Safe couplings: coupled refinement types

Abstract

We enhance refinement types with mechanisms to reason about relational properties of probabilistic computations. Our mechanisms, which are inspired from probabilistic couplings, are applicable to a rich set of probabilistic properties, including expected sensitivity, which ensures that the distance between outputs of two probabilistic computations can be controlled from the distance between their inputs. We implement our mechanisms in the type system of Liquid Haskell and we use them to formally verify Haskell implementations of two classic machine learning algorithms: Temporal Difference (TD) reinforcement learning and stochastic gradient descent (SGD). We formalize a fragment of our system for discrete distributions and we prove soundness with respect to a set-theoretical semantics.