What question did this study set out to answer?

The research aims to establish a geometric framework for privacy in anonymous shuffle experiments and derive optimal design strategies.

March 24, 2026Open Access

Anchored Likelihood-Ratio Geometry of Anonymous Shuffle Experiments: Exact Privacy Envelopes and Universal Low-Budget Design

Key Points

The research aims to establish a geometric framework for privacy in anonymous shuffle experiments and derive optimal design strategies.
Developed anchored affine likelihood-ratio laws on centered simplex polytope.
Derived exact finite-n canonical risk formulas and a trace-cap theorem.
Analyzed the performance of augmented randomized response in low-budget regimes.
Binary randomized response maximizes all convex f-divergences under the ε₀-LDP cap.
Simultaneous saturation at finite n leads to binary endpoint law.
Augmented randomized response achieves minimax-optimal results across all channels and estimators.

Abstract

We develop a geometric framework for anonymous shuffle experiments based on anchored affine likelihood-ratio laws on the centered simplex polytope. Under a common ε₀-LDP cap, binary randomized response universally maximizes all convex f-divergences and both directed hockey-stick profiles after shuffling; a rigidity converse shows that simultaneous saturation at finite n forces the pairwise law to be the binary endpoint. On the design side, we derive exact finite-n canonical risk formulas, a trace-cap theorem, and a two-orbit reduction of the global χ²-budget frontier. In the low-budget regime, augmented randomized response is asymptotically minimax-optimal to the sharp leading constant over all channels and all estimators.

Anchored Likelihood-Ratio Geometry of Anonymous Shuffle Experiments: Exact Privacy Envelopes and Universal Low-Budget Design

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Abstract

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