Are Flat Minima an Illusion?

Flat minima are strongly associated with improved generalisation in deep neural networks. However, this connection has proven 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 and examine sharpness as a function-dependent property rather than as a direct indicator of poor generalisation. We conduct extensive empirical studies ranging from single-objective optimisation, synthetic non-linear binary classification tasks, to modern image classification tasks. In single-objective optimisation, we show that flatness and sharpness are relative to the function being learned: equally optimal solutions can exhibit markedly different local geometry. In synthetic non-linear binary classification tasks, we show that increasing decision-boundary tightness can increase sharpness even when models generalise perfectly, suggesting that sharpness is not reducible to memorisation alone. Finally, in large-scale experiments, we find that sharper minima often emerge when models are regularised (e.g., via weight decay, data augmentation, or SAM), and can coincide with better generalisation, calibration, robustness, and functional consistency. Our findings are consistent with a function-centric interpretation of minima geometry: flatness should not be treated as a universal proxy for generalisation, and sharper minima can reflect desirable inductive biases associated with improved predictive behaviour, calling for a function-centric reappraisal of minima geometry.

Flat loss landscapes have long been linked to better generalization in neural networks. However, its role as a causal mechanism for generalization is less established. Grokking provides an unique testbed to understand this distinction: models are prone to fit observed data using non-generalizing structure and remain in that regime for prolonged periods, transitioning to generalization only under particular training conditions. In this work, we study whether flat loss landscapes can act as a driving mechanism in this transition. While recent work has argued for flatness as a necessary geometric condition for this transition, we find that biasing training toward flatter solutions using sharpness-aware minimization (SAM) is insufficient to reliably induce this transition, despite producing flatter solutions. However, when SAM is paired with mechanisms that drive generalization such as weight decay, an interesting property emerges: SAM can accelerate the transition to generalizing solutions by up to 4x at the epoch-level. We theoretically untangle this relationship between SAM and weight decay using a minimal interpolating two-layer ReLU model with both memorizing and generalizing solutions. We show that even in this simple setup, flatness alone cannot distinguish a memorizing solution from a generalizing one, while weight decay favors generalizing solutions. However, under a local stability analysis, there exists a window where a memorizing interpolant is locally stable under gradient descent but unstable under SAM in the low-norm regime, which can explain SAM's ability to accelerate this transition. Overall, our results provide a more interpretable account of the role of flatness in driving generalization, especially in settings where models are vulnerable to minimizing loss through learning non-generalizing structure.

The flatness of the loss landscape at a minimizer is a widely used heuristic for reasoning about neural-network generalization, yet evidence for this relation is mostly empirical and controversial. We study this relation in a teacher-student tree committee machine, where both the ERM estimator and the Hessian spectrum are analytically tractable in the proportional high-dimensional limit. First, we use a zero-temperature Gibbs formulation to obtain predictions for the observables of the typical minimizers of the empirical loss. Secondly, we use Edwards-Jones formalism to derive the limiting Hessian resolvent around these typical minimizers. All predictions agree with finite-size gradient-descent simulations. Finally, we study three measures of flatness, namely the left and right edges and the spectral mean, and check if a decrease in generalization error as the dataset size is increased corresponds to an increase in flatness. We find that the answer strongly depends on the learning task and on the ratio of the number of parameters to the number of data points. In regression, the spectral mean and right edge correlate with the generalization error, while the left edge does so only in the overparametrized regime. In classification this correlation reliably holds only in the highly overparametrized phase, while for underparametrized networks it can even reverse.