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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., Comment: Accepted to Innovations in Theoretical Computer Science (ITCS) 2023

We study the problem of reconstructing a perfect matching $M^*$ hidden in a randomly weighted $n\times n$ bipartite graph. The edge set includes every node pair in $M^*$ and each of the $n(n-1)$ node pairs not in $M^*$ independently with probability $d/n$. The weight of each edge $e$ is independently drawn from the distribution $\mathcal{P}$ if $e \in M^*$ and from $\mathcal{Q}$ if $e \notin M^*$. We show that if $\sqrt{d} B(\mathcal{P},\mathcal{Q}) \le 1$, where $B(\mathcal{P},\mathcal{Q})$ stands for the Bhattacharyya coefficient, the reconstruction error (average fraction of misclassified edges) of the maximum likelihood estimator of $M^*$ converges to $0$ as $n\to \infty$. Conversely, if $\sqrt{d} B(\mathcal{P},\mathcal{Q}) \ge 1+\epsilon$ for an arbitrarily small constant $\epsilon>0$, the reconstruction error for any estimator is shown to be bounded away from $0$ under both the sparse and dense model, resolving the conjecture in [Moharrami et al. 2019, Semerjian et al. 2020]. Furthermore, in the special case of complete exponentially weighted graph with $d=n$, $\mathcal{P}=\exp(\lambda)$, and $\mathcal{Q}=\exp(1/n)$, for which the sharp threshold simplifies to $\lambda=4$, we prove that when $\lambda \le 4-\epsilon$, the optimal reconstruction error is $\exp\left( - \Theta(1/\sqrt{\epsilon}) \right)$, confirming the conjectured infinite-order phase transition in [Semerjian et al. 2020].

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 minimax 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 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 learn on tools from the theory of Dirichlet processes, as well as a novel strategy designed for measuring 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 $P_n$ 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 R\'enyi 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.

The Metropolis-Hastings method is often used to construct a Markov chain with a given $\pi$ as its stationary distribution. The method works even if $\pi$ is known only up to an intractable constant of proportionality. Polynomial time convergence results for such chains (rapid mixing) are hard to obtain for high dimensional probability models where the size of the state space potentially grows exponentially with the model dimension. In a Bayesian context, Yang, Wainwright, and Jordan (2016) (=YWJ) used the path method to prove rapid mixing for high dimensional linear models. This paper proposes a modification of the YWJ approach that simplifies the theoretical argument and improves the rate of convergence. The new approach is illustrated by an application to an exponentially weighted aggregation estimator.

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. We develop new querying strategies and analytical techniques and use them to prove tight upper and lower bounds on the optimal query complexity. The bounds almost match across the entire parameter range, substantially improving upon existing results. We thus obtain a complete picture of the optimal query complexity as a function of the estimation accuracy and the desired levels of privacy. We also extend the results to sequential learning models in higher dimensions, and where the binary responses are noisy. Our analysis leverages a crucial insight into the nature of private learning problem, which suggests that the query trajectory of an optimal learner can be divided into distinct phases that focus on pure learning versus learning and obfuscation, respectively.

Quantile regression is a tool for learning conditional distributions. In this paper we study quantile regression in the setting where a protected attribute is unavailable when fitting the model. This can lead to "unfair'' quantile estimators for which the effective quantiles are very different for the subpopulations defined by the protected attribute. We propose a procedure for adjusting the estimator on a heldout sample where the protected attribute is available. The main result of the paper is an empirical process analysis showing that the adjustment leads to a fair estimator for which the target quantiles are brought into balance, in a statistical sense that we call $\sqrt{n}$-fairness. We illustrate the ideas and adjustment procedure on a dataset of 200,000 live births, where the objective is to characterize the dependence of the birth weights of the babies on demographic attributes of the birth mother; the protected attribute is the mother's race.

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.

A popular class of problem in statistics deals with estimating the support of a density from $n$ observations drawn at random from a $d$-dimensional distribution. The one-dimensional case reduces to estimating the end points of a univariate density. In practice, an experimenter may only have access to a noisy version of the original data. Therefore, a more realistic model allows for the observations to be contaminated with additive noise. In this paper, we consider estimation of convex bodies when the additive noise is distributed according to a multivariate Gaussian distribution, even though our techniques could easily be adapted to other noise distributions. Unlike standard methods in deconvolution that are implemented by thresholding a kernel density estimate, our method avoids tuning parameters and Fourier transforms altogether. We show that our estimator, computable in $(O(\ln n))^{(d-1)/2}$ time, converges at a rate of $ O_d(\log\log n/\sqrt{\log n}) $ in Hausdorff distance, in accordance with the polylogarithmic rates encountered in Gaussian deconvolution problems. Part of our analysis also involves the optimality of the proposed estimator. We provide a lower bound for the minimax rate of estimation in Hausdorff distance that is $\Omega_d(1/\log^2 n)$.

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 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 performs poorly 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).

We consider the sparse high-dimensional linear regression model $Y=Xb+\epsilon$ where $b$ is a sparse vector. For the Bayesian approach to this problem, many authors have considered the behavior of the posterior distribution when, in truth, $Y=X\beta+\epsilon$ for some given $\beta$. There have been numerous results about the rate at which the posterior distribution concentrates around $\beta$, but few results about the shape of that posterior distribution. We propose a prior distribution for $b$ such that the marginal posterior distribution of an individual coordinate $b_i$ is asymptotically normal centered around an asymptotically efficient estimator, under the truth. Such a result gives Bayesian credible intervals that match with the confidence intervals obtained from an asymptotically efficient estimator for $b_i$. We also discuss ways of obtaining such asymptotically efficient estimators on individual coordinates. We compare the two-step procedure proposed by Zhang and Zhang (2014) and a one-step modified penalization method.

We were trying to understand the analysis provided by Kneip (1994, Ordered Linear Smoothers). In particular we wanted to persuade ourselves that his results imply the oracle inequality stated by Tsybakov (2014, Lecture 8). This note contains our reworking of Kneip's ideas., Comment: Oracle Inequalities, Mallows' Cp, Machine Learning, Linear Smoothers, Empirical Processes