Leveraging the two timescale regime to demonstrate convergence of neural networks
This work addresses convergence guarantees for neural network training, offering a novel theoretical framework distinct from existing approaches like neural tangent kernel or mean-field regimes, though it is incremental in scope.
The authors tackled the non-convex optimization problem in training shallow neural networks by proving convergence to a global optimum in a two-timescale regime with different step sizes for inner and outer layers, demonstrating this experimentally with stochastic gradient descent.
We study the training dynamics of shallow neural networks, in a two-timescale regime in which the stepsizes for the inner layer are much smaller than those for the outer layer. In this regime, we prove convergence of the gradient flow to a global optimum of the non-convex optimization problem in a simple univariate setting. The number of neurons need not be asymptotically large for our result to hold, distinguishing our result from popular recent approaches such as the neural tangent kernel or mean-field regimes. Experimental illustration is provided, showing that the stochastic gradient descent behaves according to our description of the gradient flow and thus converges to a global optimum in the two-timescale regime, but can fail outside of this regime.