Convex and Non-convex Approaches for Statistical Inference with Noisy Labels
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We study the problem of estimation and testing in logistic regression with class-conditional noise in the observed labels, which has an important implication in the Positive-Unlabeled (PU) learning setting. With the key observation that the label noise problem belongs to a special sub-class of generalized linear models (GLM), we discuss convex and non-convex approaches that address this problem. A non-convex approach based on the maximum likelihood estimation produces an estimator with several optimal properties, but a convex approach has an obvious advantage in optimization. We demonstrate that in the low-dimensional setting, both estimators are consistent and asymptotically normal, where the asymptotic variance of the non-convex estimator is smaller than the convex counterpart. We also quantify the efficiency gap which provides insight into when the two methods are comparable. In the high-dimensional setting, we show that both estimation procedures achieve l2-consistency at the minimax optimal s log p/n rates under mild conditions. Finally, we propose an inference procedure using a de-biasing approach. To the best of our knowledge, this is the first work providing point estimation guarantees and hypothesis testing results for GLMs with non-canonical link functions, which is of independent interest. We validate our theoretical findings through simulations and a real-data example.
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