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Title: Benchmarking the Neural Linear Model for Regression

Paper content:
The neural linear model is a simple adaptive Bayesian linear regression method that has recently been used in a number of problems ranging from Bayesian optimization to reinforcement learning.
Despite its apparent successes in these settings, to the best of our knowledge there has been no systematic exploration of its capabilities on simple regression tasks.
In this work we characterize these on the UCI datasets, a popular benchmark for Bayesian regression models, as well as on the recently introduced ''gap'' datasets, which are better tests of out-of-distribution uncertainty.
We demonstrate that the neural linear model is a simple method that shows competitive performance on these tasks.
Despite the recent successes that neural networks have shown in an impressive range of tasks, they tend to be overconfident in their predictions (Guo et al., 2017) .
Bayesian neural networks (BNNs; Neal (1995) ) attempt to address this by providing a principled framework for uncertainty estimation in predictions.
However, inference in BNNs is intractable to compute, requiring approximate inference techniques.
Of these, Monte Carlo methods and variational methods, including Monte Carlo dropout (MCD) (Gal and Ghahramani, 2016) , are popular; however, the former are difficult to tune, and the latter are often limited in their expressiveness (Foong et al., 2019b; Yao et al., 2019; Foong et al., 2019a) .
The neural linear model represents a compromise between tractability and expressiveness for BNNs in regression settings: instead of attempting to perform approximate inference over the entire set of weights, it performs exact inference on only the last layer, where prediction can be done in closed form.
It has recently been used in active learning (Pinsler et al., 2019) , Bayesian optimization (Snoek et al., 2015) , reinforcement learning (Riquelme et al., 2018) , and AutoML (Zhou and Precioso, 2019), among others; however, to the best of our knowledge, there has been no systematic attempt to benchmark the model in the simple regression setting.
In this work we do so, first demonstrating the model on a toy example, followed by experiments on the popular UCI datasets (as in Hernández-Lobato and Adams (2015) ) and the recent UCI gap datasets from Foong et al. (2019b) , who identified (along with Yao et al. (2019) ) well-calibrated 'in-between' uncertainty as a desirable feature of BNNs.
In this section, we briefly describe the different models we train in this work, which are variations of the neural linear (NL) model, in which a neural network extracts features from the input to be used as basis functions for Bayesian linear regression.
The central issue in the neural linear model is how to train the network: in this work, we provide three different models, with a total of four different training methods.
For a more complete mathematical description of the models, refer to Appendix A; we summarize the models in Appendix C. Snoek et al. (2015) , we can first train the neural network using maximum a posteriori (MAP) estimation.
After this training phase, the outputs of the last hidden layer of the network are used as the features for Bayesian linear regression.
To reduce overfitting, the noise variance and prior variance (for the Bayesian linear regression) are subsequently marginalized out by slice sampling (Neal et al., 2003) according to the tractable marginal likelihood, using uniform priors.
We refer to this model as the maximum a posteriori neural linear model (which we abbreviate as MAP-L NL, where L is the number of hidden layers in the network).
We tune the hyperparameters for the MAP estimation via Bayesian optimization (Snoek et al., 2012) .
We have shown benchmark results for different variants of the neural linear model in the regression setting.
Our results show that the successes these models have seen in other areas such as reinforcement and active learning are not unmerited, with the models achieving generally good performance despite their simplicity.
Furthermore, they are not as susceptible to the the inability to express gap uncertainty as MFVI or MCD.
However, we have shown that to obtain reasonable performance extensive hyperparameter tuning is often required, unlike MFVI or MCD.
Finally, our work suggests that exact inference on a subset of parameters can perform better than approximate inference on the entire set, at least for BNNs.
We believe this broader issue is worthy of further investigation.

assistant: text: We benchmark the neural linear model on the UCI and UCI "gap" datasets.