High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation
This provides theoretical insights into the early dynamics of neural network training, which is incremental but clarifies a specific bottleneck in understanding feature learning.
The paper tackles the problem of understanding how a single gradient step on the first-layer parameters of a two-layer neural network improves feature learning in high-dimensional settings, showing that for sufficiently large learning rates, the trained features can outperform random features and linear models on certain teacher models.
We study the first gradient descent step on the first-layer parameters $\boldsymbol{W}$ in a two-layer neural network: $f(\boldsymbol{x}) = \frac{1}{\sqrt{N}}\boldsymbol{a}^\topσ(\boldsymbol{W}^\top\boldsymbol{x})$, where $\boldsymbol{W}\in\mathbb{R}^{d\times N}, \boldsymbol{a}\in\mathbb{R}^{N}$ are randomly initialized, and the training objective is the empirical MSE loss: $\frac{1}{n}\sum_{i=1}^n (f(\boldsymbol{x}_i)-y_i)^2$. In the proportional asymptotic limit where $n,d,N\to\infty$ at the same rate, and an idealized student-teacher setting, we show that the first gradient update contains a rank-1 "spike", which results in an alignment between the first-layer weights and the linear component of the teacher model $f^*$. To characterize the impact of this alignment, we compute the prediction risk of ridge regression on the conjugate kernel after one gradient step on $\boldsymbol{W}$ with learning rate $η$, when $f^*$ is a single-index model. We consider two scalings of the first step learning rate $η$. For small $η$, we establish a Gaussian equivalence property for the trained feature map, and prove that the learned kernel improves upon the initial random features model, but cannot defeat the best linear model on the input. Whereas for sufficiently large $η$, we prove that for certain $f^*$, the same ridge estimator on trained features can go beyond this "linear regime" and outperform a wide range of random features and rotationally invariant kernels. Our results demonstrate that even one gradient step can lead to a considerable advantage over random features, and highlight the role of learning rate scaling in the initial phase of training.