Abstract
The purpose of this manuscript is to derive new convergence results for several subgradient methods applied to minimizing nonsmooth convex functions with Hölderian growth. The growth condition is satisfied in many applications and includes functions with quadratic growth and weakly sharp minima as special cases. To this end there are three main contributions. First, for a constant and sufficiently small stepsize, we show that the subgradient method achieves linear convergence up to a certain region including the optimal set, with error of the order of the stepsize. Second, if appropriate problem parameters are known, we derive a decaying stepsize which obtains a much faster convergence rate than is suggested by the classical O(1/k) result for the subgradient method. Thirdly we develop a novel “descending stairs” stepsize which obtains this faster convergence rate and also obtains linear convergence for the special case of weakly sharp functions. We also develop an adaptive variant of the “descending stairs” stepsize which achieves the same convergence rate without requiring an error bound constant which is difficult to estimate in practice.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 417-450 |
| Number of pages | 34 |
| Journal | Mathematical Programming |
| Volume | 180 |
| Issue number | 1-2 |
| DOIs | |
| State | Accepted/In press - Jan 1 2019 |
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ASJC Scopus subject areas
- Software
- Mathematics(all)
Cite this
Faster subgradient methods for functions with Hölderian growth. / Johnstone, Patrick R.; Moulin, Pierre.
In: Mathematical Programming, Vol. 180, No. 1-2, 01.03.2020, p. 417-450.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Faster subgradient methods for functions with Hölderian growth
AU - Johnstone, Patrick R.
AU - Moulin, Pierre
PY - 2019/1/1
Y1 - 2019/1/1
N2 - The purpose of this manuscript is to derive new convergence results for several subgradient methods applied to minimizing nonsmooth convex functions with Hölderian growth. The growth condition is satisfied in many applications and includes functions with quadratic growth and weakly sharp minima as special cases. To this end there are three main contributions. First, for a constant and sufficiently small stepsize, we show that the subgradient method achieves linear convergence up to a certain region including the optimal set, with error of the order of the stepsize. Second, if appropriate problem parameters are known, we derive a decaying stepsize which obtains a much faster convergence rate than is suggested by the classical O(1/k) result for the subgradient method. Thirdly we develop a novel “descending stairs” stepsize which obtains this faster convergence rate and also obtains linear convergence for the special case of weakly sharp functions. We also develop an adaptive variant of the “descending stairs” stepsize which achieves the same convergence rate without requiring an error bound constant which is difficult to estimate in practice.
AB - The purpose of this manuscript is to derive new convergence results for several subgradient methods applied to minimizing nonsmooth convex functions with Hölderian growth. The growth condition is satisfied in many applications and includes functions with quadratic growth and weakly sharp minima as special cases. To this end there are three main contributions. First, for a constant and sufficiently small stepsize, we show that the subgradient method achieves linear convergence up to a certain region including the optimal set, with error of the order of the stepsize. Second, if appropriate problem parameters are known, we derive a decaying stepsize which obtains a much faster convergence rate than is suggested by the classical O(1/k) result for the subgradient method. Thirdly we develop a novel “descending stairs” stepsize which obtains this faster convergence rate and also obtains linear convergence for the special case of weakly sharp functions. We also develop an adaptive variant of the “descending stairs” stepsize which achieves the same convergence rate without requiring an error bound constant which is difficult to estimate in practice.
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UR - http://www.scopus.com/inward/citedby.url?scp=85059742309&partnerID=8YFLogxK
U2 - 10.1007/s10107-018-01361-0
DO - 10.1007/s10107-018-01361-0
M3 - Article
AN - SCOPUS:85059742309
VL - 180
SP - 417
EP - 450
JO - Mathematical Programming
JF - Mathematical Programming
SN - 0025-5610
IS - 1-2
ER -