Relative to the large literature on upper bounds on complexity of convex optimization, lesser attention has been paid to the fundamental hardness of these problems. Given the extensive use of convex optimization in machine learning and statistics, gaining an understanding of these complexity-theoretic issues is important. In this paper, we study the complexity of stochastic convex optimization in an oracle model of computation. We improve upon known results and obtain tight minimax complexity estimates for various function classes.

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While the celebrated Word2Vec technique yields semantically rich representations for individual words, there has been relatively less success in extending to generate unsupervised sentences or documents embeddings. Recent work has demonstrated that a distance measure between documents called Word Mover's Distance (WMD) that aligns semantically similar words, yields unprecedented KNN classification accuracy. However, WMD is expensive to compute, and it is hard to extend its use beyond a KNN classifier. In this paper, we propose the Word Mover's Embedding (WME), a novel approach to building an unsupervised document (sentence) embedding from pre-trained word embeddings. In our experiments on 9 benchmark text classification datasets and 22 textual similarity tasks, the proposed technique consistently matches or outperforms state-of-the-art techniques, with significantly higher accuracy on problems of short length.

Quadratic program relaxations are proposed as an alternative to linear program relaxations and tree reweighted belief propagation for the metric labeling or MAP estimation problem. An additional convex relaxation of the quadratic approximation is shown to have additive approximation guarantees that apply even when the graph weights have mixed sign or do not come from a metric. The approximations are extended in a manner that allows tight variational relaxations of the MAP problem, although they generally involve non-convex optimization. Experiments carried out on synthetic data show that the quadratic approximations can be more accurate and computationally efficient than the linear programming and propagation based alternatives.