This was a paper we presented about in Irina Rish’s neural scaling laws course (IFT6167) in winter 2022. You can view the slides we used here.

The big idea here is to use the geometric mean instead of the arithmetic mean across samples in the batch when computing the gradient for SGD. This overcomes the situation where averaging produces optima that are not actually optimal for any individual samples, as demonstrated in their toy example below: In practice, the method the authors test is not exactly the geometric mean for numerical and performance reasons, but effectively accomplishes the same thing by avoiding optima that are “inconsistent” (meaning that gradients from relatively few samples actually point in that direction).

These authors define robust error as the least upper bound on the expected loss over a family of environmental settings (including dataset, model architecture, learning algorithm, etc.): \[\sup_{e\in\mathcal F}\mathbb E_{\omega\in P^e}\left[\ell(\phi,\omega)\right]\] The fact that this is an upper bound and not an average is very important and is what makes this work unique from previous work in this direction. Indeed, what we should be concerned about is not how poorly a model performs on the average sample but on the worst-case sample.

This is a long paper, so a lot of my writing here is an attempt to condense the discussion. I’ve taken the liberty to pull exact phrases and structure from the paper without explicitly using quotes. Our main hypothesis is that deep learning succeeded in part because of a set of inductive biases, but that additional ones should be added in order to go from good in-distribution generalization in highly supervised learning tasks (or where strong and dense rewards are available), such as object recognition in images, to strong out-of-distribution generalization and transfer learning to new tasks with low sample complexity.

In the paper they use Bayes’ rule to show that the contribution of the first of two tasks is contained in the posterior distribution of model parameters over the first dataset. This is important because it means we can estimate that posterior to try to get a sense for which model parameters were most important for that first task. In this paper, they perform that estimation using a multivariate Gaussian distribution.