Edge of Stability has been a thing for a while now. We can show that most training happens at the edge of stability. This is very surprising because of a "negative " fact: Most theory does not apply (e.g., Descent Lemma). The question becomes: Does the fact that training happens at the EoS impact performance? In what way? With what mechanisms? How is that affected by the architecture? To our knowledge, these questions they are unanswered to date and our new paper is the about the first step towards this! EoS picks what part of the distribution to learn and at what speed! arxiv.org/abs/2606.04212 In particular, the consequence is that for MLPs: EoS can improve robustness or OOD behavior, but only when the relevant subset is the one selected by EoS. If boundary points dominate, EoS helps near-boundary robustness. If distributional outliers dominate, EoS helps extrapolate toward the tail. Precisely: 1. We make EoS causal. We fork training from the same state at EoS onset: one branch stays at EoS, the other exits by lowering the learning rate. Same everything, only the stability constraint changes. 2. EoS is selective: it learns some parts of the data distribution faster, while slowing others. 3. The selector is surprisingly simple. A group benefits from EoS when its gradient is big and has high alignment with v_1, the top Hessian eigenvector. Translation: to benefit from EoS, a subset must (i) point in the sharpest direction, and (ii) keep a non-vanishing gradient. This gives concrete answers to the questions above: Does EoS affect performance? Yes, but not uniformly. It reallocates learning. In what way? It prioritizes the subset with largest curvature influence. By what mechanism? Self-stabilization at the sharpness boundary couples training to v_1; only groups aligned with v_1, and whose gradients persist, get the extra progress. How does architecture matter? Architecture changes the map from input geometry to gradient geometry. Thus it can change which subset dominates, but the same predictor remains: largest curvature influence wins. With the amazing Shauna Kwag, @anakha_g, and @TomasoPoggio!

A common practitioner belief mechanistically: “weight decay stabilizes training.” We show it is wrong technically, a bit real In practice. - Training still happens at EoS with weight decay. - That said "it doesn't look like it"! The curvature at stabilization is much lower! - We properly charcaterize all we see by extending to weight decay the self-stabilization by @alex_damian_, @EshaanNichani, and @jasondeanlee and matching the result cleanly with an underdamped harmonic oscillator! - This however stabilizes/tames the dynamics in function space. Importantly: - This shows that regularized training at EoS even though "doesn't look like that", meaning curvature measures are lower than thresholds. Check out our paper: arxiv.org/pdf/2605.16622

Check the blog of Poggio Lab at MIT! We went online with some very nice blogs! The last one being about our multiagent system: poggio-lab.mit.edu/blogsupda…

Most AI for research work tries to maximize autonomy first and patch quality later. We think the near-term path is the reverse: Automating step-by-step holding the quality bar fixed. Today we’re open-sourcing PoggioAI/MSc for ML Theory Research

Does SGD really “seek flat minima”? We show that SGD has no intrinsic preference for flatness, even for stable linear networks—going against ~10 years of folklore. Flatness emerges iff label noise is isotropic; anisotropic noise drives SGD to arbitrarily sharp solutions. This reveals a new flattening–sharpening mechanism in late training, unrelated to standard progressive sharpening or Edge-of-Stability effects.