Algorithms for Sparsity-Constrained Optimization (Springer by Sohail Bahmani

By Sohail Bahmani

This thesis demonstrates ideas that offer swifter and extra exact strategies to a number of difficulties in computing device studying and sign processing. the writer proposes a "greedy" set of rules, deriving sparse ideas with promises of optimality. using this set of rules eliminates some of the inaccuracies that happened with using earlier versions.

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As stated before, for more general cost functions such as the loss functions in GLMs, however, these constants will depend on the location of the true optimum. Consequently, depending on the location of the true optimum these error bounds could be extremely large, albeit finite. When error bounds are significantly large, the sparsity of the solution obtained by `1 -regularization may not be satisfactory. This motivates investigation of algorithms that do not rely on `1 -norm to induce sparsity.

Sridharan, and A. Tewari. Learning exponential families in highdimensions: Strong convexity and sparsity. In Proceedings of the 13th International Conference on Artificial Intelligence and Statistics, volume 9 of JMLR Workshop and Conference Proceedings, pages 381–388, Sardinia, Italy, 2010. J. Liu, J. Chen, and J. Ye. Large-scale sparse logistic regression. In Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’09, pages 547–556, New York, NY, USA, 2009.

JI k2 has statistical interpretation and can determine the statistical precision of the problem. This property is easy to verify in linear regression problems. We will also show this for the logistic loss as an example in Sect. 4. Nonsmooth cost functions should be treated differently, since we do not have the luxury of working with Hessian matrices for these type of functions. 1 for nonsmooth cost functions that satisfy the SRL condition. 2. Suppose that f is a pfunction that is not necessarily smooth, but it satisfies 4s -SRL with 4s Ä 3C4 3 .

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