# News for October 2022

Yet another month that is kind of quiet! If we missed any work, please let us know in the comments.

Gaussian Mean Testing Made Simple, by Ilias Diakonikolas, Daniel Kane and Ankit Pensia (arXiv). Consider an unknown distribution distribution $$p$$ over $$\mathbb{R}^d$$ that we have sample access to. The paper studies the problem of determining whether $$p$$ is a standard Gaussian with zero mean or whether it is a Gaussian with large mean. More formally, the task is to distinguish between the case that $$p$$ is $$\mathcal{N}(0, I_d)$$ and the case that $$p$$ is a Gaussian of the form $$\mathcal{N}(\mu, \Sigma)$$, where $$||\mu||_2 \geq \epsilon$$ and $$\Sigma$$ is an unknown covariance matrix. Canonne, Chen, Kamath, Levi and Weingarten (2021) gave a sample-optimal algorithm for this problem with sample complexity $$\Theta(\sqrt{d}/\epsilon^2)$$ sample complexity. The current paper gives another sample-optimal algorithm for the same problem with a simpler analysis. In addition to being sample-optimal, the algorithm in the current paper also runs in time linear in the total sample size, which is an improvement over the work of Canonne et al.

Superpolynomial lower bounds for decision tree learning and testing, by Caleb Koch, Carmen Strassle and Li-Yang Tan (arXiv). Roughly speaking, the paper studies the problems of testing if a function has a low-depth decision tree and learning a low-depth decision tree approximating a function (provided that one such tree exists). In what follows, we summarize the testing results in the paper. Given an explicit representation of a function $$f:\{0,1\}^n \to \{0,1\}$$ and access to samples from a known distribution $$\mathcal{D}$$ over $$\{0,1\}^n$$, one can aim to determine, with probability at least $$2/3$$, if $$f$$ has a decision tree of depth at most $$d$$ or whether $$f$$ is $$\epsilon$$-far from having a decision tree of depth at most $$d\log d$$, where the distance is measured with respect to $$\mathcal{D}$$. The paper shows that, under the randomized exponential time hypothesis, this problem cannot be solved in time $$\exp(d^{\Omega(1)})$$. An immediate corollary is that the same lower bound holds for the problem of distribution-free testing of the property of having depth-$$d$$ decision trees. The bound in the current paper is an improvement over the recent work of Blais, Ferreira Pinto Jr., and Harms (2021), who give a lower bound of $$\tilde{\Omega}(2^d)$$ on the query complexity of testers for the same problem. However, the advantage of the latter result is that it is unconditional, as opposed to the result in the current paper.