THEORETICAL ANALYSIS OF SELF-TRAINING WITH DEEP NETWORKS ON UNLABELED DATA

Abstract

Self-training algorithms, which train a model to fit pseudolabels predicted by another previously-learned model, have been very successful for learning with unlabeled data using neural networks. However, the current theoretical understanding of self-training only applies to linear models. This work provides a unified theoretical analysis of self-training with deep networks for semi-supervised learning, unsupervised domain adaptation, and unsupervised learning. At the core of our analysis is a simple but realistic "expansion" assumption, which states that a lowprobability subset of the data must expand to a neighborhood with large probability relative to the subset. We also assume that neighborhoods of examples in different classes have minimal overlap. We prove that under these assumptions, the minimizers of population objectives based on self-training and input-consistency regularization will achieve high accuracy with respect to ground-truth labels. By using off-the-shelf generalization bounds, we immediately convert this result to sample complexity guarantees for neural nets that are polynomial in the margin and Lipschitzness. Our results help explain the empirical successes of recently proposed self-training algorithms which use input consistency regularization.

1. INTRODUCTION

Though supervised learning with neural networks has become standard and reliable, it still often requires massive labeled datasets. As labels can be expensive or difficult to obtain, leveraging unlabeled data in deep learning has become an active research area. Recent works in semi-supervised learning (Chapelle et al., 2010; Kingma et al., 2014; Kipf & Welling, 2016; Laine & Aila, 2016; Sohn et al., 2020; Xie et al., 2020) and unsupervised domain adaptation (Ben-David et al., 2010; Ganin & Lempitsky, 2015; Ganin et al., 2016; Tzeng et al., 2017; Hoffman et al., 2018; Shu et al., 2018; Zhang et al., 2019) leverage lots of unlabeled data as well as labeled data from the same distribution or a related distribution. Recent progress in unsupervised learning or representation learning (Hinton et al., 1999; Doersch et al., 2015; Gidaris et al., 2018; Misra & Maaten, 2020; Chen et al., 2020a; b; Grill et al., 2020) learns high-quality representations without using any labels. Self-training is a common algorithmic paradigm for leveraging unlabeled data with deep networks. Self-training methods train a model to fit pseudolabels, that is, predictions on unlabeled data made by a previously-learned model (Yarowsky, 1995; Grandvalet & Bengio, 2005; Lee, 2013) . Recent work also extends these methods to enforce stability of predictions under input transformations such as adversarial perturbations (Miyato et al., 2018) and data augmentation (Xie et al., 2019) . These approaches, known as input consistency regularization, have been successful in semi-supervised learning (Sohn et al., 2020; Xie et al., 2020) , unsupervised domain adaptation (French et al., 2017; Shu et al., 2018) , and unsupervised learning (Hu et al., 2017; Grill et al., 2020) . Despite the empirical successes, theoretical progress in understanding how to use unlabeled data has lagged. Whereas supervised learning is relatively well-understood, statistical tools for reasoning about unlabeled data are not as readily available. Around 25 years ago, Vapnik (1995) proposed the transductive SVM for unlabeled data, which can be viewed as an early version of self-training, yet there is little work showing that this method improves sample complexity (Derbeko et al., 2004) .

