Accelerated Bregman proximal gradient methods for relatively smooth convex optimization

Filip Hanzely (), Peter Richtárik () and Lin Xiao ()
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Filip Hanzely: King Abdullah University of Science and Technology (KAUST)
Peter Richtárik: King Abdullah University of Science and Technology (KAUST)
Lin Xiao: Microsoft Research

Computational Optimization and Applications, 2021, vol. 79, issue 2, No 6, 405-440

Abstract: Abstract 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 $$O(k^{-\gamma })$$ O ( k - γ ) convergence rate, where $$\gamma \in (0,2]$$ γ ∈ ( 0 , 2 ] is the triangle scaling exponent (TSE) of the Bregman distance. For the Euclidean distance, we have $$\gamma =2$$ 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 $$\gamma \le 1$$ 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 $$O(k^{-2})$$ O ( k - 2 ) rates in numerical experiments on several applications and provide posterior numerical certificates for the fast rates.

Keywords: Convex optimization; Relative smoothness; Bregman divergence; Proximal gradient methods; Accelerated gradient methods (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (4)

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DOI: 10.1007/s10589-021-00273-8

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