$ε$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics

• URL: http://arxiv.org/abs/2412.05144v3
• Date: Fri, 18 Jul 2025 14:59:07 GMT
• Title: $ε$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics
• Authors: Jiang Yang, Yuxiang Zhao, Quanhui Zhu,
• Abstract summary: We introduce the concept of $epsilon$-rank, a novel metric quantifying the effective feature of neuron functions in the terminal hidden layer.<n>We prove a negative correlation between the loss lower bound and $epsilon$-rank, demonstrating that a high $epsilon$-rank is essential for significant loss reduction.<n>We propose a novel pre-training strategy on the initial hidden layer that elevates the $epsilon$-rank of the terminal hidden layer.
• Score: 1.7056144431280509
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Understanding the training dynamics of deep neural networks (DNNs), particularly how they evolve low-dimensional features from high-dimensional data, remains a central challenge in deep learning theory. In this work, we introduce the concept of $\epsilon$-rank, a novel metric quantifying the effective feature of neuron functions in the terminal hidden layer. Through extensive experiments across diverse tasks, we observe a universal staircase phenomenon: during training process implemented by the standard stochastic gradient descent methods, the decline of the loss function is accompanied by an increase in the $\epsilon$-rank and exhibits a staircase pattern. Theoretically, we rigorously prove a negative correlation between the loss lower bound and $\epsilon$-rank, demonstrating that a high $\epsilon$-rank is essential for significant loss reduction. Moreover, numerical evidences show that within the same deep neural network, the $\epsilon$-rank of the subsequent hidden layer is higher than that of the previous hidden layer. Based on these observations, to eliminate the staircase phenomenon, we propose a novel pre-training strategy on the initial hidden layer that elevates the $\epsilon$-rank of the terminal hidden layer. Numerical experiments validate its effectiveness in reducing training time and improving accuracy across various tasks. Therefore, the newly introduced concept of $\epsilon$-rank is a computable quantity that serves as an intrinsic effective metric characteristic for deep neural networks, providing a novel perspective for understanding the training dynamics of neural networks and offering a theoretical foundation for designing efficient training strategies in practical applications.

Related papers

• Information-Theoretic Greedy Layer-wise Training for Traffic Sign Recognition [0.5024983453990065]
  layer-wise training eliminates the need for cross-entropy loss and backpropagation.<n>Most existing layer-wise training approaches have been evaluated only on relatively small datasets.<n>We propose a novel layer-wise training approach based on the recently developed deterministic information bottleneck (DIB) and the matrix-based R'enyi's $alpha$-order entropy functional.
  arXiv  Detail & Related papers  (2025-10-31T17:24:58Z)
• Global Convergence and Rich Feature Learning in $L$-Layer Infinite-Width Neural Networks under $μ$P Parametrization [66.03821840425539]
  In this paper, we investigate the training dynamics of $L$-layer neural networks using the tensor gradient program (SGD) framework.<n>We show that SGD enables these networks to learn linearly independent features that substantially deviate from their initial values.<n>This rich feature space captures relevant data information and ensures that any convergent point of the training process is a global minimum.
  arXiv  Detail & Related papers  (2025-03-12T17:33:13Z)
• Dynamical loss functions shape landscape topography and improve learning in artificial neural networks [0.9208007322096533]
  We show how to transform cross-entropy and mean squared error into dynamical loss functions. We show how they significantly improve validation accuracy for networks of varying sizes.
  arXiv  Detail & Related papers  (2024-10-14T16:27:03Z)
• Demystifying Lazy Training of Neural Networks from a Macroscopic Viewpoint [5.9954962391837885]
  We study the gradient descent dynamics of neural networks through the lens of macroscopic limits. Our study reveals that gradient descent can rapidly drive deep neural networks to zero training loss. Our approach draws inspiration from the Neural Tangent Kernel (NTK) paradigm.
  arXiv  Detail & Related papers  (2024-04-07T08:07:02Z)
• Simple and Effective Transfer Learning for Neuro-Symbolic Integration [50.592338727912946]
  A potential solution to this issue is Neuro-Symbolic Integration (NeSy), where neural approaches are combined with symbolic reasoning. Most of these methods exploit a neural network to map perceptions to symbols and a logical reasoner to predict the output of the downstream task. They suffer from several issues, including slow convergence, learning difficulties with complex perception tasks, and convergence to local minima. This paper proposes a simple yet effective method to ameliorate these problems.
  arXiv  Detail & Related papers  (2024-02-21T15:51:01Z)
• Neural Rank Collapse: Weight Decay and Small Within-Class Variability Yield Low-Rank Bias [4.829265670567825]
  We show the presence of an intriguing neural rank collapse phenomenon, connecting the low-rank bias of trained networks with networks' neural collapse properties. As the weight decay parameter grows, the rank of each layer in the network decreases proportionally to the within-class variability of the hidden-space embeddings of the previous layers.
  arXiv  Detail & Related papers  (2024-02-06T13:44:39Z)
• Elephant Neural Networks: Born to Be a Continual Learner [7.210328077827388]
  Catastrophic forgetting remains a significant challenge to continual learning for decades. We study the role of activation functions in the training dynamics of neural networks and their impact on catastrophic forgetting. We show that by simply replacing classical activation functions with elephant activation functions, we can significantly improve the resilience of neural networks to catastrophic forgetting.
  arXiv  Detail & Related papers  (2023-10-02T17:27:39Z)
• Addressing caveats of neural persistence with deep graph persistence [54.424983583720675]
  We find that the variance of network weights and spatial concentration of large weights are the main factors that impact neural persistence. We propose an extension of the filtration underlying neural persistence to the whole neural network instead of single layers. This yields our deep graph persistence measure, which implicitly incorporates persistent paths through the network and alleviates variance-related issues.
  arXiv  Detail & Related papers  (2023-07-20T13:34:11Z)
• Globally Optimal Training of Neural Networks with Threshold Activation Functions [63.03759813952481]
  We study weight decay regularized training problems of deep neural networks with threshold activations. We derive a simplified convex optimization formulation when the dataset can be shattered at a certain layer of the network.
  arXiv  Detail & Related papers  (2023-03-06T18:59:13Z)
• Online Loss Function Learning [13.744076477599707]
  Loss function learning aims to automate the task of designing a loss function for a machine learning model. We propose a new loss function learning technique for adaptively updating the loss function online after each update to the base model parameters.
  arXiv  Detail & Related papers  (2023-01-30T19:22:46Z)
• Spiking neural network for nonlinear regression [68.8204255655161]
  Spiking neural networks carry the potential for a massive reduction in memory and energy consumption. They introduce temporal and neuronal sparsity, which can be exploited by next-generation neuromorphic hardware. A framework for regression using spiking neural networks is proposed.
  arXiv  Detail & Related papers  (2022-10-06T13:04:45Z)
• Early Stage Convergence and Global Convergence of Training Mildly Parameterized Neural Networks [3.148524502470734]
  We show that the loss is decreased by a significant amount in the early stage of the training, and this decrease is fast. We use a microscopic analysis of the activation patterns for the neurons, which helps us derive more powerful lower bounds for the gradient.
  arXiv  Detail & Related papers  (2022-06-05T09:56:50Z)
• Neural Capacitance: A New Perspective of Neural Network Selection via Edge Dynamics [85.31710759801705]
  Current practice requires expensive computational costs in model training for performance prediction. We propose a novel framework for neural network selection by analyzing the governing dynamics over synaptic connections (edges) during training. Our framework is built on the fact that back-propagation during neural network training is equivalent to the dynamical evolution of synaptic connections.
  arXiv  Detail & Related papers  (2022-01-11T20:53:15Z)
• A proof of convergence for stochastic gradient descent in the training of artificial neural networks with ReLU activation for constant target functions [3.198144010381572]
  We study the gradient descent (SGD) optimization method in the training of fully-connected feedforward artificial neural networks with ReLU activation. The main result of this work proves that the risk of the SGD process converges to zero if the target function under consideration is constant.
  arXiv  Detail & Related papers  (2021-04-01T06:28:30Z)
• 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)
• Plateau Phenomenon in Gradient Descent Training of ReLU networks: Explanation, Quantification and Avoidance [0.0]
  In general, neural networks are trained by gradient type optimization methods. The loss function decreases rapidly at the beginning of training but then, after a relatively small number of steps, significantly slow down. The present work aims to identify and quantify the root causes of plateau phenomenon.
  arXiv  Detail & Related papers  (2020-07-14T17:33:26Z)
• Untangling tradeoffs between recurrence and self-attention in neural networks [81.30894993852813]
  We present a formal analysis of how self-attention affects gradient propagation in recurrent networks. We prove that it mitigates the problem of vanishing gradients when trying to capture long-term dependencies. We propose a relevancy screening mechanism that allows for a scalable use of sparse self-attention with recurrence.
  arXiv  Detail & Related papers  (2020-06-16T19:24:25Z)
• Understanding the Role of Training Regimes in Continual Learning [51.32945003239048]
  Catastrophic forgetting affects the training of neural networks, limiting their ability to learn multiple tasks sequentially. We study the effect of dropout, learning rate decay, and batch size, on forming training regimes that widen the tasks' local minima.
  arXiv  Detail & Related papers  (2020-06-12T06:00:27Z)

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.