Unifying Low Dimensional Observations in Deep Learning Through the Deep Linear Unconstrained Feature Model

• URL: http://arxiv.org/abs/2404.06106v1
• Date: Tue, 9 Apr 2024 08:17:32 GMT
• Title: Unifying Low Dimensional Observations in Deep Learning Through the Deep Linear Unconstrained Feature Model
• Authors: Connall Garrod, Jonathan P. Keating,
• Abstract summary: We study low-dimensional structure in the weights, Hessian's, gradients, and feature vectors of deep neural networks. We show how they can be unified within a generalized unconstrained feature model.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Modern deep neural networks have achieved high performance across various tasks. Recently, researchers have noted occurrences of low-dimensional structure in the weights, Hessian's, gradients, and feature vectors of these networks, spanning different datasets and architectures when trained to convergence. In this analysis, we theoretically demonstrate these observations arising, and show how they can be unified within a generalized unconstrained feature model that can be considered analytically. Specifically, we consider a previously described structure called Neural Collapse, and its multi-layer counterpart, Deep Neural Collapse, which emerges when the network approaches global optima. This phenomenon explains the other observed low-dimensional behaviours on a layer-wise level, such as the bulk and outlier structure seen in Hessian spectra, and the alignment of gradient descent with the outlier eigenspace of the Hessian. Empirical results in both the deep linear unconstrained feature model and its non-linear equivalent support these predicted observations.

Related papers

• Convergence Analysis for Learning Orthonormal Deep Linear Neural Networks [27.29463801531576]
  We provide convergence analysis for training orthonormal deep linear neural networks. Our results shed light on how increasing the number of hidden layers can impact the convergence speed.
  arXiv  Detail & Related papers  (2023-11-24T18:46:54Z)
• From Complexity to Clarity: Analytical Expressions of Deep Neural Network Weights via Clifford's Geometric Algebra and Convexity [54.01594785269913]
  We show that optimal weights of deep ReLU neural networks are given by the wedge product of training samples when trained with standard regularized loss. The training problem reduces to convex optimization over wedge product features, which encode the geometric structure of the training dataset.
  arXiv  Detail & Related papers  (2023-09-28T15:19:30Z)
• Deep Neural Collapse Is Provably Optimal for the Deep Unconstrained Features Model [21.79259092920587]
  We show that in a deep unconstrained features model, the unique global optimum for binary classification exhibits all the properties typical of deep neural collapse (DNC) We also empirically show that (i) by optimizing deep unconstrained features models via gradient descent, the resulting solution agrees well with our theory, and (ii) trained networks recover the unconstrained features suitable for DNC.
  arXiv  Detail & Related papers  (2023-05-22T15:51:28Z)
• Neural Collapse in Deep Linear Networks: From Balanced to Imbalanced Data [12.225207401994737]
  We show that complex systems with massive amounts of parameters exhibit the same structural properties when training until convergence. In particular, it has been observed that the last-layer features collapse to their class-means. Our results demonstrate the convergence of the last-layer features and classifiers to a geometry consisting of vectors.
  arXiv  Detail & Related papers  (2023-01-01T16:29:56Z)
• Multi-scale Feature Learning Dynamics: Insights for Double Descent [71.91871020059857]
  We study the phenomenon of "double descent" of the generalization error. We find that double descent can be attributed to distinct features being learned at different scales.
  arXiv  Detail & Related papers  (2021-12-06T18:17:08Z)
• An Unconstrained Layer-Peeled Perspective on Neural Collapse [20.75423143311858]
  We introduce a surrogate model called the unconstrained layer-peeled model (ULPM) We prove that gradient flow on this model converges to critical points of a minimum-norm separation problem exhibiting neural collapse in its global minimizer. We show that our results also hold during the training of neural networks in real-world tasks when explicit regularization or weight decay is not used.
  arXiv  Detail & Related papers  (2021-10-06T14:18:47Z)
• The Interplay Between Implicit Bias and Benign Overfitting in Two-Layer Linear Networks [51.1848572349154]
  neural network models that perfectly fit noisy data can generalize well to unseen test data. We consider interpolating two-layer linear neural networks trained with gradient flow on the squared loss and derive bounds on the excess risk.
  arXiv  Detail & Related papers  (2021-08-25T22:01:01Z)
• Gradient Starvation: A Learning Proclivity in Neural Networks [97.02382916372594]
  Gradient Starvation arises when cross-entropy loss is minimized by capturing only a subset of features relevant for the task. This work provides a theoretical explanation for the emergence of such feature imbalance in neural networks.
  arXiv  Detail & Related papers  (2020-11-18T18:52:08Z)
• The Neural Tangent Kernel in High Dimensions: Triple Descent and a Multi-Scale Theory of Generalization [34.235007566913396]
  Modern deep learning models employ considerably more parameters than required to fit the training data. Whereas conventional statistical wisdom suggests such models should drastically overfit, in practice these models generalize remarkably well. An emerging paradigm for describing this unexpected behavior is in terms of a emphdouble descent curve. We provide a precise high-dimensional analysis of generalization with the Neural Tangent Kernel, which characterizes the behavior of wide neural networks with gradient descent.
  arXiv  Detail & Related papers  (2020-08-15T20:55:40Z)
• Hyperbolic Neural Networks++ [66.16106727715061]
  We generalize the fundamental components of neural networks in a single hyperbolic geometry model, namely, the Poincar'e ball model. Experiments show the superior parameter efficiency of our methods compared to conventional hyperbolic components, and stability and outperformance over their Euclidean counterparts.
  arXiv  Detail & Related papers  (2020-06-15T08:23:20Z)
• Kernel and Rich Regimes in Overparametrized Models [69.40899443842443]
  We show that gradient descent on overparametrized multilayer networks can induce rich implicit biases that are not RKHS norms. We also demonstrate this transition empirically for more complex matrix factorization models and multilayer non-linear networks.
  arXiv  Detail & Related papers  (2020-02-20T15:43:02Z)

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.