  Noisy training set usually leads to the degradation of generalization and
robustness of neural networks. In this paper, we propose using a theoretically
guaranteed noisy label detection framework to detect and remove noisy data for
Learning with Noisy Labels (LNL). Specifically, we design a penalized
regression to model the linear relation between network features and one-hot
labels, where the noisy data are identified by the non-zero mean shift
parameters solved in the regression model. To make the framework scalable to
datasets that contain a large number of categories and training data, we
propose a split algorithm to divide the whole training set into small pieces
that can be solved by the penalized regression in parallel, leading to the
Scalable Penalized Regression (SPR) framework. We provide the non-asymptotic
probabilistic condition for SPR to correctly identify the noisy data. While SPR
can be regarded as a sample selection module for standard supervised training
pipeline, we further combine it with semi-supervised algorithm to further
exploit the support of noisy data as unlabeled data. Experimental results on
several benchmark datasets and real-world noisy datasets show the effectiveness
of our framework. Our code and pretrained models are released at
https://github.com/Yikai-Wang/SPR-LNL.
