Reviewers and students are sometimes greatly concerned by the distinction between:
- An open set and a closed set.
- A Supremum and a Maximum.
- An event which happens with probability 1 and an event that always happens.
I don’t appreciate this distinction in machine learning & learning theory. All machine learning takes place (by definition) on a machine where every parameter has finite precision. Consequently, every set is closed, a maximal element always exists, and probability 1 events always happen.
The fundamental issue here is that substantial parts of mathematics don’t appear well-matched to computation in the physical world, because the mathematics has concerns which are unphysical. This mismatched mathematics makes irrelevant distinctions. We can ask “what mathematics is appropriate to computation?” Andrej has convinced me that a pretty good answer to this question is constructive mathematics.
So, here’s a basic challenge: Can anyone name a situation where any of the distinctions above (or similar distinctions) matter in machine learning?