Unlabeled Data Help: Minimax Analysis and Adversarial Robustness

Abstract

The recent proposed self-supervised learning (SSL) approaches successfully demonstrate the great potential of supplementing learning algorithms with additional unlabeled data. However, it is still unclear whether the existing SSL algorithms can fully utilize the information of both labelled and unlabeled data. This paper gives an affirmative answer for the reconstruction-based SSL algorithm (Lee et al., 2020) under several statistical models. While existing literature only focuses on establishing the upper bound of the convergence rate, we provide a rigorous minimax analysis, and successfully justify the rate-optimality of the reconstruction-based SSL algorithm under different data generation models. Furthermore, we incorporate the reconstruction-based SSL into the existing adversarial training algorithms and show that learning from unlabeled data helps improve the robustness.Another motivation to study minimax lower bound is to understand the role of conditional independence (CI) for SSL. Lee et al. (2020) identifies that CI is a key factor yielding the estimation efficiency. A natural question would be how essential the CI condition is. Lee et al. (2020) suggests that the convergence rate of SSL estimate might be slower when CI does not hold. However, it is unclear whether it is caused by that the SSL is inefficient under conditional dependency, or that the fundamental information limit is worse under conditional dependency. To answer this question, minimax lower bound analysis is necessary.Besides the minimax analysis of SSL, since the aforementioned works observe a great advantage of SSL over classical supervised learning methods, it is natural