Early Stage Convergence and Global Convergence of Training Mildly Parameterized Neural Networks
This addresses the theoretical challenge of convergence in neural network training for researchers, offering insights without relying on extreme over-parameterization, though it is incremental in building on existing convergence studies.
The paper tackles the problem of understanding convergence in training mildly parameterized neural networks, proving that loss decreases significantly and quickly in early stages for common loss functions, and achieving global convergence for exponential-type losses under certain data assumptions.
The convergence of GD and SGD when training mildly parameterized neural networks starting from random initialization is studied. For a broad range of models and loss functions, including the most commonly used square loss and cross entropy loss, we prove an ``early stage convergence'' result. We show that the loss is decreased by a significant amount in the early stage of the training, and this decrease is fast. Furthurmore, for exponential type loss functions, and under some assumptions on the training data, we show global convergence of GD. Instead of relying on extreme over-parameterization, our study is based on a microscopic analysis of the activation patterns for the neurons, which helps us derive more powerful lower bounds for the gradient. The results on activation patterns, which we call ``neuron partition'', help build intuitions for understanding the behavior of neural networks' training dynamics, and may be of independent interest.