Complexity bounds for primal-dual methods minimizing the model of objective function
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We provide Frank–Wolfe (\(\equiv \) Conditional Gradients) method with a convergence analysis allowing to approach a primal-dual solution of convex optimization problem with composite objective function. Additional properties of complementary part of the objective (strong convexity) significantly accelerate the scheme. We also justify a new variant of this method, which can be seen as a trust-region scheme applying to the linear model of objective function. For this variant, we prove also the rate of convergence for the total variation of linear model of composite objective over the feasible set. Our analysis works also for quadratic model, allowing to justify the global rate of convergence for a new second-order method as applied to a composite objective function. To the best of our knowledge, this is the first trust-region scheme supported by the worst-case complexity analysis both for the functional gap and for the variation of local quadratic model over the feasible set.
KeywordsConvex optimization Complexity bounds Linear optimization oracle Conditional gradient method Trust-region method
Mathematics Subject Classification90-08 90-C25 90-C52
The author is very thankful to both anonymous reviewers for their valuable comments.
- 5.Garber, D., Hazan, E.: A linearly convergent conditional gradient algorithm with application to online and stochastic optimization (2013). arXiv:1301.4666v5
- 6.Garber, D., Hazan, E.: Faster rates for the Frank–Wolfe method over strongly-convex sets (2015). arXiv:1406.1305v2
- 8.Jaggi, M.: Revisiting Frank–Wolfe: projection-free sparse convex optimization. In: Proceedings of the 30th International Conference on Machine Learning, Atlanta, Georgia (2013)Google Scholar
- 9.Juditsky, A., Nemirovski, A.: Solving variational inequalities with monotone operators on domains given by linear minimization oracles. Math. Program. Ser. A 156, 221–256 (2015). doi: 10.1007/s10107-015-0876-3
- 10.Lacoste-Julien, S., Jaggi, M., Schmidt, M., Pletscher, P.: Block-coordinate Frank–Wolfe optimization of structural svms. In: Proceedings of the 30th International Conference on Machine Learning, Atlanta, Georgia (2013)Google Scholar
- 11.Lan, G.: The complexity of large-scale convex programming under a linear optimization oracle (2014). arXiv:1309.5550v2
- 20.Yu, Y., Zhang, X., Schuurmans, D.: Generalized conditional gradient for sparse estimation (2014). arXiv:1410.4828 (Under review in Journal of Machine Learning Research)