Convergence and Implicit Bias of Gradient Flow on Overparametrized Linear Networks
This provides theoretical insights into implicit bias in neural networks, addressing a foundational issue in machine learning, though it is incremental as it builds on prior work on linear models.
The paper tackles the problem of understanding why overparametrized neural networks generalize well by analyzing gradient flow on single-hidden-layer linear networks, showing that squared loss converges exponentially to its optimum and that proper initialization leads to the min-norm solution, with a non-asymptotic upper-bound on the distance to this solution.
Neural networks trained via gradient descent with random initialization and without any regularization enjoy good generalization performance in practice despite being highly overparametrized. A promising direction to explain this phenomenon is to study how initialization and overparametrization affect convergence and implicit bias of training algorithms. In this paper, we present a novel analysis of single-hidden-layer linear networks trained under gradient flow, which connects initialization, optimization, and overparametrization. Firstly, we show that the squared loss converges exponentially to its optimum at a rate that depends on the level of imbalance and the margin of the initialization. Secondly, we show that proper initialization constrains the dynamics of the network parameters to lie within an invariant set. In turn, minimizing the loss over this set leads to the min-norm solution. Finally, we show that large hidden layer width, together with (properly scaled) random initialization, ensures proximity to such an invariant set during training, allowing us to derive a novel non-asymptotic upper-bound on the distance between the trained network and the min-norm solution.