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.

翻译：我们研究浅层神经网络的训练动力学，采用一种双时间尺度机制，其中内层学习率远小于外层学习率。在此机制下，我们证明在简单单变量设定中，梯度流能够收敛到非凸优化问题的全局最优解。我们的结果并不要求神经元数量渐近趋于无穷，这与近期流行的神经正切核或平均场机制等方法形成区分。实验验证表明，随机梯度下降的行为与我们对梯度流的描述一致，因此在双时间尺度机制下能够收敛到全局最优解，但在此机制之外则可能失败。