Let $\Omega \subset \mathbb{R}^m$ be an open subset bounded with a smooth boundary.

**Problem** : Given any bounded continuous function $f:\Omega\to\mathbb{R}$, can we learn it to a given accuracy $\epsilon$? ($\epsilon>0$).

**Definition** : What do you mean by learning a function to a given accuracy $\epsilon$?

Using samples of $f$, at sufficiently large **but finite** number of data points that are drawn **randomly**(iid) from the set $\Omega$ (under a uniform probability distribution), and using a sufficiently large but finite number of registers whose **precision** (arithmetic) is sufficiently large but **finite** (this finite precision is an important condition), should be able to **compute** a function $F$ with only a finite number of computations (they could be additions, multiplications, and divisions but performed using registers of finite precision) such that $\|f-F\|_{L^\infty(\Omega)} \le \epsilon$.

**compute a function** $F$ : Given any query point $x$, one should give out $F(x)$.

**Conjecture**: There exists a method of learning such that one can derive a bound on required precision $p$ that depends only on $\Omega$ and $\epsilon$ and is independent of $f$.

**Question**: Has anyone formulated this problem before (any reference). Has anyone solved it? If I solve it, what is its market value? (mathematics market)

PS: solving means coming up with a method to learn such functions in the defined way.

(please feel free to tag appropriately)

randomlypart is confusing: there is no uniform distribution on a countably infinite set. $\endgroup$ – Benoît Kloeckner Jul 5 at 13:35