[Paper Review] The Power of an Example: Hidden Set Size Approximation Using Group Queries and Conditional Sampling

Dana Ron, Gilad Tsur|arXiv (Cornell University)|Apr 20, 2014

Machine Learning and Algorithms9 references4 citations

TL;DR

This paper studies the problem of approximating the size of an unknown subset $ S $ within a known universe $ U $, using two types of queries: group queries (indicating whether $ S \cap T \neq \emptyset $) and conditional sampling (returning a uniform element from $ S \cap T $ if non-empty). It establishes tight bounds on query complexity for both adaptive and non-adaptive algorithms, showing that conditional sampling offers significant advantages over group queries, especially in structured settings like interval queries.

ABSTRACT

We study a basic problem of approximating the size of an unknown set $S$ in a known universe $U$. We consider two versions of the problem. In both versions the algorithm can specify subsets $T\subseteq U$. In the first version, which we refer to as the group query or subset query version, the algorithm is told whether $T\cap S$ is non-empty. In the second version, which we refer to as the subset sampling version, if $T\cap S$ is non-empty, then the algorithm receives a uniformly selected element from $T\cap S$. We study the difference between these two versions under different conditions on the subsets that the algorithm may query/sample, and in both the case that the algorithm is adaptive and the case where it is non-adaptive. In particular we focus on a natural family of allowed subsets, which correspond to intervals, as well as variants of this family.

Motivation & Objective

• To investigate the relative power of group queries and conditional sampling in approximating the size of an unknown subset $ S \subseteq U $.
• To analyze the query complexity of approximating $ |S| $ under different constraints on the subsets $ T \subseteq U $ that may be queried or sampled.
• To compare adaptive and non-adaptive algorithms in terms of efficiency and query count for both query models.
• To establish tight upper and lower bounds on the number of queries/samples required to achieve $ (1+\epsilon) $-approximation with high probability.
• To explore the impact of structural constraints on the query sets, such as intervals or families defined by hash functions, on the efficiency of size approximation.

Proposed method

• Proposes two query models: group queries (binary answer on $ S \cap T \neq \emptyset $) and conditional sampling (uniform sample from $ S \cap T $ if non-empty).
• Designs a non-adaptive algorithm that uses a sequence of random subsets $ R^j_i $ with inclusion probability $ 1/e_{\log n - i} $, estimating $ \widehat{p}_i $ as the fraction of negative responses.
• Employs a two-stage adaptive algorithm: first, a coarse search using exponentially spaced thresholds to locate $ \ell^* $ such that $ e_{2^{\ell^*}} \approx w $, then a binary search to refine the estimate.
• Uses statistical estimation techniques to bound the error in $ \widehat{p}_i $, ensuring concentration via Chernoff bounds and logarithmic dependencies on failure probability.
• Applies a distribution over pairs $ (S_1, S_2) $ with sizes $ 2^i $ and $ 2^{i+1} $ to prove lower bounds via indistinguishability under limited queries.
• Adapts the analysis for $ \epsilon < 1 $ by modifying thresholds $ e_i $ and confidence intervals to maintain $ (1+\epsilon) $-approximation with $ O(\log(1/\delta)/\epsilon^2) $ samples per query.

Experimental results

Research questions

• RQ1 How does the power of conditional sampling compare to group queries in approximating the size of an unknown set?
• RQ2 What is the optimal number of queries/samples required for $ (1+\epsilon) $-approximation of $ |S| $ in both adaptive and non-adaptive settings?
• RQ3 How do structural constraints on query sets—such as intervals or hash-based families—affect the query complexity of set size approximation?
• RQ4 Can non-adaptive algorithms achieve similar efficiency to adaptive ones under the same query models?
• RQ5 What are the information-theoretic limits (lower bounds) on the number of queries needed for accurate set size approximation?

Key findings

• For the full family of all subsets, the paper establishes $ \tilde{O}(\sqrt{w}/\epsilon^2) $-query upper bound for conditional sampling, which is tight up to logarithmic factors.
• In the non-adaptive setting with interval queries, the paper achieves $ \tilde{O}(\log \log n / \epsilon^2) $ query complexity, independent of $ w $, under certain conditions.
• The paper proves a $ \Omega(\log \log n) $ lower bound for non-adaptive algorithms using interval queries, showing that the upper bound is nearly optimal.
• For adaptive algorithms with interval queries, the paper achieves $ \tilde{O}(\log \log w / \epsilon^2) $ query complexity, which is optimal up to logarithmic factors.
• Conditional sampling provides a significant advantage over group queries: while group queries require $ \Omega(\sqrt{w}) $ queries to approximate $ w $, conditional sampling reduces this to $ \tilde{O}(\sqrt{w}/\epsilon^2) $, with the $ \epsilon $-dependence being polynomial.
• The paper shows that for $ w > n/w $, a non-adaptive algorithm can achieve $ \tilde{O}(\log w / \epsilon^2) $ complexity by starting from $ w = n $ and descending through hypotheses, improving efficiency in sparse regimes.