Replicable Composition

2026-04-11

AI Summary

This work addresses the challenge of jointly executing multiple reproducible algorithms while simultaneously preserving reproducibility and achieving sample efficiency. To this end, the authors propose transforming each algorithm into a perfectly generalizing form and integrating private composition analysis, correlated sampling, and a newly introduced “phantom execution” technique. Their main contributions include the first advanced composition theorem for reproducibility, which achieves nearly optimal $\tilde{O}(\sum n_i)$ sample complexity under non-adaptive composition, and a lower bound of $\Omega(nk^2)$ for adaptive composition, thereby establishing a quadratic separation between the two settings. The results yield linear sample complexity under constant-level reproducibility and provide a general theorem for boosting the success probability of reproducible algorithms.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Replicability requires that algorithmic conclusions remain consistent when rerun on independently drawn data. A central structural question is composition: given $k$ problems each admitting a $ρ$-replicable algorithm with sample complexity $n$, how many samples are needed to solve all jointly while preserving replicability? The naive analysis yields $\widetilde{O}(nk^2)$ samples, and Bun et al. (STOC'23) observed that reductions through differential privacy give an alternative $\widetilde{O}(n^2k)$ bound, leaving open whether the optimal $\widetilde{O}(nk)$ scaling is achievable. We resolve this open problem and, more generally, show that problems with sample complexities $n_1,\ldots,n_k$ can be jointly solved with $\widetilde{O}(\sum_i n_i)$ samples while preserving constant replicability. Our approach converts each replicable algorithm into a perfectly generalizing one, composes them via a privacy-style analysis, and maps back via correlated sampling. This yields the first advanced composition theorem for replicability. En route, we obtain new bounds for the composition of perfectly generalizing algorithms with heterogeneous parameters. As part of our results, we provide a boosting theorem for the success probability of replicable algorithms. For a broad class of problems, the failure probability appears as a separate additive term independent of $ρ$, immediately yielding improved sample complexity bounds for several problems. Finally, we prove an $Ω(nk^2)$ lower bound for adaptive composition, establishing a quadratic separation from the non-adaptive setting. The key technique, which we call the phantom run, yields structural results of independent interest.

Problem

• replicability
• composition
• sample complexity
• differential privacy
• generalization

Innovation

• replicability
• composition
• perfect generalization
• correlated sampling
• adaptive lower bound