February had a flurry of conference deadlines, and they seem to have produced six papers for us to enjoy, including three on estimating symmetric properties of distributions.

Locally Private Hypothesis Testing, by Or Sheffet (arXiv). We now have a very mature understanding of the sample complexity of distributional identity testing — given samples from a distribution \(p\), is it equal to, or far from, some model hypothesis \(q\)? Recently,  several papers have studied this problem under the additional constraint of differential privacy. This paper strengthens the privacy constraint to local privacy, where each sample is locally noised before being provided to the testing algorithm.

Distribution-free Junta Testing, by Xi Chen, Zhengyang Liu, Rocco A. Servedio, Ying Sheng, and Jinyu Xie (arXiv). Testing whether a function is a \(k\)-junta is very well understood — when done with respect to the uniform distribution. In particular, the adaptive complexity of this problem is \(\tilde \Theta(k)\), while the non-adaptive complexity is \(\tilde \Theta(k^{3/2})\). This paper studies the more challenging task of distribution-free testing, where the distance between functions is measured with respect to some unknown distribution. The authors show that, while the adaptive complexity of this problem is still polynomial (at \(\tilde O(k^2)\)), the non-adaptive complexity becomes exponential: \(2^{\Omega(k/3)}\). In other words, there’s a qualitative gap between the adaptive and non-adaptive complexity, which does not appear when testing with respect to the uniform distribution.

The Vertex Sample Complexity of Free Energy is Polynomial, by Vishesh Jain, Frederic Koehler, and Elchanan Mossel (arXiv). This paper studies the classic question of estimating (the logarithm of) the partition function of a Markov Random Field, a highly-studied topic in theoretical computer science and statistical physics. As the title suggests, the authors show that the vertex sample complexity of this quantity is polynomial. In other words, randomly subsampling a \(\mathrm{poly}(1/\varepsilon)\)-size graph and computing its free energy gives a good approximation to the free energy of the overall graph. This is in contrast to more general graph properties, for the vertex sample complexity is super-exponential in \(1/\varepsilon\).

Entropy Rate Estimation for Markov Chains with Large State Space, by Yanjun Han, Jiantao Jiao, Chuan-Zheng Lee, Tsachy Weissman, Yihong Wu, and Tiancheng Yu (arXiv). Entropy estimation is now quite well-understood when one observes independent samples from a discrete distribution — we can get by with a barely-sublinear sample complexity, saving a logarithmic factor compared to the support size. This paper shows that these savings can also be enjoyed in the case where we observe a sample path of observations from a Markov chain.

Local moment matching: A unified methodology for symmetric functional estimation and distribution estimation under Wasserstein distance, by Yanjun Han, Jiantao Jiao, and Tsachy Weissman (arXiv). Speaking more generally of the above problem: there has been significant work into estimating symmetric properties of distributions, i.e., those which do not change when the distribution is permuted. One natural method for estimating such properties is to estimate the sorted distribution, then apply the plug-in estimator for the quantity of interest. The authors give an improved estimator for the sorted distribution, improving on the results of Valiant and Valiant.

INSPECTRE: Privately Estimating the Unseen, by Jayadev Acharya, Gautam Kamath, Ziteng Sun, and Huanyu Zhang (arXiv). One final work in this area — this paper studies the estimation of symmetric distribution properties (including entropy, support size, and support coverage), but this time while maintaining differentially privacy of the sample. By using estimators for these tasks with low sensitivity, one can additionally obtain privacy for a low or no additional cost over the non-private sample complexity.