Sorry for the delay in writing this monthly digest: we hope you didn’t spend the week frantically refreshing your browsers! We found four papers this month: let’s dive in.

A Structural Theorem for Local Algorithms with Applications to Coding, Testing, and Privacy, by Marcel Dall’Agnol, Tom Gur, and Oded Lachish (ECCC). This paper introduces the notion of “robust (local) algorithm,” an abstraction which encompasses may types of algorithms: e.g., property testing algorithms, locally decodable codes, etc. The main result of this work is that any (possibly adaptive) \(q\)-query robust local algorithm can be transformed into a non-adaptive, sample-based one making \(n^{1-1/(q^2\log q)}\) queries (where \(n\) is the input size). Here, “sample-based” means that the algorithm doesn’t get to make arbitrary queries, but just gets to observe randomly chosen coordinates of the input. As application of this result, the authors derive new upper and lower bounds for several of the types of local algorithms mentioned above, resolving open questions from previous works.

Testing Tail Weight of a Distribution Via Hazard Rate, by Maryam Aliakbarpour, Amartya Shankha Biswas, Kavya Ravichandran, Ronitt Rubinfeld (arXiv). The authors consider, from the point of view of distribution testing, the question of deciding whether data is heavy-tailed: as would, for instance, data following a power law. The paper first sets out to formalize the question, and discusses various possible definitional choices before setting on one; after which it provides a test, and analyzes its sample complexity as a function of various parameters (such as smoothness of the unknown distribution). The authors finally back these results with empirical evaluation of their algorithm.

On Testing of Samplers, by Kuldeep S. Meel, Yash Pote, Sourav Chakraborty (arXiv). Suppose you are given a (known) distribution \(p\) over some domain \(\Omega\), and want to sample from it conditioned on some predicate \(\varphi\). Now someone comes to you with an algorithm which does exactly that, efficiently, and cheaply: great! But can you easily check that you’re not getting fooled, and that this sampler actually does what it claims? This paper provides this: an algorithm which accepts if the sampled distribution is \(\varepsilon\)-close to what it should (roughly, in a multiplicative, KL divergence sense), and rejects if it’s \(\varepsilon’\)-far (in total variation distance). The number of samples required is polynomial in \(\varepsilon’-\varepsilon\), and depends on some characteristic of \(p\) and \(\varphi\), the “tilt” (ratio between max and min probability of the conditional distribution).

Finally, an omission from late September:
Sample optimal Quantum identity testing via Pauli Measurements, by Nengkun Yu (arXiv). The abstract is concise and clear enough to speak for itself: “In this paper, we show that \(\Theta(\textrm{poly}(n)\cdot\frac{4^n}{\varepsilon^2})\) is the sample complexity of testing whether two \(n\)-qubit quantum states \(\rho\) and \(\sigma\) are identical or \(\varepsilon\)-far in trace distance using two-outcome Pauli measurements.”

Please let us know if you spotted a paper we missed!