Mathematically Modeling the Lexicon Entropy of Emergent Language

We formulate a stochastic process, FiLex, as a mathematical model of lexicon entropy in deep learning-based emergent language systems. Defining a model mathematically allows it to generate clear predictions which can be directly and decisively tested. We empirically verify across four different environments that FiLex predicts the correct correlation between hyperparameters (training steps, lexicon size, learning rate, rollout buffer size, and Gumbel-Softmax temperature) and the emergent language's entropy in 20 out of 20 environment-hyperparameter combinations. Furthermore, our experiments reveal that different environments show diverse relationships between their hyperparameters and entropy which demonstrates the need for a model which can make well-defined predictions at a precise level of granularity.