Title: A Function Centric Perspective on Flat and Sharp Minima
Primary area: learning theory
Keywords: Flat Minima, Sharp Minima, Generalisation, Function, Robustness, Calibration, Safety

Abstract: Flat minima are widely believed to correlate with improved generalisation in deep neural networks. However, this connection has proven more nuanced in recent studies, with both theoretical counterexamples and empirical exceptions emerging in the literature. In this paper, we revisit the role of sharpness in model performance, proposing that sharpness is better understood as a function-dependent property rather than a reliable indicator of poor generalisation. We conduct extensive empirical studies, from single-objective optimisation to modern image classification tasks, showing that sharper minima often emerge when models are regularised (e.g., via SAM, weight decay, or data augmentation), and that these sharp minima can coincide with better generalisation, calibration, robustness, and functional consistency. Across a range of models and datasets, we find that baselines without regularisation tend to converge to flatter minima yet often perform worse across all safety metrics. Our findings demonstrate that function complexity, rather than flatness alone, governs the geometry of solutions, and that sharper minima can reflect more appropriate inductive biases (especially under regularisation), calling for a function-centric reappraisal of loss landscape geometry.

TL;DR: We investigate flat and sharp minima from a function-centric lens, characterising global minima in single-objective optimisation and scaling to large-scale tasks, we find sharp minima counterintuitively, can improve both generalisation and safety.