A Theory of Machine Learning

We critically review three major theories of machine learning and provide a new theory according to which machines learn a function when the machines successfully compute it. We show that this theory challenges common assumptions in the statistical and the computational learning theories, for it implies that learning true probabilities is equivalent neither to obtaining a correct calculation of the true probabilities nor to obtaining an almost-sure convergence to them. We also briefly discuss some case studies from natural language processing and macroeconomics from the perspective of the new theory.

Flat Posterior Does Matter For Bayesian Transfer Learning

The large-scale pre-trained neural network has achieved notable success in enhancing performance for downstream tasks. Another promising approach for generalization is Bayesian Neural Network (BNN), which integrates Bayesian methods into neural network architectures, offering advantages such as Bayesian Model averaging (BMA) and uncertainty quantification. Despite these benefits, transfer learning for BNNs has not been widely investigated and shows limited improvement. We hypothesize that this issue arises from the inability to find flat minima, which is crucial for generalization performance. To address this, we evaluate the sharpness of BNNs in various settings, revealing their insufficiency in seeking flat minima and the influence of flatness on BMA performance. Therefore, we propose Sharpness-aware Bayesian Model Averaging (SA-BMA), a Bayesian-fitting flat posterior seeking optimizer integrated with Bayesian transfer learning. SA-BMA calculates the divergence between posteriors in the parameter space, aligning with the nature of BNNs, and serves as a generalized version of existing sharpness-aware optimizers. We validate that SA-BMA improves generalization performance in few-shot classification and distribution shift scenarios by ensuring flatness.