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The Rate of Convergence of Nesterov's Accelerated Forward-Backward Method is Actually Faster Than $1/k^2$
Open accessAbstract
The forward-backward algorithm is a powerful tool for solving optimization problems with an additively separable and smooth plus nonsmooth structure. In the convex setting, a simple but ingenious acceleration scheme developed by Nesterov improves the theoretical rate of convergence for the function values from the standard $\mathcal O(k^{-1})$ down to $\mathcal O(k^{-2})$. In this short paper, we prove that the rate of convergence of a slight variant of Nesterov's accelerated forward-backward method, which produces convergent sequences, is actually $o(k^{-2})$, rather than $\mathcal O(k^{-2})$. Our arguments rely on the connection between this algorithm and a second-order differential inclusion with vanishing damping.
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Cites
A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems
Beck, A., & Teboulle, M. (2009) · SIAM Journal on Imaging Sciences · doi:10.1137/080716542 · Cited by 11,952
An iterative thresholding algorithm for linear inverse problems with a sparsity constraint.
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Introductory Lectures on Convex Optimization: A Basic Course
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Convex Analysis and Monotone Operator Theory in Hilbert Spaces
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Convex Analysis and Monotone Operator Theory in Hilbert Spaces
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Signal Recovery by Proximal Forward-Backward Splitting
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A method for solving the convex programming problem with convergence rate O(1/k^2)
Nesterov, Y. E. (1983) · Доклады Академии наук · Cited by 2,662
Smooth minimization of non-smooth functions
Nesterov, Y. (2005) · Mathematical Programming · doi:10.1007/s10107-004-0552-5 · Cited by 2,455
Weak convergence of the sequence of successive approximations for nonexpansive mappings
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Metadata
- Authors
- Attouch, H., & Peypouquet, J.
- Venue
- SIAM Journal on Optimization
- Published
- 2016-01
- Volume
- 26
- Issue
- 3
- Pages
- 1824-1834
- ISSN
- 1052-6234 (ISSN-L)
- 1095-7189
- OpenAlex
- W1837565340
- Cited by
- 273
- References
- 20
- FWCI
- 20.84
- Type
- journal article
- Language
- en
- Open access
- green
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- Attouch, H., & Peypouquet, J. (2016). The Rate of Convergence of Nesterov's Accelerated Forward-Backward Method is Actually Faster Than $1/k^2$. SIAM Journal on Optimization. https://doi.org/10.1137/15m1046095
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