Full-Batch Gradient Descent Outperforms One-Pass SGD: Sample Complexity Separation in Single-Index Learning

It is folklore that reusing training data more than once can improve the statistical efficiency of gradient-based learning. However, beyond linear regression, the theoretical advantage of full-batch gradient descent (GD, which always reuses all the data) over one-pass stochastic gradient descent (online SGD, which uses each data point only once) remains unclear. In this work, we consider learning a $d$ -dimensional single-index model with a quadratic activation, for which it is known that one-pass SGD requires $n\gtrsim d\text{log}d$ samples to achieve weak recovery. We first show that this $\text{log}d$ factor in the sample complexity persists for full-batch spherical GD on the correlation loss; however, by simply truncating the activation, full-batch GD exhibits a favorable optimization landscape at $n\simeq d$ samples, thereby outperforming one-pass SGD (with the same activation) in statistical efficiency. We complement this result with a trajectory analysis of full-batch GD on the squared loss from small initialization, showing that $n\gtrsim d$ samples and $T\gtrsim \text{log}d$ gradient steps suffice to achieve strong (exact) recovery.