The Multiple Timescales of Gradient Descent on the Edge of Stability: A Perturbative Derivation of the Central Flow

The central flow of Cohen et al. (2025) is an empirically accurate continuous-time model of gradient descent at the edge of stability in deep learning, However, its derivation is heuristic. We propose a perturbative regime in which the central flow is the limit of gradient descent: we assume that the loss decomposes as f = g + epsilon h; in the limit epsilon to 0, the dynamics of gradient descent with learning rate η converge to the gradient flow of h constrained to the minimizers of g of sharpness at most 2/η. Our approach is formal rather than rigorous; it treats gradient descent as a singularly perturbed dynamical system in epsilon. Three timescales emerge: a fast timescale of oscillations along the sharpest direction, an intermediate timescale of the self-stabilization mechanism, and a slow timescale of the dynamics along the minimizers of g-the central flow. Using the method of multiple scales, a classical formal method from singular perturbation theory, we derive the expansion of the dynamics in epsilon: the central flow emerges as the leading-order term in the expansion, while the self-stabilization mechanism appears in the next-order term. We study this mechanism beyond previous analyses: with a single eigenvalue at the edge of stability, we compute the slow drift of the energy of the fluctuations; with several eigenvalues at the edge of stability, we derive the self-stabilization system and explain why fluctuations persist.