Proper vs Improper Quantum PAC learning
Ashwin Nayak, Pulkit Sinha
TL;DR
The paper investigates whether proper quantum PAC learning can be harder than improper learning, building on classical Coupon Collector separations and prior quantum results. It delivers a near-optimal quantum Coupon Collector algorithm that meets the sharp lower bound across all $k$, and introduces the Quantum Padded Coupon Collector, whose sample complexity aligns with the classical Coupon Collector, thus reproducing the classical separation for proper vs improper quantum PAC learning in the padded setting. The results rely on a diagonalization of ensembles via modular signatures, a detailed random-walk analysis, and a careful reduction linking padding to classical coupon-collector behavior. The work suggests padding can lift classical learning behavior into the quantum regime and provides techniques broadly applicable to padded quantum data and related learning questions.
Abstract
A basic question in the PAC model of learning is whether proper learning is harder than improper learning. In the classical case, there are examples of concept classes with VC dimension $d$ that have sample complexity $Ω\left(\frac dε\log\frac1ε\right)$ for proper learning with error $ε$, while the complexity for improper learning is O$\!\left(\frac dε\right)$. One such example arises from the Coupon Collector problem. Motivated by the efficiency of proper versus improper learning with quantum samples, Arunachalam, Belovs, Childs, Kothari, Rosmanis, and de Wolf (TQC 2020) studied an analogue, the Quantum Coupon Collector problem. Curiously, they discovered that for learning size $k$ subsets of $[n]$ the problem has sample complexity $Θ(k\log\min\{k,n-k+1\})$, in contrast with the complexity of $Θ(k\log k)$ for Coupon Collector. This effectively negates the possibility of a separation between the two modes of learning via the quantum problem, and Arunachalam et al.\ posed the possibility of such a separation as an open question. In this work, we first present an algorithm for the Quantum Coupon Collector problem with sample complexity that matches the sharper lower bound of $(1-o_k(1))k\ln\min\{k,n-k+1\}$ shown recently by Bab Hadiashar, Nayak, and Sinha (IEEE TIT 2024), for the entire range of the parameter $k$. Next, we devise a variant of the problem, the Quantum Padded Coupon Collector. We prove that its sample complexity matches that of the classical Coupon Collector problem for both modes of learning, thereby exhibiting the same asymptotic separation between proper and improper quantum learning as mentioned above. The techniques we develop in the process can be directly applied to any form of padded quantum data. We hope that padding can more generally lift other forms of classical learning behaviour to the quantum setting.