Explicit regularization and implicit bias in deep network classifiers trained with the square loss

Abstract

Deep ReLU networks trained with the square loss have been observed to perform well in classification tasks. We provide here a theoretical justification based on analysis of the associated gradient flow. We show that convergence to a solution with the absolute minimum norm is expected when normalization techniques such as Batch Normalization (BN) or Weight Normalization (WN) are used together with Weight Decay (WD). The main property of the minimizers that bounds their expected error is the norm: we prove that among all the close-to-interpolating solutions, the ones associated with smaller Frobenius norms of the unnormalized weight matrices have better margin and better bounds on the expected classification error. With BN but in the absence of WD, the dynamical system is singular. Implicit dynamical regularization -- that is zero-initial conditions biasing the dynamics towards high margin solutions -- is also possible in the no-BN and no-WD case. The theory yields several predictions, including the role of BN and weight decay, aspects of Papyan, Han and Donoho's Neural Collapse and the constraints induced by BN on the network weights.

References in corpus (5)

• Prevalence of Neural Collapse during the terminal phase of deep learning training
• Theoretical Analysis of Auto Rate-Tuning by Batch Normalization
• Lexicographic and Depth-Sensitive Margins in Homogeneous and Non-Homogeneous Deep Models
• The Implicit Bias of Depth: How Incremental Learning Drives Generalization
• Implicit Regularization in ReLU Networks with the Square Loss

Cited by in corpus (4)

• Exploring Deep Neural Networks via Layer-Peeled Model: Minority Collapse in Imbalanced Training
• Neural Collapse Under MSE Loss: Proximity to and Dynamics on the Central Path
• Benign Overfitting in Multiclass Classification: All Roads Lead to Interpolation
• An Unconstrained Layer-Peeled Perspective on Neural Collapse