Study Shows Full-Batch Gradient Descent More Sample-Efficient Than One-Pass SGD in Nonlinear Learning
Researchers have proven that full-batch gradient descent (GD) can achieve strong recovery of a single-index model using only ~d samples, outperforming one-pass stochastic gradient descent (SGD), which requires ~d·log(d) samples. The work focuses on a quadratic activation function in a d-dimensional single-index model, demonstrating that a simple activation truncation removes the log(d) overhead seen in standard correlation-loss GD. This matters because it provides one of the first rigorous theoretical separations between multi-pass and one-pass gradient methods in a nonlinear, non-convex setting.
A paper accepted to ICML 2026 establishes a formal sample complexity separation between full-batch gradient descent and one-pass online SGD for learning a single-index model with a quadratic activation. While it has long been understood intuitively that reusing data improves statistical efficiency, prior theoretical work largely confined this understanding to linear regression or relied on specific loss-modification mechanisms from the first two data passes. The authors show that standard full-batch spherical GD on the correlation loss still incurs the same log(d) sample complexity penalty as one-pass SGD, but that truncating the activation resolves this by creating a more favorable optimization landscape at n ≈ d samples. A trajectory analysis further demonstrates that on the squared loss from a small initialization, n ≳ d samples and T ≳ log(d) gradient steps are sufficient for exact (strong) recovery. The result rigorously separates the statistical power of multi-pass and single-pass gradient methods in a nonlinear setting, advancing theoretical understanding of why training neural networks for multiple epochs can be fundamentally more data-efficient.
What's missing
The study is limited to a single-index model with a specific quadratic activation; it is an open question whether the sample complexity separation extends to more general activation functions, multi-index models, or deeper networks. The paper also does not address computational cost trade-offs (e.g., wall-clock time or memory) between full-batch GD and online SGD at scale, nor does it empirically validate the theoretical bounds on real-world datasets.
What different sources said
- arXiv cs.LGCenter
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