  The Lipschitz constant of the map between the input and output space
represented by a neural network is a natural metric for assessing the
robustness of the model. We present a new method to constrain the Lipschitz
constant of dense deep learning models that can also be generalized to other
architectures. The method relies on a simple weight normalization scheme during
training that ensures the Lipschitz constant of every layer is below an upper
limit specified by the analyst. A simple monotonic residual connection can then
be used to make the model monotonic in any subset of its inputs, which is
useful in scenarios where domain knowledge dictates such dependence. Examples
can be found in algorithmic fairness requirements or, as presented here, in the
classification of the decays of subatomic particles produced at the CERN Large
Hadron Collider. Our normalization is minimally constraining and allows the
underlying architecture to maintain higher expressiveness compared to other
techniques which aim to either control the Lipschitz constant of the model or
ensure its monotonicity. We show how the algorithm was used to train a
powerful, robust, and interpretable discriminator for heavy-flavor-quark
decays, which has been adopted for use as the primary data-selection algorithm
in the LHCb real-time data-processing system in the current LHC data-taking
period known as Run 3. In addition, our algorithm has also achieved
state-of-the-art performance on benchmarks in medicine, finance, and other
applications.
