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8 votes
Accepted

Distinguishing a biased coin with a small set of tests

Yes. Such families are called "averaging samplers", and there are plenty of constructions for them. You can find a more information about them (and about the more general notion of sampler) in this ...
Or Meir's user avatar
  • 5,615
7 votes
Accepted

Trivial upper bound for distribution testing

You might find this short note helpful ($\LaTeX$ code available [1] if the binary link breaks). I am reproducing the relevant part below: Theorem. (Folklore) Learning an unknown distribution over ...
Clement C.'s user avatar
  • 4,471
6 votes
Accepted

How long does it take to find a short cycle in a random graph?

No, you can't beat $\Theta(\sqrt{n})$ queries. I will explain how to formalize exfret's proof sketch of this, in a way that works for adaptive algorithms. This is all anticipated in exfret's answer; ...
D.W.'s user avatar
  • 12.1k
5 votes

How long does it take to find a short cycle in a random graph?

Let’s assume we can only query the $i$th edge of a given vertex’s adjacency list (which I am assuming is not sorted) or whether two given vertices are adjacent. In this case it should take $\sqrt n$ ...
exfret's user avatar
  • 643
5 votes

Are there distribution properties which are "maximally" hard to test?

Sorry for unearthing this post -- it is quite old, but I figured having it answered may not be that bad an idea. First, it looks like you performed your Chernoff bound with some slightly odd setting ...
Clement C.'s user avatar
  • 4,471
4 votes

Property testing in other metrics?

The work of Berman, Raskhodnikova, and Yaroslavtsev [1] introduces testing of functions $f\colon [n]^d\to \mathbb{R}$ with regard to $L_p$ distances, for $p\geq 1$. It is meant to capture situations ...
Clement C.'s user avatar
  • 4,471
3 votes

Are there hypothesis classes that are hard to learn but easy to test?

Take the class $\mathcal{M}$ of monotone boolean functions under the uniform distribution on $\{0,1\}^n$: it is known that $O(\sqrt{n}/\varepsilon^2)$ queries are sufficient to test it (even with ...
Clement C.'s user avatar
  • 4,471
3 votes

Lower bound on probability of getting two close points in a sample of $n$ points

Here's a counter-example showing your desired bound is not possible, unless I am mistaken. It's a simple variant of the example in Roei's comment. Fix any $n$ and $N\ge 4n$. Take $D$ to contain $N/...
Neal Young's user avatar
  • 10.8k
3 votes
Accepted

Testing if a distribution over $\mathbb{F}_2^n$ is heavily supported on a subspace

Unfortunately, constructing such a tester appears to be hard: at least as hard as the learning parity with noise (LPN) problem. Without loss of generality, we can focus on the problem of determining ...
D.W.'s user avatar
  • 12.1k
2 votes

Property testable in sublinear time in bounded degree graphs but not in general graphs

Chapter 10.5.1 of Oded Goldreich's Introduction to Property Testing (2017) discusses exactly that question. (See his website for an overview of the book, and (free) access to the drafts.) Now, Theorem ...
Clement C.'s user avatar
  • 4,471
2 votes

Trivial upper bound for distribution testing

Let's denote $N:=|\Omega|$. Then $\Theta(|N|/\epsilon^2)$ examples are both necessary and sufficient to learn the distribution to additive precision $\epsilon$ in total variation. This follows from ...
Aryeh's user avatar
  • 10.6k
2 votes
Accepted

Robustness to non-uniform randomness vs. one-sidedness

Only one-sided deciders are robust to bad coins, under your definition; no other decider can be robust to bad coins. Let $D$ be a decider that is not one-sided. For $x \in \mathcal{Y}$, let $\...
D.W.'s user avatar
  • 12.1k
1 vote

Are there hypothesis classes that are hard to learn but easy to test?

Please define testing precisely (under what distribution? known/unknown?). In the meantime, here is an example of what you may be looking for. Consider the example in the Kearns-Vazirani book, of ...
Aryeh's user avatar
  • 10.6k

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