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This paper extends the theory of gradient descent with large step sizes near flat minima by generalizing previous results from scalar to vector-valued outputs and addressing the dynamics around a manifold of flat minima. The authors establish a normal form for gradient descent in this broader context and derive three convergence results that are crucial for applications like matrix factorization. Key findings include the characterization of flat minima as a fiber bundle over a product of spheres and the identification of sharpness as Morse-Bott along the manifold, which has significant implications for understanding optimization in deep learning.
Flat minima form a fiber bundle over spheres, revealing new structural insights that could transform our approach to optimizing deep learning models.
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.
