Sample complexity of gradient training for multi-index targets

Inputs $x \sim N(0, I_d)$ and labels $y = g(W x)$ with $W$ an $r$ x d matrix of rank r fixed as $d \to \infty$ and $g$ a fixed link function. For a standard two-layer network trained by online or multi-pass SGD on the square loss without data preprocessing, determine the exponent $\gamma(g)$ in $n \sim d^\gamma$ required to reach test error $o(1)$, and determine whether it equals the benchmark for statistical-query and low-degree algorithms, $n \sim d^{\max(1, k_*/2)}$, with $k_*$ the generative exponent for single-index targets and the generative leap exponent for multi-index targets (Damian, Lee and Bruna, arXiv:2506.05500). The answer is $\gamma$ as a function of the Hermite structure of $g$, with proof.

Matching upper and lower bounds on the sample complexity of standard gradient training for general multi-index targets.