R. Vidal

Lachlan Ewen MacDonald, René Vidal

An important quantity in the theory of gradient descent (GD) is the \emph{sharpness}, defined as the largest eigenvalue of the objective Hessian. Classical analyses typically require the step size to be uniformly smaller than twice the reciprocal of the sharpness, but this condition is frequently violated in the training of deep neural networks. Recent work bridges this gap in the setting of overparametrised least-squares with a \emph{single scalar output}, providing a normal form for large-step GD in a neighbourhood of an \emph{isolated} flat minimum and establishing three corresponding convergence results. In this paper, we extend this theory in two directions: (1) to overparametrised least-squares with \emph{vector-valued outputs} (including regression with arbitrarily many observations), and (2) to a neighbourhood of a \emph{manifold} of flat minima (which we show is essential for applications such as matrix factorisation). We generalise both the normal form and all three convergence theorems of \cite{macdonaldeos} to this broader setting, overcoming several technical challenges, including the solution of a singular partial differential equation via a novel method that may be of independent interest. We further show that our framework applies to deep matrix factorisation under mild assumptions, yielding several new structural results. In particular, we prove that the set of flat minima forms a fibre bundle over a product of spheres, and that the sharpness is Morse-Bott along this manifold.

Konstantinos Emmanouilidis, Lachlan MacDonald, Salma Tarmoun et al.

Modern deep learning has been shown to operate at the edge of stability, routinely using learning rates far larger than those justified by classical optimization theory. Most prior analyses of the edge of stability phenomenon focus on deterministic gradient descent, leaving the stochastic setting largely unexplored. In this work, we provide sharp convergence guarantees for Stochastic Gradient Descent (SGD) applied to the multiclass cross-entropy loss, for both linear classifiers and two-layer neural networks. We show that the stochasticity of SGD may cause the dynamics to alternate between an edge-of-stability regime that is dominated by curvature-driven oscillations, and a stable regime in which the expected loss decreases at a controlled rate. Despite that, we prove that SGD self-stabilizes the dynamics, ensuring that the iterates return to stability in a fixed number of iterations and allowing convergence in the best-iterate sense even with large learning rates. Experiments validate our theoretical findings and illustrate the benefits of SGD in the large-stepsize regime.

Siyi Chen, Xiaoyan Zhang, Meng Wu et al.

Multi-agent systems communicate mostly through text, paying a lossy and expensive decode and re-encode cost. KV-cache communication is a promising alternative, yet most prior work is homogeneous, using duplicate copies of the same model, and avoids the central challenge of cross-model latent alignment; existing heterogeneous methods are also restrictive, typically assuming shared input and using transferred caches mainly for steering. We study a more fundamental question: can heterogeneous agents be aligned well enough to perform real "mind reading" and transfer both what one agent sees and how it thinks? Our information-structure analysis reveals a duality: context-aware transfer is driven by sparse reasoning signals, while context-unaware transfer, where the receiver sees no input, requires dense contextual knowledge preservation. Motivated by this, we propose dense alignment for heterogeneous KV-cache communication via a lightweight cross-model cache transformation and two-phase training: reconstruction followed by generation. Across all six directions of {Qwen3-4B, 8B, 14B} and six in-domain and out-of-domain benchmarks, our method outperforms prior heterogeneous baselines, matches or exceeds text communication in context-aware settings at roughly 2 to 3 times lower compute, and remains effective in context-unaware transfer where prior methods collapse.