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11 votes
Accepted

Is testing easier/harder than learning?

If the learning algorithm is proper (i.e. it always produces a hypothesis from the class $F_n$), then it also gives a testing algorithm -- simply run the learning algorithm, and see whether the ...
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  • 9,800
10 votes
Accepted

What is the reason for using a "Lines-Oracle" in the PCP Theorem?

The lines oracle is used to decrease the query complexity of the test from $d+1$ to $2$, at the expense of using a larger alphabet. If you don't mind making $d+1$ queries, then the lines oracle is ...
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  • 5,055
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 ...
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  • 5,055
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 ...
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  • 4,331
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; ...
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  • 10.3k
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 $\sqrt n$ ...
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  • 633
5 votes
Accepted

Lower bound on estimating $\sum_{k=1}^n a_k$ for non-increasing $(a_k)_k$

Here are the lower bounds I can show. I conjecture that for a fixed $\epsilon$, the right lower bound is $\Omega( \log n)$, but naturally I might be wrong. I am going to use a decreasing sequence (...
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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 ...
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  • 4,331
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 ...
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  • 4,331
4 votes

Lower bound on estimating $\sum_{k=1}^n a_k$ for non-increasing $(a_k)_k$

Lower bound At least $\Omega(1/\sqrt{\epsilon})$ queries are necessary to distinguish the two cases. Consider the sequence $a_1,\dots,a_n$ given by $\epsilon,2\epsilon,3\epsilon,4\epsilon,\dots$, ...
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  • 10.3k
4 votes
Accepted

Distance of arbitrary vectors to Hadamard code

What you are asking is the covering radius of the Hadamard code. I am not sure what the answer is but the covering radius of the punctured Hadamard code is at least $\frac{N}{2} - \frac{\sqrt{N}}{2}$, ...
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4 votes
Accepted

Determining the number of clusters using property testing algorithm

First. One can do better as far as the sampling - at least if $d$ is large - $O(\frac{kd \log k}{\epsilon} \log \frac{1}{\epsilon})$ should follow easily from relative approximations http://sarielhp....
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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/...
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  • 8,133
3 votes
Accepted

Hitting set of very restricted linear forms

There is a tight lower bound of size $\Omega(n/ \log n)$ by simple counting argument. Suppose there is a hitting set $H=\{\alpha_{1},\dots,\alpha_{k}\}$ of size $k$. We will show that there is always ...
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  • 1,120
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 ...
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  • 4,331
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 ...
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  • 4,331
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 $\...
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  • 10.3k
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 ...
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  • 10k
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 ...
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  • 10k

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