Several recent methods for interpretability model feature interactions by looking at the Hessian of a neural network. This poses a challenge for ReLU networks, which are piecewise-linear and thus have a zero Hessian almost everywhere. We propose SmoothHess, a method of estimating second-order interactions through Stein's Lemma. In particular, we estimate the Hessian of the network convolved with a Gaussian through an efficient sampling algorithm, requiring only network gradient calls. SmoothHess is applied post-hoc, requires no modifications to the ReLU network architecture, and the extent of smoothing can be controlled explicitly. We provide a non-asymptotic bound on the sample complexity of our estimation procedure. We validate the superior ability of SmoothHess to capture interactions on benchmark datasets and a real-world medical spirometry dataset.

• torch 1.13.1+cu117
• torchvision 0.11.1
• matplotlib 3.7.1
• numpy 1.25.5
• pandas 1.5.3
• cvxpy 1.2.1
• tqdm 4.55.0

• FourQuadrant.ipynb: Shows the Four Quadrant dataset experiment, highlighting the intuitive control offered by SmoothHess compared to the SoftPlus Hessian.
• PMSEExample.ipynb: Example notebook for the Perturbation Mean-Squared Error (PMSE) experiment, demonstrating the strong ability of SmoothHess to capture the networks local behaviour.
• AdvAttackExample.ipynb: Example notebook demonstrating the use of SmoothHess to perform adversarial attacks.

If you use this code, please cite our paper:

@article{torop2023smoothhess,
  title={SmoothHess: ReLU network feature interactions via stein's lemma},
  author={Torop, Max and Masoomi, Aria and Hill, Davin and Kose, Kivanc and Ioannidis, Stratis and Dy, Jennifer},
  journal={Advances in Neural Information Processing Systems},
  volume={36},
  pages={50697--50729},
  year={2023}
}