Improving Negative Sampling for Word Representation using Self-embedded Features

Although the word-popularity based negative sampler has shown superb performance in the skip-gram model, the theoretical motivation behind oversampling popular (non-observed) words as negative samples is still not well understood. In this paper, we start from an investigation of the gradient vanishing issue in the skipgram model without a proper negative sampler. By performing an insightful analysis from the stochastic gradient descent (SGD) learning perspective, we demonstrate that, both theoretically and intuitively, negative samples with larger inner product scores are more informative than those with lower scores for the SGD learner in terms of both convergence rate and accuracy. Understanding this, we propose an alternative sampling algorithm that dynamically selects informative negative samples during each SGD update. More importantly, the proposed sampler accounts for multi-dimensional self-embedded features during the sampling process, which essentially makes it more effective than the original popularity-based (one-dimensional) sampler. Empirical experiments further verify our observations, and show that our fine-grained samplers gain significant improvement over the existing ones without increasing computational complexity.Comment: Accepted in WSDM 201

Adaptive high-order splitting schemes for large-scale differential Riccati equations

We consider high-order splitting schemes for large-scale differential Riccati equations. Such equations arise in many different areas and are especially important within the field of optimal control. In the large-scale case, it is critical to employ structural properties of the matrix-valued solution, or the computational cost and storage requirements become infeasible. Our main contribution is therefore to formulate these high-order splitting schemes in a efficient way by utilizing a low-rank factorization. Previous results indicated that this was impossible for methods of order higher than 2, but our new approach overcomes these difficulties. In addition, we demonstrate that the proposed methods contain natural embedded error estimates. These may be used e.g. for time step adaptivity, and our numerical experiments in this direction show promising results.Comment: 23 pages, 7 figure