We investigate the training dynamics of two-layer neural networks when learning multi-index target functions. We focus on multi-pass gradient descent (GD) that reuses the batches multiple times and show that it significantly changes the conclusion about which functions are learnable compared to single-pass gradient descent. In particular, multi-pass GD with finite stepsize is found to overcome the limitations of gradient flow and single-pass GD given by the information exponent (Ben Arous et al., 2021) and leap exponent (Abbe et al., 2023) of the target function. We show that upon re-using batches, the network achieves in just two time steps an overlap with the target subspace even for functions not satisfying the staircase property (Abbe et al., 2021). We characterize the (broad) class of functions efficiently learned in finite time. The proof of our results is based on the analysis of the Dynamical Mean-Field Theory (DMFT). We further provide a closed-form description of the dynamical process of the low-dimensional projections of the weights, and numerical experiments illustrating the theory.

翻译：我们研究了两层神经网络在学习多指标目标函数时的训练动态。我们重点关注重复使用批次多次的多轮梯度下降（GD），并发现与单轮梯度下降相比，它显著改变了对哪些函数可学习的结论。特别是，具有有限步长的多轮GD能够克服由目标函数的信息指数（Ben Arous 等，2021）和跳跃指数（Abbe 等，2023）所设定的梯度流和单轮GD的局限性。我们证明，通过重复使用批次，网络即使在两个时间步内也能实现与目标子空间的重叠，即使对于不满足阶梯性质（Abbe 等，2021）的函数也是如此。我们刻画了在有限时间内高效学习的（广泛）函数类。我们的证明基于动力学平均场理论（DMFT）的分析。此外，我们提供了权重低维投影动态过程的闭式描述，以及说明该理论的数值实验。