  Interpretable machine learning has demonstrated impressive performance while
preserving explainability. In particular, neural additive models (NAM) offer
the interpretability to the black-box deep learning and achieve
state-of-the-art accuracy among the large family of generalized additive
models. In order to empower NAM with feature selection and improve the
generalization, we propose the sparse neural additive models (SNAM) that employ
the group sparsity regularization (e.g. Group LASSO), where each feature is
learned by a sub-network whose trainable parameters are clustered as a group.
We study the theoretical properties for SNAM with novel techniques to tackle
the non-parametric truth, thus extending from classical sparse linear models
such as the LASSO, which only works on the parametric truth.
  Specifically, we show that SNAM with subgradient and proximal gradient
descents provably converges to zero training loss as $t\to\infty$, and that the
estimation error of SNAM vanishes asymptotically as $n\to\infty$. We also prove
that SNAM, similar to LASSO, can have exact support recovery, i.e. perfect
feature selection, with appropriate regularization. Moreover, we show that the
SNAM can generalize well and preserve the `identifiability', recovering each
feature's effect. We validate our theories via extensive experiments and
further testify to the good accuracy and efficiency of SNAM.
