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

Alex Shvets

Abstract

We develop a geometric framework for anonymous shuffle experiments based on an anchored affine likelihood-ratio law: a mean-zero measure on the regular simplex polytope. Every finite-output d-ary channel corresponds, up to refinements, to a unique anchored law, and conversely. On privacy: among all epsilon_0-LDP channels, binary randomized response universally extremizes all convex f-divergences and hockey-stick profiles after shuffling. A rigidity converse shows that saturation of both directed envelopes at finite n forces the binary endpoint law. On design: under the pairwise chi_* budget, we prove exact trace-cap and two-orbit frontier theorems. Every frontier point is realized by a mixture of at most two orbit laws. In the low-budget regime, augmented randomized response is minimax-optimal to the sharp constant over all channels and estimators. Under the raw LDP cap, the problem reduces to subset-selection with explicit optimal subset size. The arguments are self-contained and independent of the author's trilogy.

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

Abstract

We develop a geometric framework for anonymous shuffle experiments based on an anchored affine likelihood-ratio law: a mean-zero measure on the regular simplex polytope. Every finite-output d-ary channel corresponds, up to refinements, to a unique anchored law, and conversely. On privacy: among all epsilon_0-LDP channels, binary randomized response universally extremizes all convex f-divergences and hockey-stick profiles after shuffling. A rigidity converse shows that saturation of both directed envelopes at finite n forces the binary endpoint law. On design: under the pairwise chi_* budget, we prove exact trace-cap and two-orbit frontier theorems. Every frontier point is realized by a mixture of at most two orbit laws. In the low-budget regime, augmented randomized response is minimax-optimal to the sharp constant over all channels and estimators. Under the raw LDP cap, the problem reduces to subset-selection with explicit optimal subset size. The arguments are self-contained and independent of the author's trilogy.
Paper Structure (29 sections, 33 theorems, 376 equations, 2 tables)

This paper contains 29 sections, 33 theorems, 376 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Anchored affine likelihood-ratio laws
  3. Geometry of the simplex polytope
  4. Exact representation and reconstruction
  5. Projective transport and fibers
  6. Projective maps
  7. Projective fibers and transport
  8. Pairwise privacy reduction and universal envelopes
  9. Exact one-dimensional reduction
  10. The universal binary envelope
  11. Estimation in the anchored gauge
  12. Covariance and Fisher information
  13. Canonical unbiased estimator under fixed composition
  14. Global symmetrization
  15. Exchangeable laws, trace caps, and augmented randomized response
  16. ...and 14 more sections

Key Result

Lemma 1

The map is an affine bijection from $\mathcal{X}_d$ onto $\mathcal{K}_d$, with inverse Moreover, In particular,

Theorems & Definitions (85)

  • Definition 1: Regular simplex data
  • Lemma 1: Affine identification of $\mathcal{X}_d$ and $\mathcal{K}_d$
  • proof
  • Definition 2: Anchored law
  • Definition 3: Conditionally identical refinements
  • Theorem 2.1: Exact affine representation
  • proof
  • Definition 4: Projective domains and maps
  • Proposition 1: Projective bijection
  • proof
  • ...and 75 more