Artificial Neural Networks (ANNs) are increasingly used for discrete choice analysis. But, at present, it is unknown what sample size requirements are appropriate when using ANNs in this particular context. This paper fills this knowledge gap: we empirically establish a rule-of-thumb for ANN-based discrete choice analysis based on analyses of synthetic and real data. To investigate the effect of complexity of the data generating process on the minimum required sample size, we conduct extensive Monte Carlo analyses using a series of different model specifications with different levels of model complexity, including RUM and RRM models, with and without random taste parameters. Based on our analyses we advise to use a minimum sample size of fifty times the number of weights in the ANN; it should be noted, that the number of weights is generally much larger than the number of parameters in a discrete choice model. This rule-of-thumb is considerably more conservative than the rule-of-thumb that is most often used in the ANN community, which advises to use at least ten times the number of weights.