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A Descent Lemma Beyond Lipschitz Gradient Continuity: First-Order Methods Revisited and Applications

Author

Listed:
  • Heinz H. Bauschke

    (Mathematics, University of British Columbia, Kelowna, British Columbia V1V 1V7, Canada)

  • Jérôme Bolte

    (Toulouse School of Economics, Université Toulouse 1 Capitole, 31015 Toulouse, France)

  • Marc Teboulle

    (School of Mathematical Sciences, Tel Aviv University, Ramat Aviv 69978, Israel)

Abstract

The proximal gradient and its variants is one of the most attractive first-order algorithm for minimizing the sum of two convex functions, with one being nonsmooth. However, it requires the differentiable part of the objective to have a Lipschitz continuous gradient, thus precluding its use in many applications. In this paper we introduce a framework which allows to circumvent the intricate question of Lipschitz continuity of gradients by using an elegant and easy to check convexity condition which captures the geometry of the constraints. This condition translates into a new descent lemma which in turn leads to a natural derivation of the proximal-gradient scheme with Bregman distances. We then identify a new notion of asymmetry measure for Bregman distances, which is central in determining the relevant step-size. These novelties allow to prove a global sublinear rate of convergence, and as a by-product, global pointwise convergence is obtained. This provides a new path to a broad spectrum of problems arising in key applications which were, until now, considered as out of reach via proximal gradient methods. We illustrate this potential by showing how our results can be applied to build new and simple schemes for Poisson inverse problems.

Suggested Citation

  • Heinz H. Bauschke & Jérôme Bolte & Marc Teboulle, 2017. "A Descent Lemma Beyond Lipschitz Gradient Continuity: First-Order Methods Revisited and Applications," Mathematics of Operations Research, INFORMS, vol. 42(2), pages 330-348, May.
    • Handle: RePEc:inm:ormoor:v:42:y:2017:i:2:p:330-348
    DOI: 10.1287/moor.2016.0817
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    References listed on IDEAS

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    1. Marc Teboulle, 1992. "Entropic Proximal Mappings with Applications to Nonlinear Programming," Mathematics of Operations Research, INFORMS, vol. 17(3), pages 670-690, August.
    2. Jonathan Eckstein, 1993. "Nonlinear Proximal Point Algorithms Using Bregman Functions, with Applications to Convex Programming," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 202-226, February.
    3. Unknown, 2005. "Forward," 2005 Conference: Slovenia in the EU - Challenges for Agriculture, Food Science and Rural Affairs, November 10-11, 2005, Moravske Toplice, Slovenia 183804, Slovenian Association of Agricultural Economists (DAES).
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