Relational Expectation Properties by Probabilistic Coupling

Relational properties describe how two program executions are related, while expectation properties describe averagecase behavior of probabilistic programs. We investigate formal verification techniques for relational expectation properties. This class includes key technical properties modeling stability in machine learning, and properties associated with fast mixing of Markov chains. Technically, we design a relational program logic EPRHL that is inspired by the logic PRHL, a powerful tool for proving relational properties by reasoning about probabilistic couplings. We enhance PRHL with an orthogonal, compositional reasoning principle based on premetrics; roughly, the expected distance between the outputs should be bounded as a function of the distance between the inputs. We demonstrate our logic on three classes of examples: uniform stability of variants of the Stochastic Gradient Method used in machine learning, fast mixing for a Markov chain modeling population dynamics, and fast mixing for a Markov chain from statistical physics, using the path coupling method.