Computer scientists discover limits of gradient descent (2021)
> Brouwer’s fixed point theorem. The theorem says that for any continuous function, there is guaranteed to be one point that the function leaves unchanged — a fixed point, as it’s known. This is true in daily life. If you stir a glass of water, the theorem guarantees that there absolutely must be one particle of water that will end up in the same place it started from.Wait. What if by stirring I shook the glass and at the end set it back down a foot away from its initial location. Is this author claiming that Brouwer’s theorem guarantees a particle floating somewhere where the glass used to be? Unchanged?I can imagine an unlimited number of scenarios where I can guarantee the water particles are displaced.This seems like a bad application of the theorem, right?