让下一篇重要论文，不再错过。

At the edge of stability, the iterates of gradient descent oscillate along the sharp directions of the Hessian. This poses a significant challenge for continuous-time models as the usual continuous-time approximation -- the gradient flow -- admits no oscillations. Several continuous-time models have been proposed to capture the behavior at the edge of stability, among them central flow and rod flow. Although both models can be interpreted as performing a coarse-graining in time, the connection between them remains poorly understood. Here we show that both flows can be derived from a third object, the edge coupling: a functional on consecutive iterates whose critical points are the fixed points and period-two orbits of gradient descent. It is revealed that the two flows correspond to coarse-grainings at different time scales. Rod flow coarse-grains over only the fastest time scale, while central flow coarse-grains over both the fast and intermediate time scales. Furthermore, using the edge coupling, we derive rod flows and central flows for momentum methods such as heavy ball and Adam. We verify the derived flows experimentally on representative neural network architectures, where they track the discrete iterates throughout all phases of training -- including the transition from the stable regime to the edge of stability.

Modern data systems often require aggregating information from multiple sources: some might be entirely untrustworthy, while the remaining presumably reliable sources may still suffer from a small amount of corruption. We study robust mean estimation under this practical scenario, which is formulated through the double corruption model: samples are divided among $N$ users, each with $n$ samples. $\varepsilon$ fraction of the users could be arbitrarily corrupted, while each of the remaining ``good'' users still have $α$-fraction ($α>0$) of corrupted samples. The goal is to estimate the unknown mean $μ$ with small $\ell_2$ error. This is a challenging scenario because we must handle two types of adversarial corruption: user-level corruption and sample-level corruption. When the unknown distribution has covariance matrix bounded by the identity matrix, we design a sum-of-squares (SoS) algorithm that achieves an $\ell_2$ error of $\sqrtα$ with probability at least $2/3$ as long as the total number of samples $nN=\widetildeΩ(d/α)$. The estimation error and sample complexity are min-max optimal under the mild condition. For comparison, treating all $nN$ samples as $(\varepsilon+α)$-corrupted and applying standard robust mean estimation only achieves an error of $\sqrt{\varepsilon+α}$. Thus, our new algorithm removes the adverse effect of untrustworthy users and achieves the optimal error as if only sample-level corruption is present. Also, our SoS algorithm extends to the heterogeneous setting where each user's local mean could be additionally shifted by $\sqrtβ$ in $\ell_2$ distance. We show that an error of $O(\sqrt{α+β})$ can be achieved with probability at least $2/3$ when $nN=\widetildeΩ(d/(α+β))$.