# American Institute of Mathematical Sciences

doi: 10.3934/ipi.2020047

## Some worst-case datasets of deterministic first-order methods for solving binary logistic regression

 School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC 29634, USA

* Corresponding author: Yuyuan Ouyang

Received  December 2019 Revised  April 2020 Published  August 2020

Fund Project: The authors are supported by National Science Foundation grant DMS-1913006 and Office of Naval Research grant N00014-19-1-2295

We present in this paper some worst-case datasets of deterministic first-order methods for solving large-scale binary logistic regression problems. Under the assumption that the number of algorithm iterations is much smaller than the problem dimension, with our worst-case datasets it requires at least ${{{\mathcal O}}}(1/\sqrt{\varepsilon})$ first-order oracle inquiries to compute an $\varepsilon$-approximate solution. From traditional iteration complexity analysis point of view, the binary logistic regression loss functions with our worst-case datasets are new worst-case function instances among the class of smooth convex optimization problems.

Citation: Yuyuan Ouyang, Trevor Squires. Some worst-case datasets of deterministic first-order methods for solving binary logistic regression. Inverse Problems & Imaging, doi: 10.3934/ipi.2020047
##### References:
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##### References:
 [1] F. Bach, Self-concordant analysis for logistic regression, Electronic Journal of Statistics, 4 (2010), 384-414.  doi: 10.1214/09-EJS521.  Google Scholar [2] Y. Carmon, J. C Duchi, O. Hinder and A. Sidford, Lower bounds for finding stationary points Ⅰ, Mathematical Programming, 2019, 1-50. doi: 10.1007/s10107-019-01406-y.  Google Scholar [3] Y. Carmon, J. C Duchi, O. Hinder and A. Sidford, Lower bounds for finding stationary points Ⅱ: First-order methods, Mathematical Programming, 2019, 1-41. doi: 10.1007/s10107-019-01431-x.  Google Scholar [4] J. Diakonikolas and C. Guzmán, Lower bounds for parallel and randomized convex optimization, In Conference on Learning Theory, 2019, 1132-1157. Google Scholar [5] Y. Drori, The exact information-based complexity of smooth convex minimization, Journal of Complexity, 39 (2017), 1-16.  doi: 10.1016/j.jco.2016.11.001.  Google Scholar [6] C. Guzmán and A. Nemirovski, On lower complexity bounds for large-scale smooth convex optimization, Journal of Complexity, 31 (2015), 1-14.  doi: 10.1016/j.jco.2014.08.003.  Google Scholar [7] A. Juditsky and Y. Nesterov, Deterministic and stochastic primal-dual subgradient algorithms for uniformly convex minimization, Stochastic Systems, 4 (2014), 44-80.  doi: 10.1214/10-SSY010.  Google Scholar [8] A. Nemirovski and D. Yudin, Problem Complexity and Method Efficiency in Optimization, Wiley-Interscience Series in Discrete Mathematics. John Wiley, XV, 1983.  Google Scholar [9] A. S. Nemirovski, Information-based complexity of linear operator equations, Journal of Complexity, 8 (1992), 153-175.  doi: 10.1016/0885-064X(92)90013-2.  Google Scholar [10] Y. E. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Kluwer Academic Publishers, Massachusetts, 2004. doi: 10.1007/978-1-4419-8853-9.  Google Scholar [11] Y. Ouyang and Y. Xu, Lower complexity bounds of first-order methods for convex-concave bilinear saddle-point problems, Mathematical Programming, 2019, 1-35. doi: 10.1007/s10107-019-01420-0.  Google Scholar [12] B. E Woodworth and N. Srebro, Tight complexity bounds for optimizing composite objectives, In Advances in Neural Information Processing Systems, 2016, 3639-3647. Google Scholar
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