TY - JOUR

T1 - Accelerated Bregman proximal gradient methods for relatively smooth convex optimization

AU - Hanzely, Filip

AU - Richtárik, Peter

AU - Xiao, Lin

N1 - Funding Information:
We thank Haihao Lu, Robert Freund and Yurii Nesterov for helpful conversations. We are also grateful to the anonymous referees whose comments helped improving the clarity of the paper. Peter Richtárik acknowledges the support of the KAUST Baseline Research Funding Scheme.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2021/6

Y1 - 2021/6

N2 - We consider the problem of minimizing the sum of two convex functions: one is differentiable and relatively smooth with respect to a reference convex function, and the other can be nondifferentiable but simple to optimize. We investigate a triangle scaling property of the Bregman distance generated by the reference convex function and present accelerated Bregman proximal gradient (ABPG) methods that attain an O(k-γ) convergence rate, where γ∈ (0 , 2] is the triangle scaling exponent (TSE) of the Bregman distance. For the Euclidean distance, we have γ= 2 and recover the convergence rate of Nesterov’s accelerated gradient methods. For non-Euclidean Bregman distances, the TSE can be much smaller (say γ≤ 1), but we show that a relaxed definition of intrinsic TSE is always equal to 2. We exploit the intrinsic TSE to develop adaptive ABPG methods that converge much faster in practice. Although theoretical guarantees on a fast convergence rate seem to be out of reach in general, our methods obtain empirical O(k- 2) rates in numerical experiments on several applications and provide posterior numerical certificates for the fast rates.

AB - We consider the problem of minimizing the sum of two convex functions: one is differentiable and relatively smooth with respect to a reference convex function, and the other can be nondifferentiable but simple to optimize. We investigate a triangle scaling property of the Bregman distance generated by the reference convex function and present accelerated Bregman proximal gradient (ABPG) methods that attain an O(k-γ) convergence rate, where γ∈ (0 , 2] is the triangle scaling exponent (TSE) of the Bregman distance. For the Euclidean distance, we have γ= 2 and recover the convergence rate of Nesterov’s accelerated gradient methods. For non-Euclidean Bregman distances, the TSE can be much smaller (say γ≤ 1), but we show that a relaxed definition of intrinsic TSE is always equal to 2. We exploit the intrinsic TSE to develop adaptive ABPG methods that converge much faster in practice. Although theoretical guarantees on a fast convergence rate seem to be out of reach in general, our methods obtain empirical O(k- 2) rates in numerical experiments on several applications and provide posterior numerical certificates for the fast rates.

KW - Accelerated gradient methods

KW - Bregman divergence

KW - Convex optimization

KW - Proximal gradient methods

KW - Relative smoothness

UR - http://www.scopus.com/inward/record.url?scp=85103928357&partnerID=8YFLogxK

U2 - 10.1007/s10589-021-00273-8

DO - 10.1007/s10589-021-00273-8

M3 - Article

AN - SCOPUS:85103928357

VL - 79

SP - 405

EP - 440

JO - Computational Optimization and Applications

JF - Computational Optimization and Applications

SN - 0926-6003

IS - 2

ER -