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We explore and formalize the task of synthesizing programs over noisy data, i.e., data that may contain corrupted input-output examples. By formalizing the concept of a Noise Source, an Input Source, and a prior distribution over programs, we formalize the probabilistic process which constructs a noisy dataset. This formalism allows us to define the correctness of a synthesis algorithm, in terms of its ability to synthesize the hidden underlying program. The probability of a synthesis algorithm being correct depends upon the match between the Noise Source and the Loss Function used in the synthesis algorithm's optimization process. We formalize the concept of an optimal Loss Function given prior information about the Noise Source. We provide a technique to design optimal Loss Functions given perfect and imperfect information about the Noise Sources. We also formalize the concept and conditions required for convergence, i.e., conditions under which the probability that the synthesis algorithm produces a correct program increases as the size of the noisy data set increases. This paper presents the first formalization of the concept of optimal Loss Functions, the first closed form definition of optimal Loss Functions, and the first conditions that ensure that a noisy synthesis algorithm will have convergence guarantees.
