Towards Better Generalization: Weight Decay Induces Low-rank Bias for Neural Networks

• URL: http://arxiv.org/abs/2410.02176v1
• Date: Thu, 3 Oct 2024 03:36:18 GMT
• Title: Towards Better Generalization: Weight Decay Induces Low-rank Bias for Neural Networks
• Authors: Ke Chen, Chugang Yi, Haizhao Yang,
• Abstract summary: We study the implicit bias towards low-rank weight matrices when training neural networks with Weight Decay (WD) Our work offers both theoretical and empirical insights into the strong generalization performance of SGD when combined with WD.
• Score: 9.948870430491738
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the implicit bias towards low-rank weight matrices when training neural networks (NN) with Weight Decay (WD). We prove that when a ReLU NN is sufficiently trained with Stochastic Gradient Descent (SGD) and WD, its weight matrix is approximately a rank-two matrix. Empirically, we demonstrate that WD is a necessary condition for inducing this low-rank bias across both regression and classification tasks. Our work differs from previous studies as our theoretical analysis does not rely on common assumptions regarding the training data distribution, optimality of weight matrices, or specific training procedures. Furthermore, by leveraging the low-rank bias, we derive improved generalization error bounds and provide numerical evidence showing that better generalization can be achieved. Thus, our work offers both theoretical and empirical insights into the strong generalization performance of SGD when combined with WD.

Related papers

• Towards Demystifying the Generalization Behaviors When Neural Collapse Emerges [132.62934175555145]
  Neural Collapse (NC) is a well-known phenomenon of deep neural networks in the terminal phase of training (TPT) We propose a theoretical explanation for why continuing training can still lead to accuracy improvement on test set, even after the train accuracy has reached 100%. We refer to this newly discovered property as "non-conservative generalization"
  arXiv  Detail & Related papers  (2023-10-12T14:29:02Z)
• Heavy-Tailed Regularization of Weight Matrices in Deep Neural Networks [8.30897399932868]
  Key finding indicates that the generalization performance of a neural network is associated with the degree of heavy tails in the spectrum of its weight matrices. We introduce a novel regularization technique, termed Heavy-Tailed Regularization, which explicitly promotes a more heavy-tailed spectrum in the weight matrix through regularization. We empirically show that heavytailed regularization outperforms conventional regularization techniques in terms of generalization performance.
  arXiv  Detail & Related papers  (2023-04-06T07:50:14Z)
• SGD and Weight Decay Secretly Minimize the Rank of Your Neural Network [8.79431718760617]
  Training with mini-batch SGD and weight decay induces a bias toward rank minimization in weight matrices. We show that this bias becomes more pronounced with smaller batch sizes, higher learning rates, or stronger weight decay. We empirically explore the connection between this bias and generalization, finding that it has a marginal effect on the test performance.
  arXiv  Detail & Related papers  (2022-06-12T17:06:35Z)
• Stochastic Training is Not Necessary for Generalization [57.04880404584737]
  It is widely believed that the implicit regularization of gradient descent (SGD) is fundamental to the impressive generalization behavior we observe in neural networks. In this work, we demonstrate that non-stochastic full-batch training can achieve strong performance on CIFAR-10 that is on-par with SGD.
  arXiv  Detail & Related papers  (2021-09-29T00:50:00Z)
• Distribution of Classification Margins: Are All Data Equal? [61.16681488656473]
  We motivate theoretically and show empirically that the area under the curve of the margin distribution on the training set is in fact a good measure of generalization. The resulting subset of "high capacity" features is not consistent across different training runs.
  arXiv  Detail & Related papers  (2021-07-21T16:41:57Z)
• On the Generalization of Stochastic Gradient Descent with Momentum [58.900860437254885]
  We first show that there exists a convex loss function for which algorithmic stability fails to establish generalization guarantees. For smooth Lipschitz loss functions, we analyze a modified momentum-based update rule, and show that it admits an upper-bound on the generalization error. For the special case of strongly convex loss functions, we find a range of momentum such that multiple epochs of standard SGDM, as a special form of SGDEM, also generalizes.
  arXiv  Detail & Related papers  (2021-02-26T18:58:29Z)
• The Implicit Biases of Stochastic Gradient Descent on Deep Neural Networks with Batch Normalization [44.30960913470372]
  Deep neural networks with batch normalization (BN-DNNs) are invariant to weight rescaling due to their normalization operations. We investigate the implicit biases of gradient descent (SGD) on BN-DNNs to provide a theoretical explanation for the efficacy of weight decay.
  arXiv  Detail & Related papers  (2021-02-06T03:40:20Z)
• Explicit regularization and implicit bias in deep network classifiers trained with the square loss [2.8935588665357077]
  Deep ReLU networks trained with the square loss have been observed to perform well in classification tasks. We show that convergence to a solution with the absolute minimum norm is expected when normalization techniques are used together with Weight Decay.
  arXiv  Detail & Related papers  (2020-12-31T21:07:56Z)
• Revisiting Initialization of Neural Networks [72.24615341588846]
  We propose a rigorous estimation of the global curvature of weights across layers by approximating and controlling the norm of their Hessian matrix. Our experiments on Word2Vec and the MNIST/CIFAR image classification tasks confirm that tracking the Hessian norm is a useful diagnostic tool.
  arXiv  Detail & Related papers  (2020-04-20T18:12:56Z)
• On the Generalization of Stochastic Gradient Descent with Momentum [84.54924994010703]
  momentum-based accelerated variants of gradient descent (SGD) are widely used when training machine learning models. We first show that there exists a convex loss function for which the stability gap for multiple epochs of SGD with standard heavy-ball momentum (SGDM) becomes unbounded. For smooth Lipschitz loss functions, we analyze a modified momentum-based update rule, i.e., SGD with early momentum (SGDEM) under a broad range of step-sizes.
  arXiv  Detail & Related papers  (2018-09-12T17:02:08Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.