Wide flat minima and optimal generalization in classifying high-dimensional Gaussian mixtures
This provides theoretical insights into generalization in machine learning, particularly for classification tasks, but is incremental as it builds on existing concepts of flat minima.
The paper tackles the problem of understanding why certain minimizers generalize better in high-dimensional Gaussian mixture classification, showing that Bayes-optimal solutions lie in wide flat regions of the loss landscape and achieve optimal generalization error, even for unbalanced clusters.
We analyze the connection between minimizers with good generalizing properties and high local entropy regions of a threshold-linear classifier in Gaussian mixtures with the mean squared error loss function. We show that there exist configurations that achieve the Bayes-optimal generalization error, even in the case of unbalanced clusters. We explore analytically the error-counting loss landscape in the vicinity of a Bayes-optimal solution, and show that the closer we get to such configurations, the higher the local entropy, implying that the Bayes-optimal solution lays inside a wide flat region. We also consider the algorithmically relevant case of targeting wide flat minima of the (differentiable) mean squared error loss. Our analytical and numerical results show not only that in the balanced case the dependence on the norm of the weights is mild, but also, in the unbalanced case, that the performances can be improved.