August saw a couple of conferences: RANDOM-APPROX and a seminar at the Simon’s institute. Here’s a report on the new papers we saw in August, including much progress on distribution testing.

** On sample based testers **by Oded Goldreich and Dana Ron (ECCC)

**.**The usual setting for property testing is beloved query model, where the tester gets to make any query it pleases. A much weaker setting would be to simply get random (labeled) samples of the domain. These are called sample based testers, and are more akin to setting in learning theory. Sampled based testers were also discussed in the seminal Goldreich, Goldwasser, and Ron paper. Since then, there have been variants, such as distribution-free testing and active testing. In this paper, it is shown that constant-query proximity-oblivious testers imply the existence of sublinear (polynomial dependence in size) sample based testers.

**Instance-by-instance optimal identity testing **by Gregory Valiant and Paul Valiant (ECCC). The problem of distribution testing is a problem that we all love. Given a known discrete distribution \(p\), we wish to test equality (with \(p\)) of an unknown distribution \(q\). How many independent samples are required of \(q\)? This paper gives an optimal algorithm for *each* distribution \(p\), subsuming much past work. The analysis involves a characterization of HÃ¶lder and norm-monotonicity type inequalities. Definitely on my list to read!

**Optimal algorithms for testing closeness of discrete distributions** by Siu-On Chan, Ilias Diakonikolas, Gregory Valiant, Paul Valiant (Arxiv). Now consider the setting where both \(p\) and \(q\) are unknown (over a support of size \(n\)), and we wish to test equality. (We define “far” in terms of variation distance.) A (by-now) classic paper of Batu et al. gives an \(O(n^{2/3}\log n/\varepsilon^{8/3})\) and Valiant proved an \(\Omega(n^{2/3})\) lower bound. This paper completely resolves this problem problem with an algorithm and matching lower bound of \(\max(n^{2/3}/\varepsilon^{4/3}, n^{1/2}/\varepsilon^2)\).