TY - JOUR

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

AU - Hanzely, Filip

AU - Richtárik, Peter

AU - Xiao, Lin

N1 - 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

SN - 0926-6003

VL - 79

SP - 405

EP - 440

JO - Computational Optimization and Applications

JF - Computational Optimization and Applications

IS - 2

ER -