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Noisy data, non-convex objectives, model misspecification, and numericalinstability can all cause undesired behaviors in machine learning systems. As aresult, detecting actual implementation errors can be extremely difficult. Wedemonstrate a methodology in which developers use an interactive proofassistant to both implement their system and to state a formal theorem definingwhat it means for their system to be correct. The process of proving thistheorem interactively in the proof assistant exposes all implementation errorssince any error in the program would cause the proof to fail. As a case study,we implement a new system, Certigrad, for optimizing over stochasticcomputation graphs, and we generate a formal (i.e. machine-checkable) proofthat the gradients sampled by the system are unbiased estimates of the truemathematical gradients. We train a variational autoencoder using Certigrad andfind the performance comparable to training the same model in TensorFlow.

Developing Bug-Free Machine Learning Systems With Formal Mathematics