A Neural Network Algorithm for KL Divergence Estimation with Quantitative Error Bounds
This addresses a fundamental statistical analysis problem for researchers, but it is incremental as it builds on existing neural network methods with improved theoretical guarantees.
The paper tackles the problem of estimating KL divergence for continuous random variables by proposing a neural network algorithm using random features, achieving an estimation error of O(m^{-1/2} + T^{-1/3}) with high probability.
Estimating the Kullback-Leibler (KL) divergence between random variables is a fundamental problem in statistical analysis. For continuous random variables, traditional information-theoretic estimators scale poorly with dimension and/or sample size. To mitigate this challenge, a variety of methods have been proposed to estimate KL divergences and related quantities, such as mutual information, using neural networks. The existing theoretical analyses show that neural network parameters achieving low error exist. However, since they rely on non-constructive neural network approximation theorems, they do not guarantee that the existing algorithms actually achieve low error. In this paper, we propose a KL divergence estimation algorithm using a shallow neural network with randomized hidden weights and biases (i.e. a random feature method). We show that with high probability, the algorithm achieves a KL divergence estimation error of $O(m^{-1/2}+T^{-1/3})$, where $m$ is the number of neurons and $T$ is both the number of steps of the algorithm and the number of samples.