On the Provable Advantage of Unsupervised Pretraining
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Unsupervised pretraining, which learns a useful representation using a large amount of unlabeled data to facilitate the learning of downstream tasks, is a critical component of modern large-scale machine learning systems. Despite its tremendous empirical success, the rigorous theoretical understanding of why unsupervised pretraining generally helps remains rather limited – most existing results are restricted to particular methods or approaches for unsupervised pretraining with specialized structural assumptions. This paper studies a generic framework, where the unsupervised representation learning task is specified by an abstract class of latent variable models Φ and the downstream task is specified by a class of prediction functions Ψ. We consider a natural approach of using Maximum Likelihood Estimation (MLE) for unsupervised pretraining and Empirical Risk Minimization (ERM) for learning downstream tasks. We prove that, under a mild ”informative” condition, our algorithm achieves an excess risk of 𝒪̃(√(𝒞_Φ/m) + √(𝒞_Ψ/n)) for downstream tasks, where 𝒞_Φ, 𝒞_Ψ are complexity measures of function classes Φ, Ψ, and m, n are the number of unlabeled and labeled data respectively. Comparing to the baseline of 𝒪̃(√(𝒞_Φ∘Ψ/n)) achieved by performing supervised learning using only the labeled data, our result rigorously shows the benefit of unsupervised pretraining when m ≫ n and 𝒞_Φ∘Ψ > 𝒞_Ψ. This paper further shows that our generic framework covers a wide range of approaches for unsupervised pretraining, including factor models, Gaussian mixture models, and contrastive learning.
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