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# Gradient-based Wang-Landau Algorithm: A Novel Sampler for Output Distribution of Neural Networks over the Input Space

*Proceedings of the 40th International Conference on Machine Learning*, PMLR 202:22338-22351, 2023.

#### Abstract

The output distribution of a neural network (NN) over the entire input space captures the complete input-output mapping relationship, offering in- sights toward a more comprehensive NN under- standing. Exhaustive enumeration or traditional Monte Carlo methods for the entire input space can exhibit impractical sampling time, especially for high-dimensional inputs. To make such difficult sampling computationally feasible, in this paper, we propose a novel Gradient-based Wang-Landau (GWL) sampler. We first draw the connection between the output distribution of a NN and the density of states (DOS) of a physical system. Then, we renovate the classic sampler for the DOS problem, Wang-Landau algorithm, by re-placing its random proposals with gradient-based Monte Carlo proposals. This way, our GWL sampler investigates the under-explored subsets of the input space much more efficiently. Extensive experiments have verified the accuracy of the output distribution generated by GWL and also showcased several interesting findings - for example, in a binary image classification task, both CNN and ResNet mapped the majority of human unrecognizable images to very negative logit values.