Blended Conditional Gradients: the unconditioning of conditional gradients
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| Main Authors: | , , , |
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| Format: | Preprint |
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2018
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| _version_ | 1866912284315484160 |
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| author | Braun, Gábor Pokutta, Sebastian Tu, Dan Wright, Stephen |
| author_facet | Braun, Gábor Pokutta, Sebastian Tu, Dan Wright, Stephen |
| contents | We present a blended conditional gradient approach for minimizing a smooth convex function over a polytope P, combining the Frank--Wolfe algorithm (also called conditional gradient) with gradient-based steps, different from away steps and pairwise steps, but still achieving linear convergence for strongly convex functions, along with good practical performance. Our approach retains all favorable properties of conditional gradient algorithms, notably avoidance of projections onto P and maintenance of iterates as sparse convex combinations of a limited number of extreme points of P. The algorithm is lazy, making use of inexpensive inexact solutions of the linear programming subproblem that characterizes the conditional gradient approach. It decreases measures of optimality (primal and dual gaps) rapidly, both in the number of iterations and in wall-clock time, outperforming even the lazy conditional gradient algorithms of [arXiv:1410.8816]. We also present a streamlined version of the algorithm for the probability simplex. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1805_07311 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Blended Conditional Gradients: the unconditioning of conditional gradients Braun, Gábor Pokutta, Sebastian Tu, Dan Wright, Stephen Optimization and Control Computational Complexity Machine Learning 68Q32, 90C52 We present a blended conditional gradient approach for minimizing a smooth convex function over a polytope P, combining the Frank--Wolfe algorithm (also called conditional gradient) with gradient-based steps, different from away steps and pairwise steps, but still achieving linear convergence for strongly convex functions, along with good practical performance. Our approach retains all favorable properties of conditional gradient algorithms, notably avoidance of projections onto P and maintenance of iterates as sparse convex combinations of a limited number of extreme points of P. The algorithm is lazy, making use of inexpensive inexact solutions of the linear programming subproblem that characterizes the conditional gradient approach. It decreases measures of optimality (primal and dual gaps) rapidly, both in the number of iterations and in wall-clock time, outperforming even the lazy conditional gradient algorithms of [arXiv:1410.8816]. We also present a streamlined version of the algorithm for the probability simplex. |
| title | Blended Conditional Gradients: the unconditioning of conditional gradients |
| topic | Optimization and Control Computational Complexity Machine Learning 68Q32, 90C52 |
| url | https://arxiv.org/abs/1805.07311 |