Scale-invariant Bayesian Neural Networks with Connectivity Tangent Kernel

Explaining generalizations and preventing over-confident predictions are central goals of studies on the loss landscape of neural networks. Flatness, defined as loss invariability on perturbations of a pre-trained solution, is widely accepted as a predictor of generalization in this context. However, the problem that flatness and generalization bounds can be changed arbitrarily according to the scale of a parameter was pointed out, and previous studies partially solved the problem with restrictions: Counter-intuitively, their generalization bounds were still variant for the function-preserving parameter scaling transformation or limited only to an impractical network structure. As a more fundamental solution, we propose new prior and posterior distributions invariant to scaling transformations by \textit{decomposing} the scale and connectivity of parameters, thereby allowing the resulting generalization bound to describe the generalizability of a broad class of networks with the more practical class of transformations such as weight decay with batch normalization. We also show that the above issue adversely affects the uncertainty calibration of Laplace approximation and propose a solution using our invariant posterior. We empirically demonstrate our posterior provides effective flatness and calibration measures with low complexity in such a practical parameter transformation case, supporting its practical effectiveness in line with our rationale

A modern look at the relationship between sharpness and generalization

Sharpness of minima is a promising quantity that can correlate with generalization in deep networks and, when optimized during training, can improve generalization. However, standard sharpness is not invariant under reparametrizations of neural networks, and, to fix this, reparametrization-invariant sharpness definitions have been proposed, most prominently adaptive sharpness (Kwon et al., 2021). But does it really capture generalization in modern practical settings? We comprehensively explore this question in a detailed study of various definitions of adaptive sharpness in settings ranging from training from scratch on ImageNet and CIFAR-10 to fine-tuning CLIP on ImageNet and BERT on MNLI. We focus mostly on transformers for which little is known in terms of sharpness despite their widespread usage. Overall, we observe that sharpness does not correlate well with generalization but rather with some training parameters like the learning rate that can be positively or negatively correlated with generalization depending on the setup. Interestingly, in multiple cases, we observe a consistent negative correlation of sharpness with out-of-distribution error implying that sharper minima can generalize better. Finally, we illustrate on a simple model that the right sharpness measure is highly data-dependent, and that we do not understand well this aspect for realistic data distributions. The code of our experiments is available at https://github.com/tml-epfl/sharpness-vs-generalization