Formal link between population gradient-flow time and online SGD sample complexity

Establish a rigorous correspondence between the logarithmic-time spherical population gradient flow for the single-hidden-unit autoencoder and the sample complexity of online stochastic gradient descent in the spiked cumulant model by controlling the stochastic noise in online SGD.

Background

The paper heuristically maps the time to weak recovery under population gradient flow, T = Θ(log d), to the number of samples needed by online SGD, n ≈ T d, following precedents in the single-index model literature.

A formal proof requires a careful control of the stochastic fluctuations intrinsic to online SGD and would solidify the sample-complexity predictions derived from the population dynamics.

References

A formal justification of this correspondence would require controlling the stochastic noise in online SGD, which we leave for future work.

— A solvable high-dimensional model where nonlinear autoencoders learn structure invisible to PCA while test loss misaligns with generalization  (2602.10680 - Mendes et al., 11 Feb 2026) in Section 4 (Autoencoder: population gradient flow), paragraph “Predictions for online SGD sample complexity”