Neural Additive Models (NAMs)

Updated 23 November 2025

Neural Additive Models (NAMs) are interpretable deep learning frameworks that combine the flexibility of neural networks with the transparency of generalized additive models by modeling each feature with a dedicated subnetwork.
Extensions of NAMs include higher-order interactions, sparsity with group LASSO, Bayesian uncertainty quantification, and monotonic constraints to enhance performance in complex and regulated applications.
Their modular, glass-box architecture facilitates clear visualization of feature contributions, making them ideal for sensitive domains like healthcare, finance, and survival analysis.

Neural Additive Models (NAMs) are a class of machine learning models that integrate the expressivity of deep neural networks with the innate interpretability of generalized additive models (GAMs). NAMs model each feature’s contribution using a dedicated neural network taking only the scalar feature as input, enabling direct visualization and audit of individual effects. Over the past several years, extensive research has advanced the methodology and scope of NAMs, including extensions to monotonic constraints, sparse and structural architectures, higher-order interactions, Bayesian uncertainty quantification, survival analysis, distributional regression, time-series applications, and regulated domains such as finance and healthcare.

1. Mathematical Architecture and Core Principles

A standard NAM is defined by the additive predictor: $f(x) = \beta + \sum_{j=1}^{d} f_j(x_j)$ where $x=(x_1,\dots,x_d)$ , $\beta$ is a bias term, and each $f_j$ is a one-dimensional neural network (“feature net”). For regression, $f(x)$ is used directly; for classification, a link function $\sigma(\cdot)$ , typically logistic, is applied to $f(x)$ (Agarwal et al., 2020).

Each feature net is a small multilayer perceptron, frequently employing specialized activations (e.g., ExU, ExpDive, ReLU), with regularization via dropout, weight decay, and feature dropout (Agarwal et al., 2020, Jo et al., 2022, Kim et al., 2022). The additive structure enforces exact decomposability at prediction time: each feature’s effect is given by $f_j(x_j)$ , facilitating full transparency. Extensions for multitask learning (Agarwal et al., 2020), multi-class outputs (Condor et al., 1 May 2024), and shared subnetwork libraries are readily supported.

2. Feature Interaction and Model Extensions

Although basic NAMs capture only first-order (univariate) effects, several works have generalized the additive structure:

Higher-Order NAMs (HONAM): HONAM models arbitrary orders of feature interactions by computing vector-valued representations for each feature and recursively forming element-wise products for interactions up to order $K$ (Kim et al., 2022). The final prediction is:

$\hat y = W \cdot G(x) + b$

where $G(x)$ concatenates all first-to- $K$ th order interactions. ExpDive units enable sharp shape functions.

Sparse NAMs (SNAM): Incorporates group sparsity (Group LASSO) over subnetwork parameters to perform feature selection and improve generalization. The training objective penalizes the $\ell_2$ norm of each $\theta_j$ :

$\min_{\{\theta_j\},\beta} \frac{1}{n}\sum_{i=1}^n \ell\big(y_i, \sum_{j=1}^p f_j(x_{ij};\theta_j)+\beta\big) + \lambda\sum_{j=1}^p \|\theta_j\|_2$

This achieves exact support recovery under suitable regularization (Xu et al., 2022).

Structural NAMs (SNAM): Replaces each neural subnetwork with a spline-based expansion or kernel unit, enabling statistical inference via learnable parameters $\beta_{jm}$ and classical smoothness penalties. Fast SNAM architectures employ Silverman kernel units for parameter efficiency and explicit regularization (Luber et al., 2023).
PrAda-net: Employs adaptive lasso and proximal gradient descent in a one-layer network, enabling automatic model selection of both feature and interaction components without architectural pre-specification (Allerbo et al., 2020).

3. Uncertainty Quantification and Bayesian Approaches

Bayesian extensions to NAMs improve reliability and facilitate uncertainty estimation:

Laplace-approximated NAMs (LA-NAMs): Posterior distribution over subnetwork parameters via block-diagonal Laplace approximations yields calibrated feature-wise credible intervals and supports empirical Bayes selection, pruning irrelevant features and ranking candidate interactions (Bouchiat et al., 2023).
Bayesian NAM (BayesNAM): Utilizes mean-field variational Bayesian neural networks for each $f_k$ , augmented with feature dropout during training to capture explanation inconsistency. Wide credible intervals and high explanation variance highlight regions of model or data unreliability, turning instability into a diagnostic tool (Kim et al., 10 Nov 2024).

4. Regularization, Monotonicity, and Compliance in Regulated Domains

NAMs have been adapted to satisfy strict regulatory requirements:

Monotonic NAMs (MNAM): Enforce individual and pairwise monotonicity of the prediction w.r.t. specified features via penalty terms on violations of $\partial f/\partial x_i\geq 0$ and difference monotonicity constraints. These penalties are computed on univariate grids and increase until violations are eliminated; the computational cost of monotonicity is effectively “free” (Chen et al., 2022).
Regulatory Alignment: The additive, monotonic form satisfies transparency, explainability, and fairness requirements. Empirical results in credit scoring demonstrate that MNAMs match black-box neural network accuracy and strictly adhere to business and regulatory logic (Chen et al., 2022).

5. Survival Analysis, Distributional Regression, and Time Series Extensions

NAMs have been generalized to support complex outputs and high-stakes domains:

Survival NAMs: The additive structure is embedded in proportional and non-proportional Cox models. Extensions include pairwise MLP interactions ( $f_{jk}$ ) and time-dependent feature nets for non-proportional hazards (TimeNAM) (Peroni et al., 2022). Competing risks survival is handled via CRISP-NAM, which assigns a dedicated feature net to each covariate, per event type, and constructs additive cause-specific hazards (Ramachandram et al., 27 May 2025).
Distributional Regression (NAMLSS): The model jointly fits location, scale, and shape parameters additively for regression targets beyond the mean. Each parameter $\theta_k(x)$ is a sum over feature-wise subnetworks, mapped via link functions. This enables modeling for full distributional outputs with interpretable partial dependence plots for each parameter (Thielmann et al., 2023).
Nowcasting and Time Series: In NAM-NC, one feature net per variable and time lag is used, allowing exact attribution across $(t,k)$ . Parameter-sharing variants (time-shared, feature-shared) reduce complexity. Visualization of heatmaps and shape plots enables granular temporal explanations (Jo et al., 2022).

6. Interpretability, Visualization, and Empirical Performance

The central strength of NAMs is their “glass-box” interpretability. For any prediction, each feature’s contribution is explicitly given by $f_j(x_j)$ or its generalized counterpart in higher-order or distributional variants. Visualization typically includes:

Shape plots: $x_j \mapsto f_j(x_j)$ curves for all features; in multiclass settings, per-class curves;
Feature importance: Average absolute contribution, $\frac{1}{n}\sum_i |f_j(x_{i,j})|$ ;
Interaction surfaces: For higher-order and pairwise-expansion NAMs;
Credible intervals: In Bayesian and Laplace-approximated extensions, for robust quantification;

Empirical benchmarks consistently demonstrate state-of-the-art performance on tabular regression/classification tasks, often matching or closely trailing fully connected neural networks and tree-based ensemble methods, with substantial gains in model transparency and interpretability (Agarwal et al., 2020, Xu et al., 2022, Luber et al., 2023, Bouchiat et al., 2023).

7. Practical Applications and Impact

NAMs and their extensions have proven effective in a broad spectrum of domains:

Healthcare: Survival modeling, risk estimation, and discrete treatment effect analysis, leveraging feature-level plots for clinical decision support (Peroni et al., 2022, Ramachandram et al., 27 May 2025).
Finance: Credit scoring with regulatory compliance, fraud detection, and risk management, using monotonic or sparse variants (Chen et al., 2022).
Education: Automated grading systems explicating rubric-driven features and their contributions to final scores (Condor et al., 1 May 2024).
Time Series: Multivariate nowcasting and inventory management policies with exact temporal contribution attribution (Jo et al., 2022, Siems et al., 2023).
Scientific and Industrial Domains: Large-scale spatial and temporal modeling, insurance pricing, and demand forecasting exploiting the modularity and explainability of NAM-based architectures (Thielmann et al., 2023, Allerbo et al., 2020).

The NAM paradigm has catalyzed the development of software packages, notably “dnamite,” which implements regression, classification, and survival analysis NAMs with scikit-learn style API and built-in feature selection, applicable to large-scale tabular data and clinical datasets (Ness et al., 6 Mar 2025).

Neural Additive Models represent a foundational framework for interpretable, modular, and extensible deep learning, with extensive theoretical guarantees, empirical validation, and successful deployment across regulated and high-risk domains. The ongoing development of NAMs—spanning higher-order interactions, Bayesian explainability, distributional outputs, monotonicity, and sparse selection—continues to bridge the gap between predictive power and practical interpretability in modern machine learning.