Implicit Regularization and Convergence for Weight Normalization

2019/11/18 by Xiaoxia Wu, Edgar Dobriban, Wu, Xiaoxia +13 · 4 citations
Computer Science · Engineering · #Computer Vision and Pattern Recognition (cs.CV) #FOS: Computer and information sciences #FOS: Mathematics #Face and Expression Recognition #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques #Stochastic Gradient Optimization Techniques

paper · pdf · doi:10.48550/arxiv.1911.07956

openalex publication_date 2019/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Normalization methods such as batch [Ioffe and Szegedy, 2015], weight [Salimansand Kingma, 2016], instance [Ulyanov et al., 2016], and layer normalization [Baet al., 2016] have been widely used in modern machine learning. Here, we study the weight normalization (WN) method [Salimans and Kingma, 2016] and a variant called reparametrized projected gradient descent (rPGD) for overparametrized least-squares regression. WN and rPGD reparametrize the weights with a scale g and a unit vector w and thus the objective function becomes non-convex. We show that this non-convex formulation has beneficial regularization effects compared to gradient descent on the original objective. These methods adaptively regularize the weights and converge close to the minimum l2 norm solution, even for initializations far from zero. For certain stepsizes of g and w , we show that they can converge close to the minimum norm solution. This is different from the behavior of gradient descent, which converges to the minimum norm solution only when started at a point in the range space of the feature matrix, and is thus more sensitive to initialization.

Citations

• Dropout Training as Adaptive Regularization
• Gradient Descent Converges to Minimizers
• Instance Normalization: The Missing Ingredient for Fast Stylization
• No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified\n Geometric Analysis
• Accurate, Large Minibatch SGD: Training ImageNet in 1 Hour
• A Continuous-Time View of Early Stopping for Least Squares
• Implicit Regularization in Deep Matrix Factorization
• Understanding deep learning requires rethinking generalization
• Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning
• Implementing regularization implicitly via approximate eigenvector computation
• Dropout: Explicit Forms and Capacity Control
• Optimization Theory for ReLU Neural Networks Trained with Normalization\n Layers

Cited by

• Tuning-Free Structured Sparse Recovery of Multiple Measurement Vectors using Implicit Regularization
• Multivariate Variational Autoencoder
• The Hidden Power of Normalization Layers in Neural Networks: Exponential Capacity Control
• On the Benefits of Weight Normalization for Overparameterized Matrix Sensing

Related

• Implicit Regularization for Optimal Sparse Recovery
• Understanding the Disharmony between Weight Normalization Family and Weight Decay: ε-shifted L₂ Regularizer
• L2 Regularization versus Batch and Weight Normalization
• A Unified Approach to Controlling Implicit Regularization via Mirror Descent
• Reparametrizing gradient descent