I had a similar, but slightly different picture. My picture was the tentacle covered blob.
When doing gradient descent from a random point, you usually get to a tentacle. (Ie the red trajectory)
When the system has been running for a while, it is randomly sampling from the black region. Most of the volume is in the blob. (The randomly wandering particle can easily find its way from the tentacle to the blob, but the chance of it randomly finding the end of a tentacle from within the blob is small)
Under this model, the generalization would work the same when using a valid sampling process.
When you start to include broad basins (where a small perturbation changes your loss but only slightly), and other results like Mingard et al.‘s and Valle Pérez et al.’s, I’m inclined to think this picture is more representative. But even in this picture, we can probably end up making statements about singularities in the blobby regions having disproportionate effect over all points within those regions.
Epistemic status: Highly speculative hypothesis generation.
I had a similar, but slightly different picture. My picture was the tentacle covered blob.
When doing gradient descent from a random point, you usually get to a tentacle. (Ie the red trajectory)
When the system has been running for a while, it is randomly sampling from the black region. Most of the volume is in the blob. (The randomly wandering particle can easily find its way from the tentacle to the blob, but the chance of it randomly finding the end of a tentacle from within the blob is small)
Under this model, the generalization would work the same when using a valid sampling process.
When you start to include broad basins (where a small perturbation changes your loss but only slightly), and other results like Mingard et al.‘s and Valle Pérez et al.’s, I’m inclined to think this picture is more representative. But even in this picture, we can probably end up making statements about singularities in the blobby regions having disproportionate effect over all points within those regions.