Yiding Hua

Jingqiu Ding, Yiding Hua

Consider the setting where a $ρ$-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 $n\ll ρ^2 d^{2}$ [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 $n\ll ρ^2 d^{2}$ and $ρ\gg \frac{1}{\sqrt{d}}$. 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].

Jingqiu Ding, Yiding Hua, Lucas Slot et al.

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].

Hongjie Chen, Jingqiu Ding, Tommaso d'Orsi et al.

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.

Jingqiu Ding, Yiding Hua, Kasper Lindberg et al.

We study community detection in the \emph{symmetric $k$-stochastic block model}, where $n$ nodes are evenly partitioned into $k$ clusters with intra- and inter-cluster connection probabilities $p$ and $q$, respectively. Our main result is a polynomial-time algorithm that achieves the minimax-optimal misclassification rate \begin{equation*} \exp \Bigl(-\bigl(1 \pm o(1)\bigr) \tfrac{C}{k}\Bigr), \quad \text{where } C = (\sqrt{pn} - \sqrt{qn})^2, \end{equation*} whenever $C \ge K\,k^2\,\log k$ for some universal constant $K$, matching the Kesten--Stigum (KS) threshold up to a $\log k$ factor. Notably, this rate holds even when an adversary corrupts an $η\le \exp\bigl(- (1 \pm o(1)) \tfrac{C}{k}\bigr)$ fraction of the nodes. To the best of our knowledge, the minimax rate was previously only attainable either via computationally inefficient procedures [ZZ15] or via polynomial-time algorithms that require strictly stronger assumptions such as $C \ge K k^3$ [GMZZ17]. In the node-robust setting, the best known algorithm requires the substantially stronger condition $C \ge K k^{102}$ [LM22]. Our results close this gap by providing the first polynomial-time algorithm that achieves the minimax rate near the KS threshold in both settings. Our work has two key technical contributions: (1) we robustify majority voting via the Sum-of-Squares framework, (2) we develop a novel graph bisection algorithm via robust majority voting, which allows us to significantly improve the misclassification rate to $1/\mathrm{poly}(k)$ for the initial estimation near the KS threshold.

Jingqiu Ding, Tommaso d'Orsi, Yiding Hua et al.

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.