Non-convergence of stochastic gradient descent in the training of deep neural networks
This addresses a foundational theoretical gap in machine learning by showing non-convergence under specific conditions, which is incremental as it builds on prior work on discretization parameters.
The paper tackles the problem of understanding why stochastic gradient descent (SGD) works well for training deep neural networks, demonstrating that it fails to converge for ReLU networks when depth is much larger than width and random initializations do not increase fast enough.
Deep neural networks have successfully been trained in various application areas with stochastic gradient descent. However, there exists no rigorous mathematical explanation why this works so well. The training of neural networks with stochastic gradient descent has four different discretization parameters: (i) the network architecture; (ii) the amount of training data; (iii) the number of gradient steps; and (iv) the number of randomly initialized gradient trajectories. While it can be shown that the approximation error converges to zero if all four parameters are sent to infinity in the right order, we demonstrate in this paper that stochastic gradient descent fails to converge for ReLU networks if their depth is much larger than their width and the number of random initializations does not increase to infinity fast enough.