nn2poly: An R Package for Converting Neural Networks into Interpretable Polynomials

The nn2poly package provides the implementation in R of the NN2Poly method to explain and interpret feed-forward neural networks by means of polynomial representations that predict in an equivalent manner as the original network.Through the obtained polynomial coefficients, the effect and importance of each variable and their interactions on the output can be represented. This capabiltiy of capturing interactions is a key aspect usually missing from most Explainable Artificial Intelligence (XAI) methods, specially if they rely on expensive computations that can be amplified when used on large neural networks. The package provides integration with the main deep learning framework packages in R (tensorflow and torch), allowing an user-friendly application of the NN2Poly algorithm. Furthermore, nn2poly provides implementation of the required weight constraints to be used during the network training in those same frameworks. Other neural networks packages can also be used by including their weights in list format. Polynomials obtained with nn2poly can also be used to predict with new data or be visualized through its own plot method. Simulations are provided exemplifying the usage of the package alongside with a comparison with other approaches available in R to interpret neural networks.

NN2Poly: A polynomial representation for deep feed-forward artificial neural networks

Interpretability of neural networks and their underlying theoretical behaviour remain being an open field of study, even after the great success of their practical applications, particularly with the emergence of deep learning. In this work, NN2Poly is proposed: a theoretical approach that allows to obtain polynomials that provide an alternative representation of an already trained deep neural network. This extends the previous idea proposed in arXiv:2102.03865, which was limited to single hidden layer neural networks, to work with arbitrarily deep feed-forward neural networks in both regression and classification tasks. The objective of this paper is achieved by using a Taylor expansion on the activation function, at each layer, and then using several combinatorial properties that allow to identify the coefficients of the desired polynomials. The main computational limitations when implementing this theoretical method are discussed and it is presented an example of the constraints on the neural network weights that are necessary for NN2Poly to work. Finally, some simulations are presented were it is concluded that using NN2Poly it is possible to obtain a representation for the given neural network with low error between the obtained predictions.

Towards a mathematical framework to inform Neural Network modelling via Polynomial Regression

Even when neural networks are widely used in a large number of applications, they are still considered as black boxes and present some difficulties for dimensioning or evaluating their prediction error. This has led to an increasing interest in the overlapping area between neural networks and more traditional statistical methods, which can help overcome those problems. In this article, a mathematical framework relating neural networks and polynomial regression is explored by building an explicit expression for the coefficients of a polynomial regression from the weights of a given neural network, using a Taylor expansion approach. This is achieved for single hidden layer neural networks in regression problems. The validity of the proposed method depends on different factors like the distribution of the synaptic potentials or the chosen activation function. The performance of this method is empirically tested via simulation of synthetic data generated from polynomials to train neural networks with different structures and hyperparameters, showing that almost identical predictions can be obtained when certain conditions are met. Lastly, when learning from polynomial generated data, the proposed method produces polynomials that approximate correctly the data locally.