Approximate Bayesian inference for neural networks is considered a robust alternative to standard training, often providing good performance on out-of-distribution data. However, Bayesian neural networks (BNNs) with high-fidelity approximate inference via full-batch Hamiltonian Monte Carlo achieve poor generalization under covariate shift, even underperforming classical estimation. We explain this surprising result, showing how a Bayesian model average can in fact be problematic under covariate shift, particularly in cases where linear dependencies in the input features cause a lack of posterior contraction. We additionally show why the same issue does not affect many approximate inference procedures, or classical maximum a-posteriori (MAP) training. Finally, we propose novel priors that improve the robustness of BNNs to many sources of covariate shift.

Abstract: Pre-trained language models (LMs), such as BERT (Devlin et al., 2018) and its variants, have led to significant improvements on various NLP tasks in past years. However, a theoretical framework for studying their relationships is still missing. In this paper, we fill this gap by investigating the linear dependency between pre-trained LMs. The linear dependency of LMs is defined analogously to the linear dependency of vectors. We propose Language Model Decomposition (LMD) to represent a LM using a linear combination of other LMs as basis, and derive the closed-form solution.

Pre-trained language models (LMs), such as BERT (Devlin et al., 2018) and its variants, have led to significant improvements on various NLP tasks in past years. However, a theoretical framework for studying their relationships is still missing. In this paper, we fill this gap by investigating the linear dependency between pre-trained LMs. The linear dependency of LMs is defined analogously to the linear dependency of vectors. We propose Language Model Decomposition (LMD) to represent a LM using a linear combination of other LMs as basis, and derive the closed-form solution. A goodness-of-fit metric for LMD similar to the coefficient of determination is defined and used to measure the linear dependency of a set of LMs. In experiments, we find that BERT and eleven (11) BERT-like LMs are 91% linearly dependent. This observation suggests that current state-of-the-art (SOTA) LMs are highly "correlated". To further advance SOTA we need more diverse and novel LMs that are less dependent on existing LMs.