A Mysterious Connection Between Tolerant Junta Testing and Agnostically Learning Conjunctions
Xi Chen, Shyamal Patel, Rocco A. Servedio
TL;DR
This work uncovers a deep connection between distribution-free agnostic learning of conjunctions and tolerant testing of juntas, and leverages it to achieve faster algorithms for both problems. The agnostic-learning contribution uses a ball-distribution strategy to obtain low approximate-degree bounds and a KKMS-based L1-regression procedure that runs in time $2^{\widetilde{O}(n^{1/3})}$ for constant error, overcoming prior $2^{\widetilde{O}(\sqrt{n})}$ barriers. On the testing side, the paper develops both quantum and classical tolerant-junta testers with $2^{\widetilde{O}(k^{1/3})}$ query complexity, demonstrating an adaptive advantage over non-adaptive lower bounds and establishing a superpolynomial separation. The techniques blend Fourier-analytic tools (spectral sample, normalized influences, SharpNoise, flat polynomials) with local estimators and coordinate-oracle technology to produce robust, noise-tolerant algorithms. Overall, the results push the boundaries of efficiency for core learning and testing problems in the presence of noise and illustrate a principled bridge between seemingly distinct algorithmic families.
Abstract
The main conceptual contribution of this paper is identifying a previously unnoticed connection between two central problems in computational learning theory and property testing: agnostically learning conjunctions and tolerantly testing juntas. Inspired by this connection, the main technical contribution is a pair of improved algorithms for these two problems. In more detail, - We give a distribution-free algorithm for agnostically PAC learning conjunctions over $\{\pm 1\}^n$ that runs in time $2^{\widetilde{O}(n^{1/3})}$, for constant excess error $\varepsilon$. This improves on the fastest previously published algorithm, which runs in time $2^{\widetilde{O}(n^{1/2})}$ [KKMS08]. - Building on the ideas in our agnostic conjunction learner and using significant additional technical ingredients, we give an adaptive tolerant testing algorithm for $k$-juntas that makes $2^{\widetilde{O}(k^{1/3})}$ queries, for constant "gap parameter" $\varepsilon$ between the "near" and "far" cases. This improves on the best previous results, due to [ITW21, NP24], which make $2^{\widetilde{O}(\sqrt{k})}$ queries. Since there is a known $2^{\widetildeΩ(\sqrt{k})}$ lower bound for non-adaptive tolerant junta testers, our result shows that adaptive tolerant junta testing algorithms provably outperform non-adaptive ones.