Optimal mass estimation in the conditional sampling model

TL;DR

This work develops a fundamentally faster mass-estimation method for individual elements under the fully conditional sampling model, breaking polylog barriers with a $O(\log\log N)$ component and a poly$(1/\varepsilon)$ factor, and proves a matching $Ω(\log\log N)$ lower bound. The key technique constructs a reference set as the intersection of a probabilistic target set and a random filter, then performs a binary search to select an effective conditioning that yields a stable estimate via a density ratio. The authors demonstrate broad applicability by deriving near-tight bounds for equivalence testing, distance estimation, histogram learning, and a universal tester for label-invariant properties, while establishing near-optimal lower bounds that emphasize the necessity of the full conditional model for the strongest results. Collectively, the results tighten the understanding of what is achievable with conditional sampling and offer practical estimators and testers for a wide range of distribution-property tasks. The work thus advances both the theoretical limits and the algorithmic toolkit for distribution testing in the conditional model, with potential impact on related sublinear algorithms and privacy-conscious data analysis.

Abstract

The conditional sampling model, introduced by Cannone, Ron and Servedio (SODA 2014, SIAM J. Comput. 2015) and independently by Chakraborty, Fischer, Goldhirsh and Matsliah (ITCS 2013, SIAM J. Comput. 2016), is a common framework for a number of studies concerning strengthened models of distribution testing. A core task in these investigations is that of estimating the mass of individual elements. The above mentioned works, and the improvement of Kumar, Meel and Pote (AISTATS 2025), provided polylogarithmic algorithms for this task. In this work we shatter the polylogarithmic barrier, and provide an estimator for the mass of individual elements that uses only $O(\log \log N) + O(\mathrm{poly}(1/\varepsilon))$ conditional samples. We complement this result with an $Ω(\log\log N)$ lower bound. We then show that our mass estimator provides an improvement (and in some cases a unifying framework) for a number of related tasks, such as testing by learning of any label-invariant property, and distance estimation between two (unknown) distribution. By considering some known lower bounds, this also shows that the full power of the conditional model is indeed required for the doubly-logarithmic upper bound. Finally, we exponentially improve the previous lower bound on testing by learning of label-invariant properties from double-logarithmic to $Ω(\log N)$ conditional samples, whereas our testing by learning algorithm provides an upper bound of $O(\mathrm{poly}(1/\varepsilon)\cdot\log N \log \log N)$.

Theorems & Definitions (58)

• Theorem 1.1: Informal statement of Theorem \ref{['th:estimation-task']}
• Theorem 1.2: Informal statement of Theorem \ref{['th:lbnd-tight-c-eps-task']}
• Theorem 1.2: Almost-tight upper bound for equivalence testing
• Theorem 1.2: Almost-tight upper bound for distance estimation
• Theorem 1.3: Informal statement of Theorem \ref{['th:ubnd-histogram']}
• Theorem 1.4: Informal statement of Corollary \ref{['cor:lbnd-histogram']}
• Theorem 1.5: Informal statement of Corollary \ref{['cor:universal-tester-label-invariant']}
• Theorem 1.6: Informal statement of Theorem \ref{['th:lbnd-label-invariant-testing-new']}
• Definition 3.1: Sampling oracle
• Definition 3.2: Conditional sampling oracle