Conference proceeding
A Richer Theory of Convex Constrained Optimization with Reduced Projections and Improved Rates
Proceedings of Machine Learning Research, Vol.70, pp.3901-3910
International Conference on Machine Learning, 34th
2017
Abstract
This paper focuses on convex constrained optimization problems, where the solution is subject to a convex inequality constraint. In particular, we aim at challenging problems for which both projection into the constrained domain and a linear optimization under the inequality constraint are time-consuming, which render both projected gradient methods and conditional gradient methods (a.k.a. the Frank-Wolfe algorithm) expensive. In this paper, we develop projection reduced optimization algorithms for both smooth and non-smooth optimization with improved convergence rates under a certain regularity condition of the constraint function. We first present a general theory of optimization with only one projection. Its application to smooth optimization with only one projection yields O(1/ϵ) iteration complexity, which improves over the O(1/ϵ2) iteration complexity established before for non-smooth optimization and can be further reduced under strong convexity. Then we introduce a local error bound condition and develop faster algorithms for non-strongly convex optimization at the price of a logarithmic number of projections. In particular, we achieve an iteration complexity of O˜(1/ϵ2(1−θ)) for non-smooth optimization and O˜(1/ϵ1−θ) for smooth optimization, where θ∈(0,1] appearing the local error bound condition characterizes the functional local growth rate around the optimal solutions. Novel applications in solving the constrained ℓ1 minimization problem and a positive semi-definite constrained distance metric learning problem demonstrate that the proposed algorithms achieve significant speed-up compared with previous algorithms.
Details
- Title: Subtitle
- A Richer Theory of Convex Constrained Optimization with Reduced Projections and Improved Rates
- Creators
- Tianbao YangQihang LinLijun Zhang
- Resource Type
- Conference proceeding
- Publication Details
- Proceedings of Machine Learning Research, Vol.70, pp.3901-3910
- Conference
- International Conference on Machine Learning, 34th
- Language
- English
- Date published
- 2017
- Academic Unit
- Business Analytics; Computer Science
- Record Identifier
- 9984259501502771
Metrics
13 Record Views