Nathaniel Harms

Mika Göös, Nathaniel Harms, Florian K. Richter et al.

Alice and Bob are given $n$-bit integer pairs $(x,y)$ and $(a,b)$, respectively, and they must decide if $y=ax+b$. We prove that the randomised communication complexity of this Point--Line Incidence problem is $Θ(\log n)$. This confirms a conjecture of Cheung, Hatami, Hosseini, and Shirley (CCC 2023) that the complexity is super-constant, and gives the first example of a communication problem with constant support-rank but super-constant randomised complexity.

Renato Ferreira Pinto, Nathaniel Harms

Consider two problems about an unknown probability distribution $p$: 1. How many samples from $p$ are required to test if $p$ is supported on $n$ elements or not? Specifically, given samples from $p$, determine whether it is supported on at most $n$ elements, or it is "$ε$-far" (in total variation distance) from being supported on $n$ elements. 2. Given $m$ samples from $p$, what is the largest lower bound on its support size that we can produce? The best known upper bound for problem (1) uses a general algorithm for learning the histogram of the distribution $p$, which requires $Θ(\tfrac{n}{ε^2 \log n})$ samples. We show that testing can be done more efficiently than learning the histogram, using only $O(\tfrac{n}{ε\log n} \log(1/ε))$ samples, nearly matching the best known lower bound of $Ω(\tfrac{n}{ε\log n})$. This algorithm also provides a better solution to problem (2), producing larger lower bounds on support size than what follows from previous work. The proof relies on an analysis of Chebyshev polynomial approximations outside the range where they are designed to be good approximations, and the paper is intended as an accessible self-contained exposition of the Chebyshev polynomial method.

Lorenzo Beretta, Nathaniel Harms, Caleb Koch

For a function $f \colon \{0,1\}^n \to \{0,1\}$, the junta testing problem asks whether $f$ depends on only $k$ variables. If $f$ depends on only $k$ variables, the feature selection problem asks to find those variables. We prove that these two tasks are statistically equivalent. Specifically, we show that the ``brute-force'' algorithm, which checks for any set of $k$ variables consistent with the sample, is simultaneously sample-optimal for both problems, and the optimal sample size is \[ Θ\left(\frac 1 \varepsilon \left( \sqrt{2^k \log {n \choose k}} + \log {n \choose k}\right)\right). \]

Eric Blais, Renato Ferreira Pinto, Nathaniel Harms

We consider the problem of determining which classes of functions can be tested more efficiently than they can be learned, in the distribution-free sample-based model that corresponds to the standard PAC learning setting. Our main result shows that while VC dimension by itself does not always provide tight bounds on the number of samples required to test a class of functions in this model, it can be combined with a closely-related variant that we call "lower VC" (or LVC) dimension to obtain strong lower bounds on this sample complexity. We use this result to obtain strong and in many cases nearly optimal lower bounds on the sample complexity for testing unions of intervals, halfspaces, intersections of halfspaces, polynomial threshold functions, and decision trees. Conversely, we show that two natural classes of functions, juntas and monotone functions, can be tested with a number of samples that is polynomially smaller than the number of samples required for PAC learning. Finally, we also use the connection between VC dimension and property testing to establish new lower bounds for testing radius clusterability and testing feasibility of linear constraint systems.

Nathaniel Harms, Yuichi Yoshida

We study distribution-free property testing and learning problems where the unknown probability distribution is a product distribution over $\mathbb{R}^d$. For many important classes of functions, such as intersections of halfspaces, polynomial threshold functions, convex sets, and $k$-alternating functions, the known algorithms either have complexity that depends on the support size of the distribution, or are proven to work only for specific examples of product distributions. We introduce a general method, which we call downsampling, that resolves these issues. Downsampling uses a notion of "rectilinear isoperimetry" for product distributions, which further strengthens the connection between isoperimetry, testing, and learning. Using this technique, we attain new efficient distribution-free algorithms under product distributions on $\mathbb{R}^d$: 1. A simpler proof for non-adaptive, one-sided monotonicity testing of functions $[n]^d \to \{0,1\}$, and improved sample complexity for testing monotonicity over unknown product distributions, from $O(d^7)$ [Black, Chakrabarty, & Seshadhri, SODA 2020] to $\widetilde O(d^3)$. 2. Polynomial-time agnostic learning algorithms for functions of a constant number of halfspaces, and constant-degree polynomial threshold functions. 3. An $\exp(O(d \log(dk)))$-time agnostic learning algorithm, and an $\exp(O(d \log(dk)))$-sample tolerant tester, for functions of $k$ convex sets; and a $2^{\widetilde O(d)}$ sample-based one-sided tester for convex sets. 4. An $\exp(\widetilde O(k \sqrt d))$-time agnostic learning algorithm for $k$-alternating functions, and a sample-based tolerant tester with the same complexity.

Nathaniel Harms

We present an algorithm for testing halfspaces over arbitrary, unknown rotation-invariant distributions. Using $\tilde O(\sqrt{n}ε^{-7})$ random examples of an unknown function $f$, the algorithm determines with high probability whether $f$ is of the form $f(x) = sign(\sum_i w_ix_i-t)$ or is $ε$-far from all such functions. This sample size is significantly smaller than the well-known requirement of $Ω(n)$ samples for learning halfspaces, and known lower bounds imply that our sample size is optimal (in its dependence on $n$) up to logarithmic factors. The algorithm is distribution-free in the sense that it requires no knowledge of the distribution aside from the promise of rotation invariance. To prove the correctness of this algorithm we present a theorem relating the distance between a function and a halfspace to the distance between their centers of mass, that applies to arbitrary distributions.