Scaling model capacity has been vital in the success of deep learning.

We find that the performance of the pre-training data varies substantially across distribution shifts, with no single data source dominating.

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Some recent studies attempted to explore this problemindifferentcontexts.

Lemma 5. Let S = (Z1,...,Zn) be a collection ofn independent random variables andΦ be an arbitrary random variable defined on the same probability space. Furthermore, each of these summands has zero mean. Given a deterministic algorithmf, we consider the algorithm that adds Gaussian noise to the predictionsoff: fσ(z,x,R)=f(z,x)+ξ, (151) whereξ N(0,σ2Id). Thefigureinthemiddle repeats the experiment of Figure 1a while making the training algorithm stochastic by randomizing the seed. Table 1: The architecture of the 4-layer convolutional neural network used in MNIST 4 vs 9 classification tasks.