Title: Simplifying Models with Unlabeled Output Data. Abstract: We focus on prediction problems with high-dimensional outputs that are subject to output validity constraints, e.g. a pseudocode-to-code translation task where the code must compile. For these problems, labeled input-output pairs are expensive to obtain, but "unlabeled" outputs, i.e. outputs without corresponding inputs, are freely available and provide information about output validity (e.g. code on GitHub). In this paper, we present predict-and-denoise, a framework that can leverage unlabeled outputs. Specifically, we first train a denoiser to map possibly invalid outputs to valid outputs using synthetic perturbations of the unlabeled outputs. Second, we train a predictor composed with this fixed denoiser. We show theoretically that for a family of functions with a high-dimensional discrete valid output space, composing with a denoiser reduces the complexity of a 2-layer ReLU network needed to represent the function and that this complexity gap can be arbitrarily large. We evaluate the framework empirically on several datasets, including image generation from attributes and pseudocode-to-code translation. On the SPoC pseudocode-to-code dataset, our framework improves the proportion of code outputs that pass all test cases by 3-5% over a baseline Transformer.