This paper studies the problem of inferring a $k$-factor, specifically a spanning $k$-regular graph, planted within an Erdős–Rényi random graph $\mathcal{G}(n,\lambda/n)$. We uncover an interesting “all-something-nothing” phase transition. Specifically, we show that as the average degree $\lambda$ surpasses the critical threshold of $1/k$, the inference problem undergoes a transition from almost exact recovery (“all” phase) to partial recovery (“something” phase). Moreover, as $\lambda$ tends to infinity, the accuracy of recovery diminishes to zero, leading to the onset of the “nothing” phase. This finding complements the recent result by Mossel, Niles-Weed, Sohn, Sun, and Zadik who established that for certain sufficiently dense graphs, the problem undergoes an “all-or-nothing” phase transition, jumping from near-perfect to near-zero recovery. In addition, we characterize the recovery accuracy of a linear-time iterative pruning algorithm and show that it achieves almost exact recovery when $\lambda < 1/k$. A key component of our analysis is a two-step cycle construction: we first build trees through local neighborhood exploration and then connect them by sprinkling using reserved edges. Interestingly, for proving impossibility of almost exact recovery, we construct $\Theta(n)$ many small trees of size $\Theta(1)$, whereas for establishing the algorithmic lower bound, a single large tree of size $\Theta(\sqrt{n\log n})$ suffices.

Random graph models with community structure have been studied extensively in the literature. For both the problems of detecting and recovering community structure, an interesting landscape of statistical and computational phase transitions has emerged. A natural unanswered question is: might it be possible to infer properties of the community structure (for instance, the number and sizes of communities) even in situations where actually finding those communities is believed to be computationally hard? We show the answer is no. In particular, we consider certain hypothesis testing problems between models with different community structures, and we show (in the low-degree polynomial framework) that testing between two options is as hard as finding the communities. In addition, our methods give the first computational lower bounds for testing between two different "planted" distributions, whereas previous results have considered testing between a planted distribution and an i.i.d. "null" distribution.

Convex optimization with feedback is a framework where a learner relies on iterative queries and feedback to arrive at the minimizer of a convex function. It has gained considerable popularity thanks to its scalability in large-scale optimization and machine learning. The repeated interactions, however, expose the learner to privacy risks from eavesdropping adversaries that observe the submitted queries. In this paper, we study how to optimally obfuscate the learner’s queries in convex optimization with first-order feedback, so that their learned optimal value is provably difficult to estimate for an eavesdropping adversary. We consider two formulations of learner privacy: a Bayesian formulation in which the convex function is drawn randomly, and a maximin formulation in which the function is fixed and the adversary’s probability of error is measured with respect to a minimax criterion. Suppose that the learner wishes to ensure the adversary cannot estimate accurately with probability greater than$1/L$for some$L > 0$. Our main results show that the query complexity overhead is additive in$L$in the maximin formulation, but multiplicative in$L$in the Bayesian formulation. Compared to existing learner-private sequential learning models with binary feedback, our results apply to the significantly richer family of general convex functions with full-gradient feedback. Our proofs rely on tools from the theory of Dirichlet processes, as well as a novel strategy designed for measuring information leakage under a full-gradient oracle.

We study the query complexity of a learner-private sequential learning problem, motivated by the privacy and security concerns due to eavesdropping that arise in practical applications such as pricing and Federated Learning. A learner tries to estimate an unknown scalar value, by sequentially querying an external database and receiving binary responses; meanwhile, a third-party adversary observes the learner’s queries but not the responses. The learner’s goal is to design a querying strategy with the minimum number of queries (optimal query complexity) so that she can accurately estimate the true value, while the eavesdropping adversary even with the complete knowledge of her querying strategy cannot.

Convex optimization with feedback is a framework where a learner relies on iterative queries and feedback to arrive at the minimizer of a convex function. The paradigm has gained significant popularity recently thanks to its scalability in large-scale optimization and machine learning. The repeated interactions, however, expose the learner to privacy risks from eavesdropping adversaries that observe the submitted queries. In this paper, we study how to optimally obfuscate the learner’s queries in convex optimization with first-order feedback, so that their learned optimal value is provably difficult to estimate for the eavesdropping adversary. We consider two formulations of learner privacy: a Bayesian formulation in which the convex function is drawn randomly, and a minimax formulation in which the function is fixed and the adversary’s probability of error is measured with respect to a minimax criterion. We show that, if the learner wants to ensure the probability of the adversary estimating accurately be kept below 1/L, then the overhead in query complexity is additive in L in the minimax formulation, but multiplicative in L in the Bayesian formulation. Compared to existing learner-private sequential learning models with binary feedback, our results apply to the significantly richer family of general convex functions with full-gradient feedback. Our proofs are largely enabled by tools from the theory of Dirichlet processes, as well as more sophisticated lines of analysis aimed at measuring the amount of information leakage under a full-gradient oracle.

Motivated by applications such as discovering strong ties in social networks and assembling genome subsequences in biology, we study the problem of recovering a hidden 2k-nearest neighbor (NN) graph in an n-vertex complete graph, whose edge weights are independent and distributed according to Pn for edges in the hidden 2k-NN graph and Qn otherwise. The special case of Bernoulli distributions corresponds to a variant of the Watts-Strogatz small-world graph. We focus on two types of asymptotic recovery guarantees as n→ ∞: (1) exact recovery: all edges are classified correctly with probability tending to one; (2) almost exact recovery: the expected number of misclassified edges is o(nk). We show that the maximum likelihood estimator achieves (1) exact recovery for 2 ≤ k ≤ no(1) if liminf\frac 2αnlogn > 1; (2) almost exact recovery for 1 ≤ k ≤ o(\frac lognloglogn ) if liminf\frac kD(Pn||Qn)logn > 1, where αn \triangleq -2 log∫√{d Pn d Qn} is the Rényi divergence of order \frac 12 and D(Pn||Qn) is the Kullback-Leibler divergence. Under mild distributional assumptions, these conditions are shown to be information-theoretically necessary for any algorithm to succeed. A key challenge in the analysis is the enumeration of 2k-NN graphs that differ from the hidden one by a given number of edges. We also analyze several computationally efficient algorithms and provide sufficient conditions under which they achieve exact/almost exact recovery. In particular, we develop a polynomial-time algorithm that attains the threshold for exact recovery under the small-world model.

Motivated by applications such as discovering strong ties in social networks and assembling genome subsequences in biology, we study the problem of recovering a hidden $2k$-nearest neighbor (NN) graph in an $n$-vertex complete graph, whose edge weights are independent and distributed according to $P_n$ for edges in the hidden $2k$-NN graph and $Q_n$ otherwise. The special case of Bernoulli distributions corresponds to a variant of the Watts-Strogatz small-world graph. We focus on two types of asymptotic recovery guarantees as $n\to \infty$: (1) exact recovery: all edges are classified correctly with probability tending to one; (2) almost exact recovery: the expected number of misclassified edges is $o(nk)$. We show that the maximum likelihood estimator achieves (1) exact recovery for $2 \le k \le n^{o(1)}$ if $ \liminf \frac{2\alpha_n}{\log n}>1$; (2) almost exact recovery for $ 1 \le k \le o\left( \frac{\log n}{\log \log n} \right)$ if $ \liminf \frac{kD(P_n||Q_n)}{\log n}>1, $ where $\alpha_n \triangleq -2 \log \int \sqrt{d P_n d Q_n}$ is the Rényi divergence of order $\frac{1}{2}$ and $D(P_n||Q_n)$ is the Kullback-Leibler divergence. Under mild distributional assumptions, these conditions are shown to be information-theoretically necessary for any algorithm to succeed. A key challenge in the analysis is the enumeration of $2k$-NN graphs that differ from the hidden one by a given number of edges. We also analyze several computationally efficient algorithms and provide sufficient conditions under which they achieve exact/almost exact recovery. In particular, we develop a polynomial-time algorithm that attains the threshold for exact recovery under the small-world model.

We consider the problem of selecting the best estimator among a family of Tikhonov regularized estimators, or, alternatively, to select a linear combination of these regularizers that is as good as the best regularizer in the family. Our theory reveals that if the Tikhonov regularizers share the same penalty matrix with different tuning parameters, a convex procedure based on $Q$-aggregation achieves the mean square error of the best estimator, up to a small error term no larger than $C\sigma^2$, where $\sigma^2$ is the noise level and $C>0$ is an absolute constant. Remarkably, the error term does not depend on the penalty matrix or the number of estimators as long as they share the same penalty matrix, i.e., it applies to any grid of tuning parameters, no matter how large the cardinality of the grid is. This reveals the surprising "cost-free" nature of optimally tuning Tikhonov regularizers, in striking contrast with the existing literature on aggregation of estimators where one typically has to pay a cost of $\sigma^2\log(M)$ where $M$ is the number of estimators in the family. The result holds, more generally, for any family of ordered linear smoothers; this encompasses Ridge regression as well as Principal Component Regression. The result is extended to the problem of tuning Tikhonov regularizers with different penalty matrices.

We give convergence guarantees for estimating the coefficients of a symmetric mixture of two linear regressions by expectation maximization (EM). In particular, we show that the empirical EM iterates converge to the target parameter vector at the parametric rate, provided the algorithm is initialized in an unbounded cone. In particular, if the initial guess has a sufficiently large cosine angle with the target parameter vector, a sample-splitting version of the EM algorithm converges to the true coefficient vector with high probability. Interestingly, our analysis borrows from tools used in the problem of estimating the centers of a symmetric mixture of two Gaussians by EM. We also show that the population of EM operator for mixtures of two regressions is anti-contractive from the target parameter vector if the cosine angle between the input vector and the target parameter vector is too small, thereby establishing the necessity of our conic condition. Finally, we give empirical evidence supporting this theoretical observation, which suggests that the sample-based EM algorithm may not converge to the target vector when initial guesses are drawn accordingly. Our simulation study also suggests that the EM algorithm performs well even under model misspecification (i.e., when the covariate and error distributions violate the model assumptions).