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Dynamics in Deep Classifiers Trained with the Square Loss: Normalization, Low Rank, Neural Collapse, and

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This study explores training deep neural networks with square loss, revealing how weight decay and gradient descent influence solutions. Findings include improved bounds for convolutional layers and a bias toward low-rank matrices, enhancing generalization and predicting neural collapse.

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Area of Science:

  • Machine Learning
  • Deep Learning Theory
  • Optimization

Background:

  • Overparameterized deep networks trained with square loss exhibit complex dynamics.
  • Understanding convergence properties and error bounds is crucial for network performance.
  • Existing bounds for dense networks do not fully capture the benefits of specific architectures like CNNs.

Purpose of the Study:

  • To analyze the properties of training deep homogeneous rectified linear unit networks under square loss.
  • To derive novel norm-based error bounds for convolutional layers.
  • To investigate the generalization properties of solutions obtained via stochastic gradient descent with weight decay.

Main Methods:

  • Modeling gradient flow dynamics under square loss.
  • Utilizing Lagrange multipliers and weight decay with gradient descent variants.
  • Deriving norm-based bounds for convolutional and dense layers.
  • Analyzing quasi-interpolating solutions from stochastic gradient descent.

Main Results:

  • Achieved convergence to a minimum Frobenius norm product (ρ) with normalized weight decay.
  • Derived significantly improved norm-based bounds for convolutional layers compared to dense networks.
  • Demonstrated that stochastic gradient descent with weight decay introduces a bias towards low-rank weight matrices, enhancing generalization.
  • Predicted and experimentally verified inherent stochastic gradient descent noise and neural collapse phenomena.
  • Showcased the advantage of deep networks for problems suited to sparse architectures.

Conclusions:

  • Deep networks, particularly sparse architectures like CNNs, offer advantages for specific problems by avoiding the curse of dimensionality.
  • The derived bounds and analysis provide theoretical insights into the generalization capabilities of deep learning models.
  • The study confirms the effectiveness of weight decay and gradient descent in achieving desirable network properties.