Adversarial Low Degree Testing

TL;DR

This paper introduces a new query-efficient tester for low-degree polynomials in the online erasure model, nearly matching classical testing complexities and applicable to functions over finite fields.

Contribution

It provides the first nearly optimal tester for low-degree functions in the online erasure model, extending property testing capabilities in adversarial settings.

Findings

The tester distinguishes low-degree from far-from-low-degree functions with polylogarithmic query complexity.

The approach uses random affine subspace queries to efficiently test low-degree properties.

The method applies to functions over finite fields and can be adapted to related models.

Abstract

In the $t$-online-erasure model in property testing, an adversary is allowed to erase $t$ values of a queried function for each query the tester makes. This model was recently formulated by Kalemaj, Raskhodnikova andVarma, who showed that the properties of linearity of functions as well as quadraticity can be tested in$O_t(1)$ many queries: $O(\log (t))$ for linearity and $2^{2^{O(t)}}$ for quadraticity. They asked whether the more general property of low-degreeness can be tested in the online erasure model, whether better testers exist for quadraticity, and if similar results hold when ``erasures'' are replaced with ``corruptions''. We show that, in the $t$-online-erasure model, for a prime power $q$, given query access to a function $f: \mathbb{F}_q^n \xrightarrow[]{} \mathbb{F}_q$, one can distinguish in $\mathrm{poly}(\log^{d+q}(t)/\delta)$ queries between the case that $f$ is degree at most $d$, and the case that $f$ is $\delta$-far from any degree $d$ function (with respect to the fractional hamming distance). This answers the aforementioned questions and brings the query complexity to nearly match the query complexity of low-degree testing in the classical property testing model. Our results are based on the observation that the property of low-degreeness admits a large and versatile family of query efficient testers. Our testers operates by querying a uniformly random, sufficiently large set of points in a large enough affine subspace, and finding a tester for low-degreeness that only utilizes queries from that set of points. We believe that this tester may find other applications to algorithms in the online-erasure model or other related models, and may be of independent interest.