TRIPODS Winter School & Workshop- Guido Montufar
Title- Implicit Bias of Gradient Descent for Mean Squared Error Regression with Wide Neural Networks
Abstract- We investigate gradient descent training of wide neural networks and the corresponding implicit bias in function space. For 1D regression, we show that the solution of training a width-n shallow ReLU network is within n−1/2 of the function which fits the training data and whose difference from initialization has smallest 2-norm of the weighted second derivative with respect to the input. The curvature penalty function 1/ζ is expressed in terms of the probability distribution that is utilized to initialize the network parameters, and we compute it explicitly for various common initialization procedures. For instance, asymmetric initialization with a uniform distribution yields a constant curvature penalty, and hence the solution function is the natural cubic spline interpolation of the training data. While similar results have been obtained in previous works, the analysis clarifies important details and allows us to obtain significant generalizations. In particular, the result generalizes to multivariate regression and different activation functions. Moreover, we show that the training trajectories are captured by trajectories of spatially adaptive smoothing splines with decreasing regularization strength. (Joint work with Hui Jin.)