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Great visualizations. Really enjoyed having a well-written example where mathematical proofs directly help with understanding a practical application.

I wonder what would happen with this analysis if a momentum term was added to the gradient descent. It seems that it would fix the specific failure modes in the examples, but I wonder if there's a corresponding mathematical way of categorizing what kinds of functions can(not) be quickly optimized with GD + momentum.

The animation is very good, making the article easy to understand

There is one very clear example that I ran across due to the reasons outlined in the article. If you have a wavelet and you're trying to slide it around to make it fit, that will fail spectacularly. There are lots of problems that boil down to basically the above.

The neural net answer is being able to spawn a wavelet at any position, as opposed to tweaking the position of an existing one.

This is a great article and its super helpful, thanks to whoever wrote it!

It frustrates me when math explainers, and textbooks, seem to start from the "here's why our methods are insufficient to solve our problem" and fail to provide an example of the problem they are trying to solve.

What's the question this method is attempting to answer? What does an answer look like? How does this method lead to it?

> If you have ever tried to minimize a function with gradient descent

"and if otherwise, go kick sand," I guess.

Simplex methods can handle those tough situations, though.

Kudos for beatiful formulae rendering.