  As machine learning systems are increasingly used to make real world legal
and financial decisions, it is of paramount importance that we develop
algorithms to verify that these systems do not discriminate against minorities.
We design a scalable algorithm for verifying fairness specifications. Our
algorithm obtains strong correctness guarantees based on adaptive concentration
inequalities; such inequalities enable our algorithm to adaptively take samples
until it has enough data to make a decision. We implement our algorithm in a
tool called VeriFair, and show that it scales to large machine learning models,
including a deep recurrent neural network that is more than five orders of
magnitude larger than the largest previously-verified neural network. While our
technique only gives probabilistic guarantees due to the use of random samples,
we show that we can choose the probability of error to be extremely small.
