Abstract: The minimum mean-square error (MMSE) achievable by optimal estimation of a random variable $S$ given another random variable $T$ is of much interest in a variety of statistical contexts. Motivated by a growing interest in auditing machine learning models for unintended information leakage, we propose a neural network-based estimator of this MMSE. We derive a lower bound for the MMSE based on the proposed estimator and the Barron constant associated with the conditional expectation of $S$ given $T$. Since the latter is typically unknown in practice, we derive a general bound for the Barron constant that produces order optimal estimates for canonical distribution models.