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Labeling data for modern machine learning is expensive and time-consuming.Latent variable models can be used to infer labels from weaker,easier-to-acquire sources operating on unlabeled data. Such models can also betrained using labeled data, presenting a key question: should a user invest infew labeled or many unlabeled points? We answer this via a framework centeredon model misspecification in method-of-moments latent variable estimation. Ourcore result is a bias-variance decomposition of the generalization error, whichshows that the unlabeled-only approach incurs additional bias undermisspecification. We then introduce a correction that provably removes thisbias in certain cases. We apply our decomposition framework to three scenarios-- well-specified, misspecified, and corrected models -- to 1) choose betweenlabeled and unlabeled data and 2) learn from their combination. We observetheoretically and with synthetic experiments that for well-specified models,labeled points are worth a constant factor more than unlabeled points. Withmisspecification, however, their relative value is higher due to the additionalbias but can be reduced with correction. We also apply our approach to studyreal-world weak supervision techniques for dataset construction.

Comparing the Value of Labeled and Unlabeled Data in Method-of-Moments Latent Variable Estimation