Least-time Gradient 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 $f = g + \varepsilon h$; in the limit $\varepsilon \to 0$, 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 $\varepsilon$. 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 $\varepsilon$: 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.

We revisit the convergence guarantees of the Extragradient (EG) method for unconstrained biaffine min-max optimization. It is known that EG with a fixed stepsize achieves a $Θ(T^{-1/2})$ last-iterate convergence rate, which is slower than the optimal $\mathcal{O}(T^{-1})$ rate attainable by incorporating additional mechanisms such as anchoring. Motivated by recent advances showing that dynamic stepsizes alone can significantly accelerate gradient descent, we ask whether dynamic stepsizes can similarly accelerate the last-iterate convergence of EG. We present the first positive result in this direction. Specifically, we provide a deterministic dynamic stepsize schedule that accelerates the convergence rate of EG to $\mathcal{O}(T^{-2/3+\varepsilon})$ for any $\varepsilon > 0$. We also show that this rate is tight when the extrapolation and update steps of EG use the same stepsize. We then show that allowing different stepsizes for the extrapolation and update steps further improves the convergence rate to the near-optimal $\mathcal{O}(T^{-1+\varepsilon})$. Our analysis reduces stepsize scheduling to an optimization problem, whose solution leads to a stepsize schedule that follows (a discretization of) a power-law distribution. Our proposed stepsize schedules and analysis extend to other methods, such as Optimistic Gradient (OG), and suggest broader applicability to general min-max optimization problems.

Gradient descent in deep learning may operate at the edge of stability (EoS), a regime in which the largest eigenvalue of the loss Hessian hovers near the stability threshold $2/η$, where $η$ is the learning rate. Classical analysis tools such as gradient flow and the descent lemma do not apply here, motivating the search for a continuous-time model valid at EoS. We propose Edge Flow, a system of three coupled ordinary differential equations that provides a tractable, faithful, and predictive model of gradient descent dynamics at EoS. Edge Flow decomposes the dynamics into a center, an oscillation direction, and an oscillation magnitude. The center follows a modified gradient flow on a symmetrized loss; the direction tracks a top eigenvector of the Hessian via Rayleigh quotient dynamics; and the magnitude grows or decays exponentially depending on whether the sharpness exceeds or falls below the threshold $2/η$. Crucially, sharpness stabilization emerges from the coupled dynamics via a self-stabilization feedback loop. Discretizing Edge Flow only requires two gradient evaluations and one Hessian--vector product at each iteration. We demonstrate empirically that Edge Flow tracks the dynamics of gradient descent at least as faithfully as previously proposed continuous-time EoS models, while in addition resolving the oscillation of the sharpness at the onset of EoS, and that it provides a principled framework for understanding and mitigating instabilities in this regime.