SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Analogously to the above toy examples, we present a trade-off curve between RMSE and concurvity, averaging over 10 random initialization seeds per regularization strength  $\lambda$ . In this experiment, we choose
$k = 400$  components for both daily and weekly seasonality, to allow concavity to occur and fit the data almost exactly. Our findings are identical to the toy examples, demonstrating a steep decline in concavity when increasing  $\lambda$  with only a small increase in RMSE.
Finally, we note that concavity can often be identified by visual inspection for additive univariate time-series models as each component is a function of the same variable, see Figure 1. In contrast, on multivariate tabular data, concavity 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 – a 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 [23], California Housing [37], Adult [18], MIMIC-II [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 B.1. Additionally, we provide a 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 concavity measure  $R_{\perp}$  and validation fit quality when employing the proposed concavity regularization. Figure 5 displays the results for the tabular datasets, including the pyGAM baseline. It is clear that the concavity regularizer effectively reduces the concavity measure  $R_{\perp}$  without significantly compromising the model fit quality across all 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  $R_{\perp}$  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, we
Figure 5: Trade-off curves between model fit quality and measured concavity  $R_{\perp}$  for 50 levels of concavity regularization strength  $\lambda$ . Each regularization strength is evaluated over 10 initialization seeds to account for training variability, particularly noticeable in smaller datasets. The results of a conventional GAM are shown for comparison.
observe the variation in the scale of  $R_{\perp}$  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 concavity and model accuracy can serve as a valuable tool for identifying the optimal level of regularization strength. As expected, pyGAM obtains similar levels of concavity as the unregularized NAM at equally similar levels of model performance. We refer to Appendix E.3.1 for more verbose trade-off curves.
Case study: California Housing Our preceding experiments demonstrated that concavity reduction can be achieved when training NAMs on 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 concavity regularization. More specifically, we evaluate  $\lambda = 0.1$  (determined based on Figure 5) and  $\lambda = 0.0$  both for 60 random weight initializations.
First, we 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, as shown in Figure 6d. Note that the result in the non-regularized case is on par with the original NAM evaluation [2] 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 6a (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, as seen in the left correlation matrix. Conversely, the right correlation matrix reveals that concavity regularization effectively reduces the correlation of transformed features. This effect is especially pronounced for previously highly correlated features such as Longitude and Latitude, or Total Bedrooms and Households.
Third, we investigate how concavity impacts the estimation of the individual feature importances, which is of key interest for interpretable models such as NAMs. Following [2], we measure the importance of feature  $i$  as  $\frac{1}{N} \sum_{j=1}^{N} |f_i(x_{i,j}) - \overline{f_i}|$  where  $\overline{f_i}$  denotes the average of the shape function  $f_i$  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, a pattern which is
<sup>7</sup>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.
(a) Average feature correlations of the features (lower left) and non-linearly transformed features (upper right).
(c) Average and individual shape functions of 3 out of the 9 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.
(b) Combined box and strip plot of the models' feature.
Figure 6: Results for the California Housing dataset. The considered NAMs were trained with and without concavity regularization using 60 model initialization seeds each.
(d) Test RMSEs.
particularly clear for the strongly correlated features which we identified in Figure 6a. Such variance in feature importance can detrimentally impair the interpretability of 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 on the respective features, two observations are particularly interesting: (1) Features that are mostly uncorrelated remain unaffected by the regularization – an effect we have previously seen in Toy Example 1 – which can, for example, be observed in the case of the Median income feature. (2) Input correlations lead to a 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 are 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 concavity in the data, the shape functions of the pyGAM model exhibit large variance which impedes their interpretability [40]. A similar behavior can be observed for the remaining feature contributions, which are depicted in Figure 11 in Appendix E.3.3.
In a supplementary experiment in Appendix E.3.4, we compare concavity 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 are more gently preserved for concavity regularization.
In summary, our case study on the California Housing dataset establishes that concavity regularization significantly enhances interpretability and consistency of a GAM in terms of shape functions and feature importance, whilst maintaining high model accuracy.
Classical works on concavity in GAMs The term concavity was first introduced in [9]; for a well-written introduction to concavity, we refer the reader to [40]. Numerous subsequent works have developed techniques to address concavity, such as improving numerical stability in spline-based GAM fitting [25, 48] and adapting Lasso regularization for GAMs [5]. Partial GAMs [21] were proposed to address concavity 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 concavity, a property first noticed in [22]. In addition, [28] proposes a novel feature selection algorithm that chooses a minimal subset to deal with concavity. We refer to [28] for a thorough comparison of different concavity measures. In contrast, our proposed regularizer adds no additional constraints on the feature set size and does not explicitly enforce feature sparsity.
Modern neural approaches to GAMs Recent advancements in neural approaches to GAMs, such as Neural Additive Models (NAMs) [2] and Neural Prophet [46], have provided more flexible and powerful alternatives to classical methods [24]. These have spurred interest in the subject leading to 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 an automatic differentiation framework.
Regularization via decorrelation Similar types of decorrelation regularizers have previously been proposed in the machine learning literature but in different contexts. [13] found that regularizing the cross-covariance of hidden activations significantly increases generalization performance and proposed DeCov. OrthoReg [41] 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 [51], leverage decorrelation as a self-supervised learning technique to learn representations that are invariant to different transformations of the input data.
In this paper, we have introduced a differentiable concavity regularizer, designed to mitigate the often overlooked issue of concavity in differentiable Generalized Additive Models (GAMs). Through comprehensive empirical evaluations, we demonstrated that our regularizer effectively reduces concavity in differentiable GAMs such as Neural Additive Models and Neural Prophet. 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 interpretability-accuracy trade-off is an inherent aspect of concavity 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 concavity regularizer does not directly enforce feature sparsity. Thus, a comparison between sparsity regularizers and our concavity regularizer in unbalanced datasets would be of high interest. In addition, future work could explore how the joint optimization of concavity and model fit could be improved by framing it as a multi-objective problem. Finally, our concavity regularizer could be adapted to differentiable GAMs that incorporate pairwise interactions. Specifically, it would be interesting to 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 concavity in GAM modeling workflows. We believe this will lead to more interpretable models and hence more reliable and robust analyses, potentially avoiding false conclusions.
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  $\mathcal{H}_{\perp}$  Rules Out Concavity
In this section, we formalize our claim from Section 3 that the space  $\mathcal{H}_{\perp}$  provably rules out concavity. In the following,  $\langle \cdot ,\cdot \rangle$  and  $\| \cdot \| _2$  denote the Euclidean scalar product and norm, respectively. For  $v = (v_{1},\ldots ,v_{N})\in \mathbb{R}^{N}$ , we set  $\bar{v} := \frac{1}{N}\sum_{l = 1}^{N}v_{l}$  and denote the all-one vector by  $\mathbb{1}:= (1,\dots ,1)\in \mathbb{R}^N$ . The standard (empirical) Pearson's correlation coefficient is then given by
$$
\operatorname {C o r r} (v, w) := \left\{ \begin{array}{l l} \frac {\langle v - \bar {v} \mathbb {1} , w - \bar {w} \mathbb {1} \rangle}{\| v - \bar {v} \mathbb {1} \| _ {2} \cdot \| w - \bar {w} \mathbb {1} \| _ {2}} & \text {i f v a n d w a r e n o n - c o n s t a n t}, \\ \infty & \text {o t h e r w i s e}, \end{array} \right. \qquad v, w \in \mathbb {R} ^ {N}.
$$
Lemma A.1. Let  $X_{1},\ldots ,X_{p}\in \mathbb{R}^{N}$  be a set of feature variables with  $p > 1$  and let  $\mathcal{H}\subset \{(f_1,\dots ,f_p)\mid f_i:\mathbb{R}\to \mathbb{R}\}$  be a class of functions. Consider the following subclass of  $\mathcal{H}$ :
$$
\mathcal {H} _ {\perp} := \left\{\left(f _ {1}, \dots , f _ {p}\right) \in \mathcal {H} \mid \operatorname {C o r r} \left(f _ {i} \left(X _ {i}\right), f _ {j} \left(X _ {j}\right)\right) = 0 \text {f o r a l l} i \neq j \right\} \subset \mathcal {H}.
$$
Then we do not have concavity w.r.t.  $X_{1},\ldots ,X_{p}$  and  $\mathcal{H}_{\perp}$
Proof. Towards a contradiction, assume that there are  $(g_1,\ldots ,g_p)\in \mathcal{H}_{\perp}$  and  $c_{0}\in \mathbb{R}$  such that  $c_{0}\mathbb{1} + \sum_{i = 1}^{p}g_{i}(X_{i}) = 0$
We set  $v_{i} \coloneqq g_{i}(X_{i})$  and trivially add the mean vectors to the linear combination:
$$
0 = c _ {0} \mathbb {1} + \sum_ {i = 1} ^ {p} v _ {i} = \left(c _ {0} + \sum_ {i = 1} ^ {p} \bar {v} _ {i}\right) \mathbb {1} + \sum_ {i = 1} ^ {p} \left(v _ {i} - \bar {v} _ {i} \mathbb {1}\right).
$$
Using that  $\operatorname{Corr}(v_i, v_j) = 0$  for  $i \neq j$ , we then obtain
$$
\begin{array}{l} 0 = \langle 0, v _ {j} - \bar {v} _ {j} \mathbb {1} \rangle = \left(c _ {0} + \sum_ {i = 1} ^ {p} \bar {v} _ {i}\right) \langle \mathbb {1}, v _ {j} - \bar {v} _ {j} \mathbb {1} \rangle + \sum_ {i = 1} ^ {p} \langle v _ {i} - \bar {v} _ {i} \mathbb {1}, v _ {j} - \bar {v} _ {j} \mathbb {1} \rangle \\ = \left(c _ {0} + \sum_ {i = 1} ^ {p} \bar {v} _ {i}\right) \cdot \left(N \bar {v} _ {j} - N \bar {v} _ {j}\right) + \| v _ {j} - \bar {v} _ {j} \mathbb {1} \| _ {2} ^ {2} = \| v _ {j} - \bar {v} _ {j} \mathbb {1} \| _ {2} ^ {2}. \\ \end{array}
$$
We conclude that  $v_{j} = \bar{v}_{j}\mathbb{1}$ , i.e.,  $v_{j}$  is a constant vector. But this contradicts the definition of  $\mathcal{H}_{\perp}$  because we would have  $\operatorname{Corr}(v_i, v_j) = \infty$  with any  $i \neq j$ .
A.2 Additional Remarks on Concavity and our Regularizer
(1) The definitions of multicollinearity (Definition 2.1) and concavity (Definition 2.2) are based on a fixed (deterministic) feature design, but one could also formulate probabilistic versions, cf. [44]. The latter typically facilitates a 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 a non-linearity. Two simple (toy) examples are presented in Section 4.1.
(3) Our concavity regularizer  $R_{\perp}$  does not automatically affect the predictive performance of a GAM. For example, assuming that the input features are drawn from stochastically independent random variables, we can conclude that  $\mathrm{Corr}(f_i(X_i), f_j(X_j)) \approx 0$  for a large enough sample size  $N$ , since non-linear transforms of independent random variables remain independent. Consequently, we have that  $R_{\perp}(\{f_i\}_i, \{X_i\}_i) \approx 0$ , so that no (in this case undesirable) regularization takes effect.
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 A.1.
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 a 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 1 and the hyperparameters per dataset are shown in Figure 7.
Hyperparameter	Value / Range	Scaling
Learning Rate	[1e-4, 1e-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 1: Hyperparameter Search Space
Hyperparameter	Value
Learning Rate	1e-3
Weight Decay	0.0
Activation	GELU
# of neurons per layer	128
# of hidden layers	3
Num. Epochs	50
Batch Size	128
Correlation Denominator ε	1e-12
Start Conc. Reg. after x% of steps	5
(a) Toy Example 1&2
Hyperparameter	Value
Learning Rate	7.93e-4
Weight Decay	1.79e-2
Activation	ELU
# of neurons per layer	75
# of hidden layers	6
Num. Epochs	91
Batch Size	128
Correlation Denominator ε	1e-12
Start Conc. Reg. after x% of steps	5
(b) Boston Housing
Hyperparameter	Value
Learning Rate	9.46e-3
Weight Decay	3.73e-3
Activation	ReLU
# of neurons per layer	72
# of hidden layers	5
Num. Epochs	39
Batch Size	512
Correlation Denominator ε	1e-12
Start Conc. Reg. after x% of steps	5
(c) California Housing
Hyperparameter	Value
Learning Rate	2.64e-3
Weight Decay	1.64e-3
Activation	GELU
# of neurons per layer	204
# of hidden layers	4
Num. Epochs	200
Batch Size	512
Correlation Denominator ε	1e-12
Start Conc. Reg. after x% of steps	5
(d) Adult
Hyperparameter	Value
Learning Rate	3.31e-3
Weight Decay	1.08e-3
Activation	GELU
# of neurons per layer	190
# of hidden layers	3
Num. Epochs	20
Batch Size	512
Correlation Denominator ε	1e-12
Start Conc. Reg. after x% of steps	5
(e) MIMIC-II
Hyperparameter	Value
Learning Rate	3.88e-3
Weight Decay	1.10e-3
Activation	GELU
# of neurons per layer	168
# of hidden layers	3
Num. Epochs	23
Batch Size	512
Correlation Denominator ε	1e-12
Start Conc. Reg. after x% of steps	5
(f) MIMIC-III
Hyperparameter	Value
Learning Rate	3.42e-3
Weight Decay	1.05e-2
Activation	GELU
# of neurons per layer	127
# of hidden layers	3
Num. Epochs	30
Batch Size	512
Correlation Denominator ε	1e-12
Start Conc. Reg. after x% of steps	5
(g) Support2
Figure 7: Hyperparameters per dataset for NAM experiments.
B.2 pyGAM