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Private graphon estimation via sum-of-squares

Hongjie Chen, Jingqiu Ding, Tommaso d'Orsi, Yiding Hua, Chih-Hung Liu, David Steurer

TL;DR

This work delivers the first polynomial-time, node-differentially private algorithms for learning stochastic block models and graphon estimation with fixed numbers of blocks, achieving statistical utilities comparable to the best exponential-time private mechanisms. The core idea is to run an exponential mechanism over a sum-of-squares relaxation of the underlying combinatorial score, with the sos level determined by the target block count. A key technical contribution is showing that graphon distances reduce to quadratic optimization over the Birkhoff polytope, enabling low-degree sos certificates and multiplicative approximation guarantees. The authors also develop robust Lipschitz-extension techniques within the sos framework to extend privacy guarantees beyond maximum-degree constraints and establish lower bounds on sample complexity, as well as non-private improvements in the same computational regime. Overall, the results bridge computational tractability and privacy, yielding practical, provably-private graphon and SBM learning algorithms with strong theoretical guarantees.

Abstract

We develop the first pure node-differentially-private algorithms for learning stochastic block models and for graphon estimation with polynomial running time for any constant number of blocks. The statistical utility guarantees match those of the previous best information-theoretic (exponential-time) node-private mechanisms for these problems. The algorithm is based on an exponential mechanism for a score function defined in terms of a sum-of-squares relaxation whose level depends on the number of blocks. The key ingredients of our results are (1) a characterization of the distance between the block graphons in terms of a quadratic optimization over the polytope of doubly stochastic matrices, (2) a general sum-of-squares convergence result for polynomial optimization over arbitrary polytopes, and (3) a general approach to perform Lipschitz extensions of score functions as part of the sum-of-squares algorithmic paradigm.

Private graphon estimation via sum-of-squares

TL;DR

This work delivers the first polynomial-time, node-differentially private algorithms for learning stochastic block models and graphon estimation with fixed numbers of blocks, achieving statistical utilities comparable to the best exponential-time private mechanisms. The core idea is to run an exponential mechanism over a sum-of-squares relaxation of the underlying combinatorial score, with the sos level determined by the target block count. A key technical contribution is showing that graphon distances reduce to quadratic optimization over the Birkhoff polytope, enabling low-degree sos certificates and multiplicative approximation guarantees. The authors also develop robust Lipschitz-extension techniques within the sos framework to extend privacy guarantees beyond maximum-degree constraints and establish lower bounds on sample complexity, as well as non-private improvements in the same computational regime. Overall, the results bridge computational tractability and privacy, yielding practical, provably-private graphon and SBM learning algorithms with strong theoretical guarantees.

Abstract

We develop the first pure node-differentially-private algorithms for learning stochastic block models and for graphon estimation with polynomial running time for any constant number of blocks. The statistical utility guarantees match those of the previous best information-theoretic (exponential-time) node-private mechanisms for these problems. The algorithm is based on an exponential mechanism for a score function defined in terms of a sum-of-squares relaxation whose level depends on the number of blocks. The key ingredients of our results are (1) a characterization of the distance between the block graphons in terms of a quadratic optimization over the polytope of doubly stochastic matrices, (2) a general sum-of-squares convergence result for polynomial optimization over arbitrary polytopes, and (3) a general approach to perform Lipschitz extensions of score functions as part of the sum-of-squares algorithmic paradigm.
Paper Structure (70 sections, 66 theorems, 248 equations, 10 algorithms)

This paper contains 70 sections, 66 theorems, 248 equations, 10 algorithms.

Table of Contents

  1. Introduction
  2. Parameter estimation and privacy
  3. Distance measure for block-connectivity matrices
  4. State of the art and computational complexity
  5. Robustness and graphons
  6. Results
  7. Stochastic block model
  8. Graphon
  9. Lower bound on the sample complexity for private mechanisms
  10. Improvement in non-private setting
  11. Techniques
  12. Privacy
  13. Utility (without sum-of-squares)
  14. Utility with sum-of-squares
  15. Approximation scheme for polynomial minimization over (near-)integral polytopes
  16. ...and 55 more sections

Key Result

Theorem 1.2

Consider a graph $\bm G$ sampled from the $({B_0}, d, n)$-block model defined in definition:sbm-balanced. Suppose $\lVert{B_0}\rVert_{\max}\leqslant R\leqslant \sqrt{\frac{n\varepsilon}{\mathop{\mathrm{polylog}}\nolimits(n)}}$ for some $R\in \varmathbb R_+$, and $\varepsilon^4 n^2\geqslant \mathop{\

Theorems & Definitions (132)

  • Definition 1.1: Balanced stochastic block model
  • Theorem 1.2: Learning SBMs with node differential privacy
  • Theorem 1.3: Graphon estimation with node differential privacy, formal statement in \ref{['thm:formalmaintheorem']}
  • Theorem 1.4: Sample complexity lower bound for private estimation of stochastic block model
  • Theorem 1.5
  • Definition 3.1: Doubly stochastic matrix
  • Definition 3.2: Birkhoff polytope
  • Theorem 3.3: Birkhoff--von Neumann theorem
  • Definition 3.4: Node-differential privacy KNRS13
  • Theorem 3.5: Exponential Mechanism mcsherry2007mechanism
  • ...and 122 more