Low degree conjecture implies sharp computational thresholds in stochastic block model

We investigate implications of the (extended) low-degree conjecture (recently formalized in [MW23]) in the context of the symmetric stochastic block model. Assuming the conjecture holds, we establish that no polynomial-time algorithm can weakly recover community labels below the Kesten-Stigum (KS) threshold. In particular, we rule out polynomial-time estimators that, with constant probability, achieve correlation with the true communities that is significantly better than random. Whereas, above the KS threshold, polynomial-time algorithms are known to achieve constant correlation with the true communities with high probability[Mas14,AS15]. To our knowledge, we provide the first rigorous evidence for the sharp transition in recovery rate for polynomial-time algorithms at the KS threshold. Notably, under a stronger version of the low-degree conjecture, our lower bound remains valid even when the number of blocks diverges. Furthermore, our results provide evidence of a computational-to-statistical gap in learning the parameters of stochastic block models. In contrast to prior work, which either (i) rules out polynomial-time algorithms for hypothesis testing with 1-o(1) success probability [Hopkins18, BBK+21a] under the low-degree conjecture, or (ii) rules out low-degree polynomials for learning the edge connection probability matrix [LG23], our approach provides stronger lower bounds on the recovery and learning problem. Our proof combines low-degree lower bounds from [Hopkins18, BBK+21a] with graph splitting and cross-validation techniques. In order to rule out general recovery algorithms, we employ the correlation preserving projection method developed in [HS17].

Private Edge Density Estimation for Random Graphs: Optimal, Efficient and Robust

We give the first polynomial-time, differentially node-private, and robust algorithm for estimating the edge density of Erd\H{o}s-R\'enyi random graphs and their generalization, inhomogeneous random graphs. We further prove information-theoretical lower bounds, showing that the error rate of our algorithm is optimal up to logarithmic factors. Previous algorithms incur either exponential running time or suboptimal error rates. Two key ingredients of our algorithm are (1) a new sum-of-squares algorithm for robust edge density estimation, and (2) the reduction from privacy to robustness based on sum-of-squares exponential mechanisms due to Hopkins et al. (STOC 2023).

Private graphon estimation via sum-of-squares

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.

Reaching Kesten-Stigum Threshold in the Stochastic Block Model under Node Corruptions

We study robust community detection in the context of node-corrupted stochastic block model, where an adversary can arbitrarily modify all the edges incident to a fraction of the $n$ vertices. We present the first polynomial-time algorithm that achieves weak recovery at the Kesten-Stigum threshold even in the presence of a small constant fraction of corrupted nodes. Prior to this work, even state-of-the-art robust algorithms were known to break under such node corruption adversaries, when close to the Kesten-Stigum threshold. We further extend our techniques to the $Z_2$ synchronization problem, where our algorithm reaches the optimal recovery threshold in the presence of similar strong adversarial perturbations. The key ingredient of our algorithm is a novel identifiability proof that leverages the push-out effect of the Grothendieck norm of principal submatrices.

SQ Lower Bounds for Random Sparse Planted Vector Problem

Consider the setting where a $\rho$-sparse Rademacher vector is planted in a random $d$-dimensional subspace of $R^n$. A classical question is how to recover this planted vector given a random basis in this subspace. A recent result by [ZSWB21] showed that the Lattice basis reduction algorithm can recover the planted vector when $n\geq d+1$. Although the algorithm is not expected to tolerate inverse polynomial amount of noise, it is surprising because it was previously shown that recovery cannot be achieved by low degree polynomials when $n\ll \rho^2 d^{2}$ [MW21]. A natural question is whether we can derive an Statistical Query (SQ) lower bound matching the previous low degree lower bound in [MW21]. This will - imply that the SQ lower bound can be surpassed by lattice based algorithms; - predict the computational hardness when the planted vector is perturbed by inverse polynomial amount of noise. In this paper, we prove such an SQ lower bound. In particular, we show that super-polynomial number of VSTAT queries is needed to solve the easier statistical testing problem when $n\ll \rho^2 d^{2}$ and $\rho\gg \frac{1}{\sqrt{d}}$. The most notable technique we used to derive the SQ lower bound is the almost equivalence relationship between SQ lower bound and low degree lower bound [BBH+20, MW21].