When data is collected in an adaptive manner, even simple methods like ordinary least squares can exhibit non-normal asymptotic behavior. As an undesirable consequence, hypothesis tests and confidence intervals based on asymptotic normality can lead to erroneous results. We propose an online debiasing estimator to correct these distributional anomalies in least squares estimation. Our proposed method takes advantage of the covariance structure present in the dataset and provides sharper estimates in directions for which more information has accrued. We establish an asymptotic normality property for our proposed online debiasing estimator under mild conditions on the data collection process, and provide asymptotically exact confidence intervals. We additionally prove a minimax lower bound for the adaptive linear regression problem, thereby providing a baseline by which to compare estimators. There are various conditions under which our proposed estimator achieves the minimax lower bound up to logarithmic factors. We demonstrate the usefulness of our theory via applications to multi-armed bandit, autoregressive time series estimation, and active learning with exploration.

We argue that the next frontier in natural language understanding (NLU) and generation (NLG) will include models that can efficiently access external structured knowledge repositories. In order to support the development of such models, we release the VisualSem knowledge graph (KG) which includes nodes with multilingual glosses and multiple illustrative images and visually relevant relations. We also release a neural multi-modal retrieval model that can use images or sentences as inputs and retrieves entities in the KG. This multi-modal retrieval model can be integrated into any (neural network) model pipeline and we encourage the research community to use VisualSem for data augmentation and/or as a source of grounding, among other possible uses. VisualSem as well as the multi-modal retrieval model are publicly available and can be downloaded in: https://github.com/iacercalixto/visualsem.

We provide the first information theoretical tight analysis for inference of latent community structure given a sparse graph along with high dimensional node covariates, correlated with the same latent communities. Our work bridges recent theoretical breakthroughs in detection of latent community structure without nodes covariates and a large body of empirical work using diverse heuristics for combining node covariates with graphs for inference. The tightness of our analysis implies in particular, the information theoretic necessity of combining the different sources of information. Our analysis holds for networks of large degrees as well as for a Gaussian version of the model.

We provide the first information theoretical tight analysis for inference of latent community structure given a sparse graph along with high dimensional node covariates, correlated with the same latent communities. Our work bridges recent theoretical breakthroughs in detection of latent community structure without nodes covariates and a large body of empirical work using diverse heuristics for combining node covariates with graphs for inference. The tightness of our analysis implies in particular, the information theoretic necessity of combining the different sources of information. Our analysis holds for networks of large degrees as well as for a Gaussian version of the model.

Maximum A posteriori Probability (MAP) inference in graphical models amounts to solving a graph-structured combinatorial optimization problem. Popular inference algorithms such as belief propagation (BP) and generalized belief propagation (GBP) are intimately related to linear programming (LP) relaxation within the Sherali-Adams hierarchy. Despite the popularity of these algorithms, it is well understood that the Sum-of-Squares (SOS) hierarchy based on semidefinite programming (SDP) can provide superior guarantees. Unfortunately, SOS relaxations for a graph with $n$ vertices require solving an SDP with $n^{\Theta(d)}$ variables where $d$ is the degree in the hierarchy. In practice, for $d\ge 4$, this approach does not scale beyond a few tens of variables. In this paper, we propose binary SDP relaxations for MAP inference using the SOS hierarchy with two innovations focused on computational efficiency. Firstly, in analogy to BP and its variants, we only introduce decision variables corresponding to contiguous regions in the graphical model. Secondly, we solve the resulting SDP using a non-convex Burer-Monteiro style method, and develop a sequential rounding procedure. We demonstrate that the resulting algorithm can solve problems with tens of thousands of variables within minutes, and outperforms BP and GBP on practical problems such as image denoising and Ising spin glasses. Finally, for specific graph types, we establish a sufficient condition for the tightness of the proposed partial SOS relaxation.

Estimators computed from adaptively collected data do not behave like their non-adaptive brethren. Rather, the sequential dependence of the collection policy can lead to severe distributional biases that persist even in the infinite data limit. We develop a general method decorrelation procedure -- W-decorrelation -- for transforming the bias of adaptive linear regression estimators into variance. The method uses only coarse-grained information about the data collection policy and does not need access to propensity scores or exact knowledge of the policy. We bound the finite-sample bias and variance of the W-estimator and develop asymptotically correct confidence intervals based on a novel martingale central limit theorem. We then demonstrate the empirical benefits of the generic W-decorrelation procedure in two different adaptive data settings: the multi-armed bandits and autoregressive time series models.

Maximum A posteriori Probability (MAP) inference in graphical models amounts to solving a graph-structured combinatorial optimization problem. Popular inference algorithms such as belief propagation (BP) and generalized belief propagation (GBP) are intimately related to linear programming (LP) relaxation within the Sherali-Adams hierarchy. Despite the popularity of these algorithms, it is well understood that the Sum-of-Squares (SOS) hierarchy based on semidefinite programming (SDP) can provide superior guarantees. Unfortunately, SOS relaxations for a graph with $n$ vertices require solving an SDP with $n^{\Theta(d)}$ variables where $d$ is the degree in the hierarchy. In practice, for $d\ge 4$, this approach does not scale beyond a few tens of variables. In this paper, we propose binary SDP relaxations for MAP inference using the SOS hierarchy with two innovations focused on computational efficiency. Firstly, in analogy to BP and its variants, we only introduce decision variables corresponding to contiguous regions in the graphical model. Secondly, we solve the resulting SDP using a non-convex Burer-Monteiro style method, and develop a sequential rounding procedure. We demonstrate that the resulting algorithm can solve problems with tens of thousands of variables within minutes, and outperforms BP and GBP on practical problems such as image denoising and Ising spin glasses. Finally, for specific graph types, we establish a sufficient condition for the tightness of the proposed partial SOS relaxation.

Given a large data matrix $A\in\mathbb{R}^{n\times n}$, we consider the problem of determining whether its entries are i.i.d. with some known marginal distribution $A_{ij}\sim P_0$, or instead $A$ contains a principal submatrix $A_{{\sf Q},{\sf Q}}$ whose entries have marginal distribution $A_{ij}\sim P_1\neq P_0$. As a special case, the hidden (or planted) clique problem requires to find a planted clique in an otherwise uniformly random graph. Assuming unbounded computational resources, this hypothesis testing problem is statistically solvable provided $|{\sf Q}|\ge C \log n$ for a suitable constant $C$. However, despite substantial effort, no polynomial time algorithm is known that succeeds with high probability when $|{\sf Q}| = o(\sqrt{n})$. Recently Meka and Wigderson \cite{meka2013association}, proposed a method to establish lower bounds within the Sum of Squares (SOS) semidefinite hierarchy. Here we consider the degree-$4$ SOS relaxation, and study the construction of \cite{meka2013association} to prove that SOS fails unless $k\ge C\, n^{1/3}/\log n$. An argument presented by Barak implies that this lower bound cannot be substantially improved unless the witness construction is changed in the proof. Our proof uses the moments method to bound the spectrum of a certain random association scheme, i.e. a symmetric random matrix whose rows and columns are indexed by the edges of an Erd\"os-Renyi random graph.

In sparse principal component analysis we are given noisy observations of a low-rank matrix of dimension $n\times p$ and seek to reconstruct it under additional sparsity assumptions. In particular, we assume here that the principal components $\bv_1,\dots,\bv_r$ have at most $k_1, \cdots, k_q$ non-zero entries respectively, and study the high-dimensional regime in which $p$ is of the same order as $n$. In an influential paper, Johnstone and Lu \cite{johnstone2004sparse} introduced a simple algorithm that estimates the support of the principal vectors $\bv_1,\dots,\bv_r$ by the largest entries in the diagonal of the empirical covariance. This method can be shown to succeed with high probability if $k_q \le C_1\sqrt{n/\log p}$, and to fail with high probability if $k_q\ge C_2 \sqrt{n/\log p}$ for two constants $0 < C_1,C_2 < \infty$. Despite a considerable amount of work over the last ten years, no practical algorithm exists with provably better support recovery guarantees. Here we analyze a covariance thresholding algorithm that was recently proposed by Krauthgamer, Nadler and Vilenchik \cite{KrauthgamerSPCA}. We confirm empirical evidence presented by these authors and rigorously prove that the algorithm succeeds with high probability for $k$ of order $\sqrt{n}$. Recent conditional lower bounds \cite{berthet2013computational} suggest that it might be impossible to do significantly better. The key technical component of our analysis develops new bounds on the norm of kernel random matrices, in regimes that were not considered before.

Estimating a vector from noisy quadratic observations is a task that arises naturally in many contexts, from dimensionality reduction, to synchronization and phase retrieval problems. It is often the case that additional information is available about the unknown vector (for instance, sparsity, sign or magnitude of its entries). Many authors propose non-convex quadratic optimization problems that aim at exploiting optimally this information. However, solving these problems is typically NP-hard. We consider a simple model for noisy quadratic observation of an unknown vector $\bvz$. The unknown vector is constrained to belong to a cone $\Cone \ni \bvz$. While optimal estimation appears to be intractable for the general problems in this class, we provide evidence that it is tractable when $\Cone$ is a convex cone with an efficient projection. This is surprising, since the corresponding optimization problem is non-convex and --from a worst case perspective-- often NP hard. We characterize the resulting minimax risk in terms of the statistical dimension of the cone $\delta(\Cone)$. This quantity is already known to control the risk of estimation from gaussian observations and random linear measurements. It is rather surprising that the same quantity plays a role in the estimation risk from quadratic measurements.