We provide statistical and computational analysis of sparse Principal Component Analysis (PCA) in high dimensions. The sparse PCA problem is highly nonconvex in nature. Consequently, though its global solution attains the optimal statistical rate of convergence, such solution is computationally intractable to obtain. Meanwhile, although its convex relaxations are tractable to compute, they yield estimators with suboptimal statistical rates of convergence.

Information extraction (IE) systems aim to automatically extract structured information, such as named entities, relations between entities, and events, from unstructured texts. While most existing work addresses a particular IE task, universally modeling various IE tasks with one model has achieved great success recently. Despite their success, they employ a one-stage learning strategy, i.e., directly learning to extract the target structure given the input text, which contradicts the human learning process. In this paper, we propose a unified easy-to-hard learning framework consisting of three stages, i.e., the easy stage, the hard stage, and the main stage, for IE by mimicking the human learning process. By breaking down the learning process into multiple stages, our framework facilitates the model to acquire general IE task knowledge and improve its generalization ability. Extensive experiments across four IE tasks demonstrate the effectiveness of our framework. We achieve new state-of-the-art results on 13 out of 17 datasets. Our code is available at \url{https://github.com/DAMO-NLP-SG/IE-E2H}.

We provide statistical and computational analysis of sparse Principal Component Analysis (PCA) in high dimensions. The sparse PCA problem is highly nonconvex in nature. Consequently, though its global solution attains the optimal statistical rate of convergence, such solution is computationally intractable to obtain. Meanwhile, although its convex relaxations are tractable to compute, they yield estimators with suboptimal statistical rates of convergence. In this paper, we propose a two-stage sparse PCA procedure that attains the optimal principal subspace estimator in polynomial time. The main stage employs a novel algorithm named sparse orthogonal iteration pursuit, which iteratively solves the underlying nonconvex problem.

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We provide statistical and computational analysis of sparse Principal Component Analysis (PCA) in high dimensions. The sparse PCA problem is highly nonconvex in nature. Consequently, though its global solution attains the optimal statistical rate of convergence, such solution is computationally intractable to obtain. Meanwhile, although its convex relaxations are tractable to compute, they yield estimators with suboptimal statistical rates of convergence. On the other hand, existing nonconvex optimization procedures, such as greedy methods, lack statistical guarantees. In this paper, we propose a two-stage sparse PCA procedure that attains the optimal principal subspace estimator in polynomial time. The main stage employs a novel algorithm named sparse orthogonal iteration pursuit, which iteratively solves the underlying nonconvex problem. However, our analysis shows that this algorithm only has desired computational and statistical guarantees within a restricted region, namely the basin of attraction. To obtain the desired initial estimator that falls into this region, we solve a convex formulation of sparse PCA with early stopping. Under an integrated analytic framework, we simultaneously characterize the computational and statistical performance of this two-stage procedure. Computationally, our procedure converges at the rate of $1/\sqrt{t}$ within the initialization stage, and at a geometric rate within the main stage. Statistically, the final principal subspace estimator achieves the minimax-optimal statistical rate of convergence with respect to the sparsity level $s^*$, dimension $d$ and sample size $n$. Our procedure motivates a general paradigm of tackling nonconvex statistical learning problems with provable statistical guarantees.