Abstract
Decentralized optimization algorithms have received much attention due to the recent advances in network information processing. However, conventional decentralized algorithms based on projected gradient descent are incapable of handling high-dimensional constrained problems, as the projection step becomes computationally prohibitive. To address this problem, this paper adopts a projection-free optimization approach, a.k.a. the Frank-Wolfe (FW) or conditional gradient algorithm. We first develop a decentralized FW (DeFW) algorithm from the classical FW algorithm. The convergence of the proposed algorithm is studied by viewing the decentralized algorithm as an inexact FW algorithm. Using a diminishing step size rule and letting t be the iteration number, we show that the DeFW algorithm's convergence rate is O(1/t) for convex objectives; is O(1/t2) for strongly convex objectives with the optimal solution in the interior of the constraint set; and is O(1√t) toward a stationary point for smooth but nonconvex objectives. We then show that a consensus-based DeFW algorithm meets the above guarantees with two communication rounds per iteration. We demonstrate the advantages of the proposed DeFW algorithm on low-complexity robust matrix completion and communication efficient sparse learning. Numerical results on synthetic and real data are presented to support our findings.
| Original language | English (US) |
|---|---|
| Article number | 7883821 |
| Pages (from-to) | 5522-5537 |
| Number of pages | 16 |
| Journal | IEEE Transactions on Automatic Control |
| Volume | 62 |
| Issue number | 11 |
| DOIs | |
| State | Published - Nov 1 2017 |
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Keywords
- Communication efficient algorithms
- consensus algorithms
- decentralized optimization
- Frank-Wolfe (FW) algorithm
- high-dimensional optimization
- least absolute shrinkage and selection operator (LASSO)
- matrix completion
ASJC Scopus subject areas
- Control and Systems Engineering
- Computer Science Applications
- Electrical and Electronic Engineering
Cite this
Decentralized Frank-Wolfe Algorithm for Convex and Nonconvex Problems. / Wai, Hoi To; Lafond, Jean; Scaglione, Anna; Moulines, Eric.
In: IEEE Transactions on Automatic Control, Vol. 62, No. 11, 7883821, 01.11.2017, p. 5522-5537.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Decentralized Frank-Wolfe Algorithm for Convex and Nonconvex Problems
AU - Wai, Hoi To
AU - Lafond, Jean
AU - Scaglione, Anna
AU - Moulines, Eric
PY - 2017/11/1
Y1 - 2017/11/1
N2 - Decentralized optimization algorithms have received much attention due to the recent advances in network information processing. However, conventional decentralized algorithms based on projected gradient descent are incapable of handling high-dimensional constrained problems, as the projection step becomes computationally prohibitive. To address this problem, this paper adopts a projection-free optimization approach, a.k.a. the Frank-Wolfe (FW) or conditional gradient algorithm. We first develop a decentralized FW (DeFW) algorithm from the classical FW algorithm. The convergence of the proposed algorithm is studied by viewing the decentralized algorithm as an inexact FW algorithm. Using a diminishing step size rule and letting t be the iteration number, we show that the DeFW algorithm's convergence rate is O(1/t) for convex objectives; is O(1/t2) for strongly convex objectives with the optimal solution in the interior of the constraint set; and is O(1√t) toward a stationary point for smooth but nonconvex objectives. We then show that a consensus-based DeFW algorithm meets the above guarantees with two communication rounds per iteration. We demonstrate the advantages of the proposed DeFW algorithm on low-complexity robust matrix completion and communication efficient sparse learning. Numerical results on synthetic and real data are presented to support our findings.
AB - Decentralized optimization algorithms have received much attention due to the recent advances in network information processing. However, conventional decentralized algorithms based on projected gradient descent are incapable of handling high-dimensional constrained problems, as the projection step becomes computationally prohibitive. To address this problem, this paper adopts a projection-free optimization approach, a.k.a. the Frank-Wolfe (FW) or conditional gradient algorithm. We first develop a decentralized FW (DeFW) algorithm from the classical FW algorithm. The convergence of the proposed algorithm is studied by viewing the decentralized algorithm as an inexact FW algorithm. Using a diminishing step size rule and letting t be the iteration number, we show that the DeFW algorithm's convergence rate is O(1/t) for convex objectives; is O(1/t2) for strongly convex objectives with the optimal solution in the interior of the constraint set; and is O(1√t) toward a stationary point for smooth but nonconvex objectives. We then show that a consensus-based DeFW algorithm meets the above guarantees with two communication rounds per iteration. We demonstrate the advantages of the proposed DeFW algorithm on low-complexity robust matrix completion and communication efficient sparse learning. Numerical results on synthetic and real data are presented to support our findings.
KW - Communication efficient algorithms
KW - consensus algorithms
KW - decentralized optimization
KW - Frank-Wolfe (FW) algorithm
KW - high-dimensional optimization
KW - least absolute shrinkage and selection operator (LASSO)
KW - matrix completion
UR - http://www.scopus.com/inward/record.url?scp=85036465623&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85036465623&partnerID=8YFLogxK
U2 - 10.1109/TAC.2017.2685559
DO - 10.1109/TAC.2017.2685559
M3 - Article
AN - SCOPUS:85036465623
VL - 62
SP - 5522
EP - 5537
JO - IEEE Transactions on Automatic Control
JF - IEEE Transactions on Automatic Control
SN - 0018-9286
IS - 11
M1 - 7883821
ER -