [le-3, 1e3).
C Experimental Details
C.1 NAM Experiments
In all of our NAM experiments, we use the Adam optimizer 33 and adjust our learning rate using Cosine Annealing (32 and decay to O. For all regression problems
the Mean Squared Error (MSE) and for all binary classfication problems the Binary Cross Entropy (BCE) as loss function L.
In all of our experiments, we apply the concurvity regularization after warm-up phase consisting of 5% of the total optimization steps for stability reasons. After this warm-up phase, regularization performed at each step.
15

---

C.2 Toy Examples
We sample 10,000 datapoints from the model and use 7,000, 2,000, 1,000 for training, validation and testing respectively. We use the validation split to find adequate hyperparameters via small manual search. Our NAM has 3 layers per feature NN with 128 hidden units and uses the GeLU [26] activation function. We use no weight decay, train for 50 epochs at an initial learning rate of 1e-3 and a batch size 128. Otherwise, the experimental setup is the same as described earlier in Section C.1
D Dataset Details
The Boston Housing Dataset [23], compiled by Harrison and Rubinfeld in the 1970s, is benchmark dataset employed in machine learning and statistical modeling for housing price prediction. Consisting of 506 neighborhoods in the Boston metropolitan area, it features 13 attributes, including crime rate, zoning information, industrial acreage, Charles River proximity, air quality, housing characteristics, accessibility to employment centers, highways, and education, as well as demographic factors. The primary objective to predict the median value of owner-occupied homes using these features.
Boston Housing
California Housing The California Housing Dataset 37 a widely-used benchmark dataset for machine learning and statistical modeling, particularly in the domain of housing price prediction. Originally derived from the 1990 California Census, it consists of 20,640 samples, each representing census block group. The dataset contains information as median income, housing median age, average number of rooms, average number of bedrooms, population and average household size. It also includes the geographical location (latitude and longitude) of each block group. The objective is to predict the median house value. We obtained the dataset from [18]
various housing- attributes, such
Adult The Adult Dataset [8] is also known as the "Census Income" dataset. Extracted from 1994 United States Census Bureau data, it comprises 48,842 records, each representing an individual. The dataset contains 14 features, including age, work class, education, marital status, occupation, relationship, race, sex, capital gain, capital loss, hours worked per week, and native country. The objective is to predict whether an individual's 's annual income exceeds $50,000.
MIMIC-II
The MIMIC-II (Multiparameter Intelligent Monitoring in Intensive Care) dataset 29 a public database, offering clinical data from multitude of Intensive Care Unit (ICU) patients. is managed by the MIT Lab for Computational Physiology and encompasses wide variety of data points, such as patient demographics, vital signs, lab test results, medications, procedures, caregiver notes, and imaging reports, well mortality rates both in and out of the hospital.
MIMIC-III MIMIC-III (Medical Information Mart for Intensive Care III) (27 is the successor to the MIMIC-II database. It contains additional, more recent patient records and provides more detailed data, including free-text interpretation of imaging reports, allowing for more granular research and
improved application in areas like machine learning, health informatics, and predictive modeling.
SUPPORT2 The Support2 [4] (Study to Understand Prognoses and Preferences for Outcomes and Risks of Treatments) dataset is a clinical database that contains detailed medical information from large cohort of seriously ill hospitalized adults. The dataset created part of multi-center study designed to understand the outcomes of decisions made in the course of medical treatment. provides extensive variables, including demographic data, physiological measurements, diagnostic information, treatment plans, and outcomes such as survival and quality of life. The data collected span a diverse set of medical conditions, making the SUPPORT2 dataset a valuable resource for researchers seeking to study clinical decision-making prognosis evaluation, and healthcare outcomes.
16

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E Additional results
E.1
Toy Example 1
Figure shows an additional scatter plot comparing the contributions of the transformed features.
691
032
300
6:3
+15
(a) w/o concurvity regularization
(b) w/ concurvity regularization (A = 0.1)
Figure 8: (Toy Example 1) Pair plot demonstrating the difference in shape functions learned without and with concurvity regularization. The features X and X2 are fully correlated in this example. Each subplot is generated by scatter plot over a large set of random samples from the input feature pair X2). As expected, the model trained without regularization shows strong (almost perfect) linear correlation between the transformed features fi (X1), fx()) and the target variable Y = X1, while our regularizer effectively decorrelates (X, and f2(X)). But note that (X,) and (X2) are not stochastically independent, as the plots in (b) clearly show non-linear functional relationship between the two features.
17

---

E.2 Additional Toy Example from [28
In response suggestion to include more complex toy example during the review process, we have replicated the toy example from (28 using our NAM setup. To recap, this example contains 7 input features:
X1 X2 X3 U(,,)
X4 X3 +
X5 X3 oN(0,)))
X6 = X3 +x- N
X7 X1 .x2 01),
Y 2x + X3 + 2 ((),
where the standard deviations are sufficiently small to create severe concurvity among the features (a) = 0.05, 02 0.5). We simulated 10, 000 data points from this model and created a 7:3 train/test split. We fitted 20 random initializations of NAM in the unregularized, concurvity-regulaizeel and I.1 regularized setting. The regularization parameter was determined separately for each regularization type based trade-off curves. For concurvity regularization, used X 0.1, and for LI, we used 0.05.
The results, reported as the 2² on the test set, are reported in Table 2 with the top three rows from
28 for comparison. Confidence intervals the mean are estimated on 10, 000 bootstrap samples. Features are presented in descending order of their importance (as defined in the main paper) for the best-fitting model in each setting, while the actual importances are reported below.
Model
Selected features
!l (%) (5%, 95%)
R_ (5%. 95%)
Full model 28
Full model
84.99
Stepwisc 28
X1, Xa
85.11
Hybrid
X1, Xx
85.31
Unregularized (ours)
X, Xg. N4. X2, Xs. X3.
80.77 (80.31 / 80.95)
0.22 (0.20 /0.23)
+ Feature importance
0.129, 0.097, 0.066, 0.053, 0.043, 0.013 0.004
Concurvity Reg. (ours)
X1.Xa Xo. X2. X4.
79.28 (78.52 79.88)
0.03 (0.02 0.003)
Feature importance
0.132, 0.125, 0.088, 0.070, 0.002, 0.002
LI Reg. (ours)
X X1, Xs. Xr. X,, X3, X₂ 0.147, 0.106. 0.037, 0.009, 0.008, 0.005,
79.12, (78.50 79.47)
0,21, (0.20/ 0.21)
+ Feature importance
Table 2: Results of replication experiment of the toy example proposed in 28 All results are calculated over the test split, with our results reported as mean values and confidence intervals. In addition to our three NAM settings (unregularized, concurvity regularization, and regularization), we report the values from the original paper 28 for comparison.
We note that both concurvity regularization and 1.1 regularization correctly identify the three predictive features X1, X5 and X6. on which directly depends. This is not the case without regularization. Furthermore, we find that concurvity regularization effectively reduces R1, unlike L1 regularization.
We also note that the R² values all NAM implementations lower than those reported (28). which we believe is due to the inductive biases in spline-based models being particularly well-suited the mostly polynomial-basee based problem. This notwithstanding, our results further underline the effectiveness of concurvity regularization.
18

---

E.3 Tabular Data
E.3.1 Verbose Trade-off Curves
In addition to the trade-off curves presented in the main text (see Figure 5 Figure shows a more verbose version, separating the visualization for the measured validation concurvity and validation accuracy as a function of the regularization strength A. This explicit visualization allows us for an easier choice of X, for example, by determining the elbow in the validation accuracies, after which stronger regularization would lead to a significant loss of performance. For example, this can be nicely observed in the case of the Boston Housing dataset
1.
Boston Housing
California Housing
Adult
0,100
0.3
0,075
RMSE R,
Val. R
0.050
Individaal sceds
Scad avange
1.0
Val. RMSE [100k
0.9
Val. DCT
0.40
0.8
0.35
0.7
0.6
10
10
10'
Reg. strength MIMIC2
Reg. strength A
Reg. strength A Suppori2
MIMIC3
0.05
0.04
Val. R,
0.05
0.03
Val.
0.03
0.04
0.02
0.02
0.00
0.34
0.30
Val. BCE
Val. BCE
Vol. BCE
0.32
0.46
0.28
0.44
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Figure 9: Verbose trade-off curves between the model fit quality and measured concurvity R for 50 levels of the concurvity regularization strength X. Each regularization strength is evaluated over 10 initialization seeds to account for training variability, which is particularly noticeable in smaller datasets.
E.3.2 Runtime Analysis
Due to calculating pair-wise correlations, our proposed regularizer has a quadratic computational complexity. However, using an efficient implementation leveraging vectorization, the scaling constant can be well controlled. To investigate the computational overhead of our regularizer, we measure the mean absolute runtime across 100 repetitions for different numbers of input features and batch sizes. Furthermore, we evaluate the relative overhead of our regularizer compared to NAM model. The relative overhead clearly depends on the size and implementation details of the NAM. Here, we compare with NAM for which each feature-dependen network consists of 3-layer MLP with 128 neurons each. The calculation over the features vectorized using batched matmuls. The results are shown in Figure 10 all obtained with MI MacBook Pro. As expected, observe power-law
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scaling of the runtime of the regularizer, which is in particular observable for an increasing numbers of input features. In particular, for moderate numbers of input features between 64 and 256, the absolute runtime is about ms. Compared with the forward pass of 3-layer NAM, the proposed regularizer typically implies a relative overhead of well below 10 Note that the results compare only with the forward pass of the NAM model and do not include data loading, batch collation, loss calculation, and backward passes. We note that especially for larger batch sizes, the relative overhead introduced by our regularizer is well below 5 %. On the other hand, for smaller batch sizes, the relative overhead can become large, due to the reduced effectiveness of vectorization. In absolute terms, however, the overhead is very small for smaller batch sizes well. We therefore conclude that the overhead introduced by regularization is negligible in commonly considered model setups.
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Figure 10: Left plot: Absolute runtime of the proposed regularizer for different numbers input features and batch sizes. Right plot: Computational overhead relative the forward pass of a 3 3-layer NAM, with 128 neurons each, for different numbers of input features and batch sizes.
E.3.3 In-detail Results for the Remaining Tabular Datasets
In addition to the results of our case study with the California Housing dataset in Section 45 we here present the shape functions of the complete set of features, see Figure [1]
Furthermore, we show the analogous results for the remaining datasets Adult (Figure 12 Boston Housing (Figure 13). MIMIC-II (Figure 14), MIMIC-III (Figure 15 & (6), and Support2 (Fig- ure
E.3.4 Comparing Concurvity Regularization to L1 Regularization
Although we chose pyGAM our baseline in the main paper, we have additionally compared our concurvity regularizer LI regularization, as a popular alternative to eliminating redundant features. Results the California Housing dataset are included in Figure 19 We performed LI regularization on the terms w. (x,,) in Equation (GAM), with regularization strength À = 0.1, as determined by trade-off curves on feature LI norms versus RMSE.
The figure suggests that L1 regularization prunes features more strongly than concurvity regularization, which may be desirable in some situations. However, this comes at cost of larger test RMSE, as well as some seemingly predictive features being pruned. For example, L1 regularization removes the two local peaks in the Latitude dimension corresponding to San Francisco and Los Angeles. In addition, unlike I.1 regularization, concurvity regularization does not prune uncorrelated features.
Finally, we note that there may be different motivations behind choosing sparsity regularization (as few features as possible) or concurvity regularization (features become as decorrelated possible). If the main goal is to address concurvity, then sparsity regularization is a blunt tool, and vice versa.
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Figure 11: Shape functions all input features in the case of the California Housing dataset. Depicted the average as well as individual contributions with and without concurvity regularization using 60 initialization seeds each. The PyGAM results plotted for comparison include the estimated 95 confidence intervals.
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ag
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Figure 12: Results for the Adult dataset. The considered NAMs were trained with and without concurvity regularization using 60 model initialization seeds each.
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Figure 13: Results for the Boston Housing dataset. The considered NAMs were trained with and without concurvity regularization using 60 model initialization seeds each.
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Figure 14: Results for the MIMIC-II dataset. The considered NAMs were trained with and without concurvity regularization using 60 model initialization seeds each.

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Figure 15: Results (part 1) for the MIMIC-III dataset. The considered NAMs were trained with and without concurvity regularization using 60 model initialization seeds each.
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Figure 16: Results (part 2) for the MIMIC-III dataset. The considered NAMs were trained with and without concurvity regularization using 60 model initialization seeds each.
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Figure 17: Results (part 1) for the Support2 dataset. The considered NAMs were trained with and without concurvity regularization using 60 model initialization seeds each.
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Figure 18: Results (part 2) for the Support2 dataset. The considered NAMs were trained with and without concurvity regularization using 60 model initialization seeds each.
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Comparison of concurvity and LI regularization on the California Housing dataset. The considered NAMs were trained using 60 model initialization seeds cach. Note that for LI, the regularization strength is set to À = 0.1.
Figure 19:
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Reviews:
[{"review_id": "MxmqXFp4G6", "reviewer": "Reviewer_5zAd", "summary": "The paper proposes a regularization using pairwise correlations of shape functions for differentiable GAM, aimed at reducing concurvity between shape functions", "questions": "1.\tIn Figure 2(a), we see the model with regularization chooses a random point on a curve. This is because both features are identical in the setting, making the model unidentifiable. What if the two features are correlated but not perfectly correlated? Is the regularizer able to identify the true predictive variable X1, and not so affected by random seeds?\n2.\tHow to determine a proper level of the strength of regularization (lambda)?", "limitations": "yes", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "1.\tThe idea of the paper is well-explained and straightforward.  The correlated terms can be self-cancelled to avoid to unnecessary complexity. So reducing concurvity may potentially enhance generalizability. \n2.\tThe paper’s story, theory and the numerical experiments all demonstrate that regularization effectively controls the correlations of the non-linearly transformed features.\n3.\tThe proposed component is easy to implement and optimize and as far as I understand, it can be incorporated into any differentiable GAM. \n", "weaknesses": "1.\tThe foundation of this paper – the reason why correlated terms should be cancelled out is still beyond me. For the time series scenario in the introduction, I understand that high frequency terms of feature pairs hinder interpretation. But what if the correlation does lie in the ground truth? Especially for the tabular data, the visualized results in the public dataset show that most of the features have almost no prediction power for the target variable (e.g. Population, households, housing age before 30 years, total bedrooms and total rooms, etc.), which seem to be counterintuitive. Little concurvity is allowed, but it happens in the real world. Is it possible that restricting concurvity may in general push the shape functions away from their natural relationship (trained to be not effective but actually affects the target)? The author may provide further evidence and explanation regarding the negative effects of concurvity, to make the article more convincing. \n2.\tWhile previous literature found that regularizing the cross-covariance increases generalization performance, the numerical results presented in this paper indicate that controlling the concurvity will sacrifice the generalization ability anyhow. The accuracy is slightly worse than the one without regularization, in exchange of the stated ‘interpretability’, which unfortunately is not straightforward in the figures. Therefore, it appears that the trade-off between accuracy and correlation may not be worth it in this case.\n3.\tThe paper showcases the motivation using a time-series example, but no visual results for time series are presented to address the issue. It is doubtful whether the regularization does control the frequency or some uninterpretable parts, rather than merely force terms to be uncorrelated (in an unnecessary manner).\n4.\tIt is not shown how to do the optimization. Is the calculation of regularization time-consuming and unstable? What would happen if the regularization is implemented in every iteration rather than only 5% of the total optimization steps as introduced in the article? It seems a mismatch between the design and the realization.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer 5zAd,\n\nThank you for your comprehensive review and constructive feedback on our paper. We appreciate your recognition of the clarity of our idea, the effectiveness of our regularization in controlling the correlations of the non-linearly transformed features, and the ease of implementation and optimization of our proposed component.\n\nWe have carefully considered your concerns and questions and have addressed them as follows:\n\n**Weaknesses:**\n1. On the concern about the foundation of the paper and the negative effects of concurvity: Thank you for you for this insightful comment. Since related points were brought up by the other reviewers as well, we have added a global statement; please see comment C1. Regarding your specific concern: While we agree that ground-truth input features often exhibit natural relationships due to correlation, we do not think that the behavior of our regularization approach is counterintuitive. In fact, the idea of “suppressing” redundant features is well-established in the field of feature selection and widely-accepted in the community. These modes intend to provide a particularly simple predictive model, relying on as few input features as possible. However, this does not mean that the other features have no predictive power (they almost always have). Our approach aligns well with this approach, as it provides a particularly simple model by decorrelation the target features.\nHaving said this, we agree that it is useful to provide more evidence that ignoring concurvity can be very problematic. To this end, we have added a spline-based GAM (via pyGAM (Servén et al. 2018)) as a baseline to our experiments (see also global comment C3), which demonstrates that the resulting shape functions exhibit a large variance – an issue commonly reported in the concurvity literature (please see the review pdf). In other words, the shape functions do not exhibit a natural relationship, since the space of shape functions is degenerate with possibly infinitely many equivalent solutions. We demonstrate that our method can greatly reduce this ambiguity, and in that sense, make the prediction models more interpretable.\nWe hope that this clarifies your concern about the foundation of our approach and we are happy to make this point clearer in the camera-ready version.\n\nServén D., Brummitt C. (2018). pyGAM: Generalized Additive Models in Python. Zenodo. DOI: 10.5281/zenodo.1208723\n\n2. On the trade-off between accuracy and correlation:\nWe are assuming that you are referring to the work by Cogswell 2016 by “regularizing the cross-covariance increases generalization performance”. While the work by Cogswell et al. indeed proposes a decorrelation approach, it operates in a fairly different regime, namely decorrelating _hidden representations_ to reduce overfitting of deep neural networks.  Although regularization often improves generalization of deep neural networks by reducing overfitting, we did not expect the same improvement in generalization in GAMs since overfitting is typically not a major issue in this type of models. Rather, regularization is employed here to constrain the properties of the model with the aim of improving interpretability.  The trade-off between accuracy and concurvity mirrors the well-established sparsity-accuracy trade-off in classical feature selection paradigms, and our experiments demonstrate that the sacrifice in accuracy is fairly small in all considered cases. We are not aware of any previous works where regularization has improved generalization in GAMs; if the reviewer has any such works in mind we would be happy for a pointer to these references. Regarding the gain of interpretability, we kindly refer to our answer to your first concern as well as global comment C1.\n\nM Cogswell, et al. “Reducing Overfitting in Deep Networks by Decorrelating Representations.” In: International Conference on Learning Representations, 2016.\n\n3. On the lack of visual results for time series: \nWe believe this to be a misunderstanding: the example we present in Figure 1 shows actual experimental results obtained for the three settings detailed in Section 4.2. The only learnable parameters of the model are the coefficients of the Fourier terms of which each has a distinct frequency. As a result, the decorrelation does directly influence the dominance of a frequency since the model has no other parameters to adapt. \n\n4. On the optimization process: \nWe thank the reviewer for raising this concern. We elaborate more on the efficiency and scaling of the regularization in the general comment C5. We want to clarify that the regularization is added after a warm-up phase of 5% of the total optimization steps, but afterwards used *in every optimization step*. We will clarify this in Appendix C.1.\n\n**Questions:**\n1. On the questions about Figure 2(a) and what happens if two features are correlated but not perfectly correlated.\nThank you for suggesting this experiment which was similarly suggested by reviewer 17dg. We added an additional result for toy example 1 in Figure 4 of the rebuttal pdf (center row) where the features have a correlation of 0.9. We find that w/o regularization the model converges to the wrong solution in every case while w/ regularization the model converges to the right solution in almost all cases.\n\n2. How to determine a proper level of the strength of regularization (lambda)? We address this in our global comment C4.\n\n\nWe hope that these responses satisfactorily address your concerns and provide further clarification on our work. We are open to continuing this dialogue to further refine our paper. We truly appreciate your insightful feedback and will incorporate your suggestions in the final version of our paper.\n"}, {"review_id": "u1zOjR7p0O", "reviewer": "Reviewer_17dg", "summary": "Generalized additive models (GAMs) offer greater interpretability than other machine learning methods while having greater flexibility than generalized linear models. This paper addresses how to account for concurvity (the non-linear analog to multicollinearity in linear regression). The authors suggest directly penalizing correlations between the features during training. Concurivity poses problems with interpretability (which (transformed) feature to attribute the contributions to) and can increase the variance in the model. The proposed regularizer was found to reduce concurvity while maintaining strong predictive performance. The viability and success of this method are shown in time series forecasting as well as tabular regression tasks. ", "questions": "- How much do the $f_i$ differ depending on $\\lambda$? How fair is it to compare feature importances across different levels of regularization if the resulting $f_i$ are not the same? (Possibly addressed in figure 6.c?)\n\n- line 126.5: in the $gam-fit_\\perp$ what is the $l$ index for? It does not show up in any of the other terms\n\n- What is the additional computational burden of computing the regularizer? How well does this scale?\n\n- How does gam-fit perform in the presence of covariates that are neither perfectly correlated nor entirely uncorrelated (as they were in the toy examples)?", "limitations": "Not applicable", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- Clear motivation for the problem and general organization of the paper\n- Method is light on assumptions regarding the feature transformations, $f_i$ so this can be applied in many settings\n- Improves model interpretability while reducing variance in fitted functions", "weaknesses": "- The figures were very information dense (particularly fig 6). More discussion of these results and how to interpret them would aid the reader\n- Not much space was spent on the time-series experiments relative to how much was dedicated to it in the introduction\n\n\nMinor\n- line 75: (linear) regression\n- line 83: minimize over $\\beta$ as well?\n- line 126: minimize over $\\beta$?\n\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer 17dg,\n\nThank you for your constructive feedback and recognition of the clear motivation for our problem, the broad applicability of our method, and its contribution to improving model interpretability and reducing variance in fitted functions. We have addressed your concerns and questions as follows:\n\n1. On the information density of the figures:\nWe agree that, given the space constraints of the submission, some figures contain quite a lot of insights. We happily extend our discussion in the camera-ready version.\n\n2. On the space dedicated to time-series experiments: \nThe main reason why we did not devote more space to time-series experiments is their “degenerated” nature in the context of GAMs. Indeed, in the NeuralProphet example, time is used as the only input feature for all additive components, so that perfect multicollinearity is present. While our approach can be very useful in such scenarios, it should be considered an extreme case of our tabular data experiments, where the input feature relationships are more complex.\nOn the other hand, the special form of the time-series example in Fig. 1 makes it intuitive and therefore well-suited for an introduction to our approach without prior knowledge of concurvity. Hopefully, this clarifies the mismatch in dedicated space. We are happy to point out this aspect in the camera-ready version.\n\n3. On the minor errors:\nWe appreciate your attention to detail and have adjusted the paper accordingly.\n4. On the difference in $f_i$ depending on $\\lambda$:\nWe fully agree that this is tricky in general. As also addressed in our global comment C2, quantifying increased interpretability is challenging and there exists no gold standard yet. We choose to compare the shape functions $f_i$ and aggregated feature importances of NAMs trained with and without concurvity regularization in our case study. After all, the shape functions describe the prediction mechanism of GAMs and are used to gain insights or even guide practitioners in making decisions alongside the aggregated feature importances. The main purpose of Fig. 6(b) is therefore not a direct quantitative comparison of feature importantes between different regularization levels, but it should be rather used to extract qualitative insights, such as larger variances or bi-modality. We are happy to clarify this in the camera-ready version. \n5. On the additional computational burden of computing the regularizer: \nWe address this in our global remark C5. We will clarify the computational scaling in more detail in the camera-ready paper.\n6. On the performance of gam-fit in the presence of covariates that are neither perfectly correlated nor entirely uncorrelated: \nWe thank the reviewer for suggesting this experiment which was similarly suggested by reviewer 5zAd. We added an additional result for toy example 1 in Figure 4 of the rebuttal pdf (center row) where the features have a correlation of 0.9. We find that w/o regularization the model converges to the wrong solution in every case while w/ regularization the model converges to the right solution in almost all cases.\n\nWe hope that these responses address your concerns and we are open to further discussions. We appreciate your feedback and will make the necessary adjustments in the camera-ready version of the paper.\n"}, {"review_id": "MWh3zsLmll", "reviewer": "Reviewer_aFW2", "summary": "The authors highlight concurvity as a relevant concern when developing additive models. They introduce a differentiable regularizer that aims to reduce concurvity (which will in general raise RMSE, though there may be a regularization strength that gives a good trade-off).\n\nThe proposed regularizer is applied to a neural additive model, which is demonstrated on three toy examples (two 2-variable examples where $Y$ equals one of the variables, one time series example with different weekly and daily step functions) and three UCI data sets (2x regression, 1x classification).\n", "questions": "### Substantial questions:\n\nQ1. Time series with multiple seasonalities are in some sense a “degenerate” example as all functions depend on the same feature $t$. How, if at all, would time series have to be treated differently from “regular” data sets with multiple input features $X_1, \\dots, X_p$?\n\nQ2. Definition 2.2: you claim this “revised formal definition” as a main contribution of your work. How is it different from (and what are the similarities to) previous definitions?\n\nQ3. Section 3: your description sounds like concurvity is purely a *model* issue, but is it not a property of the *data*?\n\nQ4. Proof in Appendix A.1: this seems to hinge on your definition of correlation which assigns $\\infty$ to the correlation with a constant vector. However, this correlation is simply ill-defined - and one could equally define it to be zero. Does this not invalidate your proof?\n\nQ5. Your proposed regularizer scales quadratically with the number of additive components: how would you efficiently address this by parallelization? (Does the scaling not remain quadratically?)\n\nQ6. “Our concurvity regularizer is agnostic to the function class …” (lines 140-142): “metrics proposed in the literature [42, 25] are not directly applicable.” [42] (Wood 2001) does not mention concurvity at all and while [25] (Kovács 2022) discusses concurvity, I could not find any mention of score/metric proposed in there. Moreover, what is the similarity of your proposed metric to that of [35] (Ramsay et al)?\n\nQ7.a) Toy Example 2: Fig. 3 (a) suggests that without regularization, the model learns that $f_1(X_1) = 0$ *more* accurately (range of $f_1$ ~ 0.0015) than with concurvity regularization (range of $f_1$ ~ 0.02), how does this support the claimed benefit of regularization?\nQ7.b) How do you see that $f_2$ approximates $|f_1|$ (lines 192-193)?\n\nQ8. lines 238-241: could it be that this is due to it being classification rather than regression?\n\nQ9. You discuss the distinction between multicollinearity and concurvity. Is it possible to have multicollinearity, but *not* have concurvity?\n\n### Notation and clarity:\n\nC10. Regarding the origin of concurvity (lines 100-101, 293), you might want to cite the work by Buja, Donnell, Stuetzle (1986) cited within your reference [9].\n\nC11. The sum $\\sum_{l=1}^N$ in the (GAM-Fit) and (GAM-Fit$_\\perp$ equations seems to be inconsistent with the vector notation (the subscript $l$ is not used anywhere).\n\nQ12. line 85: the notation used in the equation $\\mathcal{H} \\subset \\dots$ is not immediately clear to me. Is $\\mathcal{H}$ supposed to be a space of $p$-tuples?\n\nQ13. “every suitable linear combination of features can be modified by adding a trivial linear combination” (lines 93-94) can you clarify what you mean here? Having gone through some of your references I understand, but your description on its own is rather confusing.\nLikewise, “any non-trivial zero-combination of features can be added to a solution of (GAM-Fit)” (line 109-110): what does this mean?\n\nC14. Simply writing “(GAM-Fit)” e.g. in line 110 made me take a while to realize that this is referring to an equation and to find where it was. How about having the equation label include a number, e.g. “(2; GAM-Fit)”, and then you can explicitly refer to “a solution of Eq. (2; GAM-Fit)” for example?\n\nC15. Additional remarks in Appendix A.2: these are very helpful, would be great to at least summarize the points in the main text.\n\nQ16. Figure 2 (b) is great to show that there is a value of $\\lambda$ that reduces the concurvity measure without affecting RMSE perceptibly, but how sensitive is this to the value?\nThis might be easier to see in two separate plots of $\\lambda$ vs $R_\\perp$ and $\\lambda$ vs RMSE overlaid on top of each other (to identify the range of $\\lambda$ in which both are low simultaneously).\n\nQ17. line 172: what is Fig. 8 supposed to demonstrate? This is not clear to me.\n\nQ18. lines 175, 177: what would be the “scale” of $\\lambda$? What does “moderate” or “considerably high” mean?\n\nQ19. line 243: Are you aware of any other references that discuss the “elbow technique”? Thorndike’s “Who belongs in the family?” does not even mention “elbow”, and I would hope a better reference in this context is available (though it was a thoroughly enjoyable read which I thank the authors for bringing to my attention!)\n\nC20. The references are inconsistent.", "limitations": "The authors acknowledge the limited validation of their approach (though I would not call it \"diverse\"). They also acknowledge that there is an interpretability-accuracy trade-off in concurvity regularization (though I find their demonstration of \"increased interpretability\" lacking).", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The paper provides a nice overview of concurvity and related works and why it is an important issue to be aware of. It introduces a novel regularizer that can easily be combined with any differentiable models.\n", "weaknesses": "The authors include a large set of references to related work on concurvity. Yet there is no comparison to other models (with the same proposed regularizer) or other methods of dealing with concurvity (despite mentioning multiple related approaches).\nThe paper claims that their concurvity regularization encourages feature selection (line 169) but there is no demonstration of this nor comparison to actual feature selection approaches.\n\nVery limited evaluation on real-world data sets; see for example [b] in the same space which has an extensive comparison both on many data sets and across multiple methods. Cited work such as [25] (Kovács 2022) uses toy examples to much better effect; a comparison of the proposed NAM/regularizer approach on the same toy example would make for a much stronger submission.\n\nOf course minimizing a given metric (such as the proposed concurvity regularizer) reduces that same metric, but I am missing a stronger demonstration of the practical benefits/relevance of this. The paper claims the resulting model is more interpretable, but does not really demonstrate this. See [a] for a discussion of interpretability of GAM features.\n\nThough the discussion is all about concurvity and non-linearity, only scalar correlation values are shown in Fig. 6 (a). Pair plots of both $X_i$ vs $X_j$ and $f_i(X_i)$ vs $f_j(X_j)$ might demonstrate the effect of the regularization more easily. Claims such as “their feature contributions largely [cancel] each other out” (lines 282-284) could easily be shown by showing their sum across the data set.\n\n### References:\n- [a] *\"How Interpretable and Trustworthy are GAMs?\"* Chang, Tan, Lengerich, Goldenberg, Caruana (2021)\n- [b] *\"Additive Gaussian Processes Revisited\"* Lu, Boukouvalas, Hensman (2022)\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer aFW2,\n\nWe appreciate the time and effort you have invested in reviewing our submission. Your feedback is valuable and we are happy to clarify and expand on the main points raised in your review.\nDue to the character limit we were unable to include our answers to every question, but we will happily provide them upon additional request. The remaining questions are answered in our global comments. \n\nWe have addressed your concerns as follows:\n\n**Weaknesses:**\n- Regarding the lack of comparison to other models and methods of dealing with concurvity:\nOur regularizer is only applicable to differentiable GAMs, with Neural Additive Models as the most prominent example. On the other hand, previous approaches dealing with concurvity investigate specific feature selection strategies (Kovács 2022), which are not straightforward to apply to NAMs (where all feature functions are fitted jointly). Since our work specifically focuses on differentiable GAMs, we have intentionally not included a direct comparison with previous concurvity reduction techniques. We are happy to point this out more clearly in the camera-ready version. Nevertheless, we agree that reporting a classical GAM baseline is useful. Therefore, we have added a spline-based GAM (via pyGAM (Servén et al. 2018)) as a baseline to our experiments. Please see also our global comment C3.\n\n- On the lack of demonstration of the claimed feature selection: \nIndeed, the concurvity regularizer encourages feature selection where features are strongly correlated, as demonstrated in Fig. 2a and 6. While feature selection is a possible consequence of concurvity regularization (when features are correlated) it is not the main aim and hence we do not compare with sparsity seeking methods, as detailed in our global comment C1.\n\n**Substantial questions:**\n1. Indeed, we do not need to treat time series and regular tabular data differently in our approach. However, since time series form a data modality of independent interest, we have decided to put this example into a separate part. But we agree that it can be also considered a “degenerated” extreme case of our tabular data experiments.\n2. Previous works do not give a precise definition. Our definition is loosely inspired by the one of [Ramsay et al., 2003], with an important difference: Ramsay et al. only consider model spaces of a cartesian form $\\mathcal{H} = \\mathcal{H}\\_1 \\times … \\times \\mathcal{H}\\_p$, while we allow for any subset of $p$-tuples. This might appear like a minor detail but is the key to our insights in Sec. 3 and theoretically justifies why our regularizer is useful. In particular, spaces like $\\mathcal{H}\\_\\perp$ would be incompatible with previous definitions. \n3. Yes, one could say that concurvity is primarily a model issue. As we show, restricting the model space to  $\\mathcal{H}\\_\\perp$ in Def. 2.2, concurvity can be ruled out, regardless of multicollinearity in the data. But of course, the data also plays an important role. For example, if the inputs $X_i$ are stochastically independent, so are any non-linear transformations. This implies that the regularization term is zero and does not affect the GAM-fit at all (see Remark (3) in Appendix A.2). \n4. The assignment of infinity for constant features is just for convenience and we agree that a minor adaptation of the proof would be required for a different convention. This could be addressed by adding another constraint to $\\mathcal{H}\\_\\perp$, excluding constant features. However, this technicality might cause confusion in the main body. Hence, we decided on the simplified version and address the ill-definedness of the correlation in Footnote 5.\n5. We elaborate on this issue in general comment C5. \n6. We agree that the paper introducing mgcv [42] (Wood 2001) does not mention concurvity, however, the concurvity indices are implemented in mgcv and are described in its manual. Hence, we find the reference appropriate. (Kovács 2022) has a good description of the concurvity indices implemented in mgcv on Page 7. Regarding the similarity of our metric compared to Ramsay et al.: They compute for each feature the correlation between $f_i$  and the sum of all other functions (excluding $f_i$), whereas we compute the pairwise correlation between all $f_i$ and $f_j$.\n7. a.) We agree that the range of $f_1$ is closer to zero in the unregularized case; however, we also find that the unregularized model has introduced an almost perfect correlation between $f_1(X_1)$ and $f_2(X_2)$, which isn’t present in the data. Note that both models have the same validation RMSE. b.) From Figure 3 (a) one can see that $f_2(X_2)$ approximates $|f_1(X_1)|$ up to an affine transformation.\n8. $R_\\perp$ is measured in the (cartesian product) target space, so the regression target space or the space of raw logits in the case of classification. In both cases, the scale will depend on the target specifics and not on the target type. That is, generally, this does not result in $R\\_\\perp$ being on a much smaller scale in classification as opposed to regression tasks.\n9. Yes,an example is the NeuralProphet experiment in Fig. 1, where time is used as input feature for all input components, implying perfect multicollinearity. Our approach demonstrates that the non-linearly transformed features can be decorrelated anyway. More generally, the idea of ruling out concurvity in the presence of (perfect) multicollinearity was precisely our motivation when deriving our regularizer in Section 3. \n\nServén D., Brummitt C. (2018). pyGAM: Generalized Additive Models in Python. Zenodo. DOI: 10.5281/zenodo.1208723\n\nLászló Kovács. Feature selection algorithms in generalized additive models under concurvity. Computational Statistics, pages 1–33, 2022.\n\nT O Ramsay, R T Burnett, and D Krewski. The Effect of Concurvity in Generalized Additive Models Linking Mortality to Ambient Particulate Matter. Epidemiology, 14(1):18–23, 2003.\n\nWe appreciate your feedback."}, {"review_id": "oXdsA6UOmr", "reviewer": "Reviewer_arAb", "summary": "The paper concerns the issue of concurvity in the context of Generalized Additive Models (GAMs), which can be considered as an extension of multicolinearity to GAMs. The authors propose a regularization scheme to reduce the concurvity between learned functions, hoping to improve interpretability of the model.  ", "questions": "- Connections to sparse regularization: As pointed out by the authors, the issue of multicolinearity can also arise in the context of linear models. It is well-known that regularization schemes such as sparse regularization can help to deal with correlation. For example, see [1,2]. The notion of variable selection in GAMs is also well-explored, see [3] and references therein. A natural question would be how such penalization schemes deal with concurvity. Intuitively, if one can reduce the total number of features used, a more compact model can be obtained that is less likely to suffer from concurvity.\n\n- How should one choose the parameter $\\lambda$? The authors mention the elbow method but I think this will need more exploration. It seems that the concurvity regularization can reduce validation accuracy. So, if one uses validation performance to choose a model, they'll end up with a model with no regularization. It is not clear to me how a practitioner can use the concurvity regularization if they have to sacrifice accuracy. Going back to the previous point, in the context of linear models, sparsity regularization usually comes with improved out-of-sample performance.\n\n- Extensions: Another interesting direction is that GAMs might be too simple for more complex datasets, and methods that use higher-order interaction terms have been developed [3,4]. Would it be possible to extend the concurvity regularization to such interaction models? This definitely falls out of the focus of this paper, but a short discussion from the authors would be appreciated.\n\n\n\n\n[1] Figueiredo, M., & Nowak, R.. (2016). Ordered Weighted L1 Regularized Regression with Strongly Correlated Covariates: Theoretical Aspects.\n\n[2] Hazimeh, H., & Mazumder, R. (2020). Fast best subset selection: Coordinate descent and local combinatorial optimization algorithms. Operations Research, 68(5), 1517-1537.\n\n[3] Chang, Chun-Hao, Rich Caruana, and Anna Goldenberg. \"NODE-GAM: Neural Generalized Additive Model for Interpretable Deep Learning.\" International Conference on Learning Representations.\n\n[4] Enouen, J., & Liu, Y. Sparse Interaction Additive Networks via Feature Interaction Detection and Sparse Selection. In Advances in Neural Information Processing Systems.\n", "limitations": "n/a", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- Multicolinearity/concurvity are indeed  serious problems in statistics, and any method that can address such issues is of interest to the community.\n- The proposed concurvity penalty is new and has not been discussed before.\n- The numerical experiments show the promise of the proposed methodology.", "weaknesses": "- I think the numerical results although interesting, are rather limited. As far as I could tell, authors only discuss 4 tabular datasets, which is not enough. More datasets and baselines should be added---see [1] as an example.\n\n[1] Chang, Chun-Hao, Rich Caruana, and Anna Goldenberg. \"NODE-GAM: Neural Generalized Additive Model for Interpretable Deep Learning.\" International Conference on Learning Representations.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer arAb,\n\nThank you for the thorough review of the paper, as well as for acknowledging the importance of the issue of concurvity and the novelty of our approach to mitigating this issue. We have addressed your concerns and questions as follows:\n\n**1. On the scope of evaluation:** \nWe address this shared concern in more detail in our general comments C2 and C3. More specifically, as we argue, quantifying interpretability is yet an open question in the literature and a problem for the community to address. As such, we believe that simply evaluating accuracy and possibly sparsity or concurvity measures across a multitude of additional datasets provides limited additional insights.\n\nFor these reasons we favored depth over width in our evaluation and chose a more detailed approach to assessing interpretability, illustrated in Figure 5 and, particularly, Fig. 6. We argue that understanding the impact on interpretability requires specific background knowledge on each dataset as showcased by our investigation of the California Housing dataset, limiting the value of evaluating a large benchmark.\n\nWhile quantifying interpretability is an important open question for the community, we argue that it is important to provide practitioners with tools to potentially deal with limited interpretability in their applications now, instead of waiting for progress on measuring interpretability.\n \t\n**2. On the connection to sparse regularization:**\nWe agree with the reviewer that there is an interesting connection between our proposed concurvity regularization and sparse regularization and elaborate in more detail in general comment C2. However, a detailed comparison between these two regularization paradigms are outside the scope of our paper. For a thorough review of dealing with concurvity in spline-based GAMs via feature selection algorithms we refer the reviewer to Kovács 2022 [1].\n\n[1] László Kovács. Feature selection algorithms in generalized additive models under concurvity. Computational Statistics, pages 1–33, 2022.\n\n**3. On the choice of the regularization strength $\\lambda$:**\nWhile we agree that model accuracy is paramount for a prediction model, there exist additional desiderata for interpretable models. Our concurvity regularizer enforces the requirement to eliminate self-canceling feature contributions thereby helping to avoid drawing false conclusions. While this leads to a slight decrease in model accuracy we feel that this is justified considering the increase in interpretability. Please see our global comment C1 for further elaboration of this argument. We propose to use the elbow method (or L-curve) to estimate the correct level of concurvity regularization but depending on the use-case, one could also choose e.g. a maximal 5% decrease in model performance to find the optimal trade-off. We further address this issue in our global comment C4.\n\n**4. On extending the approach to GAMs with higher-order interactions:**\nWe thank the reviewer for the suggestion. Using concurvity regularization in higher-order interaction models would be straightforward, in the sense that any additional feature term can also be considered when calculating the pairwise correlation. However, it is currently unclear to us what kind of decomposition should be expected. We will be adding the following to the conclusion: “Moreover, it would be interesting to see how the concurvity regularizer works in differentiable GAMs that incorporate pairwise or higher-order interactions. Specifically, contrasting this with the ANOVA decomposition proposed by Lengerich et al. (2020) [2], in terms of single and pairwise interactions, could unveil some interesting insights.”\n\n\n[2] Lengerich, B., Tan, S., Chang, C., Hooker, G. &amp; Caruana, R.. (2020). Purifying Interaction Effects with the Functional ANOVA: An Efficient Algorithm for Recovering Identifiable Additive Models. Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 108:2402-2412\n\n\nOnce again, we sincerely thank you for your constructive feedback and valuable insights. We look forward to the opportunity to further improve our work through continued dialogue.\n"}]

Author rebuttal:
We thank all reviewers for carefully reading our manuscript as well as their thoughtful comments and suggestions. We are happy that the reviewers acknowledge that “multicollinearity/concurvity are indeed serious problems in statistics” (R.arAb) and that our approach is novel (R.arAb), light on assumptions (R.12dg), easy to implement and optimize (R.5zAd), can be combined with any differentiable GAM (R.aFW2 & R.17dg), and that it improves the model interpretability (R17dg). Thus, we believe our contribution to be of value to the NeurIPS community.

We summarize and address some questions shared amongst the reviewers in the following.

### C1. Motivation of our contribution and clarifications on concurvity in general
While all reviewers agree on the efficacy of our regularizer, there were some questions as to the motivation to reduce concurvity, which we address here.

Although concurvity may be a property of datasets (as illustrated in our two toy examples), we argue that it is an undesirable property of a model. Akin to linear models in the presence of multicollinearity, concurvity in GAMs produces a large variance in model fits (see Fig. 2c att. pdf) and may introduce spurious correlations between the transformed features. Both phenomena may be observed in the California Housing case study. In particular, some features with positive correlation in the input space (e.g., “Total bedrooms” and “Population”) become negatively correlated in the transformed feature space of the unregularized NAM, leading to canceling contributions. In conventional statistics, it is not uncommon to drop features or employ sparsity regularization in order to remove multicollinearity (Dormann et al., 2013). However, sparsity regularization penalizes large vector norms of transformed features, thus also affecting model fit in the absence of concurvity. Thus, if the primary goal is to remove spurious correlations from a model rather than to reduce the number of features, concurvity regularization may be preferable.

Restricting concurvity reduces variance at the expense of increasing bias, generally resulting in a reduced accuracy. As we do not regularize in order to reduce overfitting, we did not expect improved generalization from regularization.

### C2. On quantifying interpretability
We note that quantifying interpretability is an unsolved issue in the community. While Chang et al. [a] attempt to quantify interpretability using sparsity and fidelity metrics, these cannot be considered gold standard or best-practice as they pose additional concerns and limitations. Whereas data fidelity can only be measured on synthetic datasets, sparsity “can hide data bias and discriminate against minority groups.” [a]. 

Therefore, we chose a more detailed approach to assessing interpretability, illustrated in figures 5 and 6 and the detailed discussion of the California Housing case study. Here, we note that regularization mitigates inflated feature importances in correlated features, removes spurious correlations, reduces variance in the shape functions, and prunes some correlated features while leaving uncorrelated features intact.

While quantitative measures of interpretability are still lacking, we argue that practical tools to deal with this issue in the meantime are still needed. We believe our contribution serves as an alternative to sparsity-based regularization, thus contributing to the toolkit of building interpretable models.

[a] C Chang, S Tan, B Lengerich, A Goldenberg, and R Caruana. “How Interpretable and Trustworthy are GAMs?” In ACM SIGKDD Conference on Knowledge Discovery & Data Mining, p 95–105, 2021. 

### C3. On the evaluation of our regularizer
A concern shared between most reviewers is the scope of our evaluation, currently comprising three toy examples  and four real-world tabular datasets. In response to these concerns, we have conducted additional experiments on three additional tabular datasets. Moreover, we have added evaluation of a conventional spline-based GAM across all tabular datasets, which is depicted in the attached PDF (Fig. 1&2c) and will be included in our camera-ready version. The results are clearly in line with our previous findings and further demonstrate the applicability of our method.

Finally, we will provide further evaluation of the investigated datasets, similar to our in-depth analysis presented in Fig. 6, in the appendix of the camera-ready version.

### C4. On regularization strength lambda in practice
As some reviewers pointed out, choosing the regularization strength lambda is of great practical importance. We briefly mention the elbow technique (or L-curve) in our paper, using the provided tradeoff curves (e.g. Fig 2b, 3b, 4 and 5). While these curves are well suited to identify a good tradeoff point between gains (in terms of lower concurvity) and losses (in terms of less accuracy), identifying the corresponding lambda is arguably tricky in the printed version (but trivial in an interactive digital plot). Alternatively, we now also provide separate curves for concurvity and accuracy over lambda, as suggested by R.aFW2 – see the attached PDF.

### C5. On computational complexity and overhead
A last general concern among reviewers was computational overhead, given the quadratic scaling of the regularizer in the number of features mentioned in the paper. An important point here is that the calculation of the pairwise correlations can be parallelized via vectorization, making it efficient to calculate while keeping the scaling constant controllable (i.e. via increased parallel compute). As an example, even for a dataset of around 1000 columns/features (which is way beyond most typical datasets) at a batch size of 512 our implementation of the proposed concurvity regularizer has a negligible average runtime of 6.9 ms (tested on an M1 MacBook Pro averaged over 1000 runs). We will provide a small analysis of the runtime and overhead in the appendix of our revised paper.


Meta-review: The reviewers praise the important problem, original idea, writing of the paper, ease of implementation, improved interpretability. The main criticisms are the somewhat unclear motivation for the method, tradeoff between performance and interpretability, computational considerations, choice of hyperparameters, limited scope of experiments, and lack of baselines. However, these concerns have been mostly addressed by the rebuttal.
Decision: Accept (poster)