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On Differentially Private Linear Algebra

Haim Kaplan, Yishay Mansour, Shay Moran, Uri Stemmer, Nitzan Tur

Abstract

We introduce efficient differentially private (DP) algorithms for several linear algebraic tasks, including solving linear equalities over arbitrary fields, linear inequalities over the reals, and computing affine spans and convex hulls. As an application, we obtain efficient DP algorithms for learning halfspaces and affine subspaces. Our algorithms addressing equalities are strongly polynomial, whereas those addressing inequalities are weakly polynomial. Furthermore, this distinction is inevitable: no DP algorithm for linear programming can be strongly polynomial-time efficient.

On Differentially Private Linear Algebra

Abstract

We introduce efficient differentially private (DP) algorithms for several linear algebraic tasks, including solving linear equalities over arbitrary fields, linear inequalities over the reals, and computing affine spans and convex hulls. As an application, we obtain efficient DP algorithms for learning halfspaces and affine subspaces. Our algorithms addressing equalities are strongly polynomial, whereas those addressing inequalities are weakly polynomial. Furthermore, this distinction is inevitable: no DP algorithm for linear programming can be strongly polynomial-time efficient.

Paper Structure

This paper contains 18 sections, 32 theorems, 106 equations, 10 algorithms.

Table of Contents

  1. Introduction
  2. Overview of Main Results
  3. Linear Equalities (\ref{['sec:equalitiesmain']})
  4. Linear Programming and Convex Combinations (\ref{['sec:inequalitiesmain']})
  5. Learning
  6. Open Questions
  7. Subspaces and Linear Equalities
  8. Synthetic Linear Equalities and Affine Hulls Generation
  9. Stable Partitions
  10. Synthetic Linear Spans
  11. Synthetic Affine Spans
  12. Synthetic Systems of Linear Equations
  13. Polyhedra and Linear Inequalities
  14. When the feasible region is not fully dimensional
  15. Point in the Convex Hull via Linear Programming and Affine Spans
  16. ...and 3 more sections

Key Result

Proposition 2

Given an input sequence $\{\boldsymbol{v_i}\}_{i=1}^m \subseteq \mathbb{F}^d$ to alg:vector_partition, let $\{A_t\}_{t=1}^k$ be the output partition. Then, there exist at most $d$ nested subspaces $V_1\subseteq \ldots \subseteq V_d$ such that each $\mathtt{span}(A_j)$ is one of these subspaces. Furt

Theorems & Definitions (71)

  • Remark 1
  • Proposition 2
  • proof
  • Remark 3
  • Theorem A: Private Linear Span
  • proof
  • Theorem B: Private Affine Span
  • proof
  • Theorem C: Private Synthetic Linear Equations
  • proof
  • ...and 61 more