Convergence Rates in Forward-Backward Splitting

By George H.-G. Chen, R. Tyrrell Rockafellar. 1997

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In Optimization & Training

Citations in the corpus

Stochastic Gradient Descent-Ascent and Consensus Optimization for Smooth Games: Convergence Analysis under Expected Co-coercivity (2021, methodology): “1 is the first convergence theorem of SGDA that includes the deterministic GDA originally provided by Chen and Rockafellar [1997] as a special case.”
A Variational Inequality Perspective on Generative Adversarial Networks (2018, background): “These iterates are known to converge linearly under an additional assumption on the operator 4 (Chen and Rockafellar, 1997), but oscillate for a bilinear operator as shown in Fig.”
A Proximal Stochastic Gradient Method with Progressive Variance Reduction (2014, methodology): “This method can be viewed as a special case of splitting algorithms [20, 7, 32], and its accelerated variants have been proposed and analyzed in [1, 25].”
Optimization with Sparsity-Inducing Penalties (2011, background): “As of today, they have been extended to various settings [36, 39, 38, 139].”
Stochastic Variance Reduction Methods for Saddle-Point Problems (2016, positive): “This algorithm is known not to converge in general [8], but is linearly convergent for strongly-convex-concave problems, when σ = 1/L2, with the rate (1− 1/(1 + L2))t [19] (see simple proof in Appendix B.”
Fixed-Point Continuation for l1-Minimization: Methodology and Convergence (2008, background): “General convergence theory exists for forward-backward splitting methods [8, 30, 45].”
A Variational Inequality Perspective on Generative Adversarial Nets (2018, methodology): “The iterates of the gradient method are known to converge linearly when the operator is strongly monotone3 (Chen and Rockafellar, 1997), but oscillates for a bilinear operator as showed in Fig.”
On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems (2019, background): “…concave in y , there is a vast literature establishing asymptotic and nonasymptotic convergence for the average iterates generated by GDA with the equal stepsizes ( η x = η y ); [see, e.g., Korpelevich, 1976, Chen and Rockafellar, 1997, Nedi´c and Ozdaglar, 2009, Nemirovski, 2004, Du and Hu, 2018].”
Convergence of a splitting inertial proximal method for monotone operators (2003, methodology): “[20] R.T. Rockafellar Monotone operator and the proximal point algorithm,SIAM J. Control.”
On Auxiliary Principle for Equilibrium Problems (2003, methodology): “Many are the possible developments of the analysis: t he connections with the proximal methods for equilibrium problems [12]; the applications to decomposition algorithms [3, 6, 2]; the extensions to vector equilibrium problems (see e.”
Proximal Algorithms (2013, unknown): “Important papers on forward-backward splitting include those by Passty [159], Lions and Mercier [129], Fukushima and Mine [88], Gabay [90], Lemaire [120], Eckstein [78], Chen [54], Chen and Rockafellar [55], Tseng [184, 185, 187], Combettes and Wajs [62], and Beck and Teboulle [17, 18].”
Proximal Splitting Methods in Signal Processing (2009, methodology): “thm ﬁnds its roots in the projected gradient method [91] and in decomposition methods for solving variational inequalities [96, 112]. More recent forms of the algorithm and reﬁnements can be found in [20, 35, 45, 82, 122]. Let us note that, on the one hand, when f 1 = 0, (11) reduces to the gradient”
A Generalized Forward-Backward Splitting (2011, unknown): “This algorithm, which finds its roots in numerical analysis for PDEs, has been well studied for solving monotone inclusion and convex optimization problems [9, 19, 27, 45, 57, 62, 73, 75].”