Implicit regularization for deep neural networks driven by an Ornstein-Uhlenbeck like process
This provides theoretical insight into implicit regularization mechanisms in deep learning, which is incremental but clarifies a known bottleneck for researchers in optimization and generalization.
The paper analyzes how adding noise to training labels during stochastic gradient descent leads to implicit regularization in deep neural networks, characterized by a term that penalizes the squared gradient norm of the model with respect to parameters, and demonstrates this effect drives networks toward simpler models in specific settings like matrix sensing and two-layer networks.
We consider networks, trained via stochastic gradient descent to minimize $\ell_2$ loss, with the training labels perturbed by independent noise at each iteration. We characterize the behavior of the training dynamics near any parameter vector that achieves zero training error, in terms of an implicit regularization term corresponding to the sum over the data points, of the squared $\ell_2$ norm of the gradient of the model with respect to the parameter vector, evaluated at each data point. This holds for networks of any connectivity, width, depth, and choice of activation function. We interpret this implicit regularization term for three simple settings: matrix sensing, two layer ReLU networks trained on one-dimensional data, and two layer networks with sigmoid activations trained on a single datapoint. For these settings, we show why this new and general implicit regularization effect drives the networks towards "simple" models.