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.

首次开发纯节点差分隐私算法，用于学习随机块模型和多项式时间估计图论，其统计效能保证与先前最佳的信息理论（指数时间）节点私有机制相匹配。该算法基于一个指数机制，用于基于二次优化的得分函数，其水平取决于区块数。我们结果的关键要素是：（1）在可双重随机矩阵的多面体上进行的一个二次优化问题，描述了区块图模式之间的距离；（2）多项式优化在任意多面体上的通用二次收敛结果；（3）对得分函数进行利普希茨扩展的通用方法，作为二次优化算法范例的一部分。

基于和平方的私有图信息估计