High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation. (arXiv:2205.01445v1 [stat.ML])

Web: http://arxiv.org/abs/2205.01445

We study the first gradient descent step on the first-layer parameters
$\boldsymbol{W}$ in a two-layer neural network: $f(\boldsymbol{x}) =
\frac{1}{\sqrt{N}}\boldsymbol{a}^\top\sigma(\boldsymbol{W}^\top\boldsymbol{x})$,
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 …

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