Necessity of transversal perturbations for central-flow validity

Establish whether a perturbation with a nonzero transversal gradient component—such as transversal perturbations in the loss, noise, or numerical errors—is necessary for the central-flow prediction to hold in sharp-valley gradient descent.

Background

The minimal example shows that removing the transversal term from the perturbation can allow the iterates to leave the stable valley for a substantial slow time, after which the central-flow approximation fails.

The authors formulate a conjecture that some transversal perturbation is necessary to maintain the central-flow behavior, but do not establish this necessity in general.

References

More generally, we conjecture that a perturbation in $x$ (transversal gradient of $h$, noise, numerical errors, etc.) is necessary for the central flow prediction to hold.

— The Multiple Timescales of Gradient Descent on the Edge of Stability: A Perturbative Derivation of the Central Flow  (2609.01034 - Berthier, 1 Sep 2026) in Remark 2.2, “Central flow failure”

We leave it for future work to systematically study SGD-momentum in river-valley.

— Towards Understanding Momentum Acceleration in River-Valley Loss Landscape  (2609.30957 - Lu et al., 25 Sep 2026) in Appendix, Section 2, “Further Discussions,” paragraph “The stochastic setting”