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Forward--Backward Splitting with Deviations for Monotone Inclusions
Optimization and Control · Submitted 1 Dec 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This approach provides great flexibility and opens up for the design of new and improved forward--backward-based algorithms, while retaining global convergence guarantees.
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The abstract, sentence by sentence
We propose and study a weakly convergent variant of the forward--backward algorithm for solving structured monotone inclusion problems. Our algorithm features a per-iteration deviation vector which provides additional degrees of freedom. The only requirement on the deviation vector to guarantee convergence is that its norm is bounded by a quantity that can be computed online. This approach provides great flexibility and opens up for the design of new and improved forward--backward-based algorithms, while retaining global convergence guarantees. These guarantees include linear convergence of our method under a metric subregularity assumption without the need to adapt the algorithm parameters. Choosing suitable monotone operators allows for incorporating deviations into other algorithms, such as Chambolle--Pock and Krasnoselsky--Mann iterations. We propose a novel inertial primal--dual algorithm by selecting the deviations along a momentum direction and deciding their size using the norm condition. Numerical experiments demonstrate our convergence claims and show that even this simple choice of deviation vector can improve the performance, compared, e.g., to the standard Chambolle--Pock algorithm.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We propose and study a weakly convergent variant of the forward--backward algorithm for solving structured monotone inclusion problems. | contribution 1.00 | 0.94 | 0.23 | asserted 1.00 |
| s1 | Our algorithm features a per-iteration deviation vector which provides additional degrees of freedom. | method 0.80 | 0.88 | 0.06 | asserted 1.00 |
| s2 | The only requirement on the deviation vector to guarantee convergence is that its norm is bounded by a quantity that can be computed online. | result 0.72 | 0.77 | 0.18 | asserted 1.00 |
| s3 | This approach provides great flexibility and opens up for the design of new and improved forward--backward-based algorithms, while retaining global convergence guarantees. | interpretation 0.99 | 0.65 | 0.40 | asserted 0.78 |
| s4 | These guarantees include linear convergence of our method under a metric subregularity assumption without the need to adapt the algorithm parameters. | result 0.98 | 0.90 | 0.22 | asserted 1.00 |
| s5 | Choosing suitable monotone operators allows for incorporating deviations into other algorithms, such as Chambolle--Pock and Krasnoselsky--Mann iterations. | contribution 0.47 | 0.45 | 0.07 | hedged possibility 0.67 |
| s6 | We propose a novel inertial primal--dual algorithm by selecting the deviations along a momentum direction and deciding their size using the norm condition. | contribution 0.64 | 0.94 | 0.12 | asserted 1.00 |
| s7 | Numerical experiments demonstrate our convergence claims and show that even this simple choice of deviation vector can improve the performance, compared, e.g., to the standard Chambolle--Pock algorithm. | result 1.00 | 0.96 | 0.19 | asserted 1.00 |
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The paper
Original abstract
We propose and study a weakly convergent variant of the forward--backward algorithm for solving structured monotone inclusion problems. Our algorithm features a per-iteration deviation vector which provides additional degrees of freedom. The only requirement on the deviation vector to guarantee convergence is that its norm is bounded by a quantity that can be computed online. This approach provides great flexibility and opens up for the design of new and improved forward--backward-based algorithms, while retaining global convergence guarantees. These guarantees include linear convergence of our method under a metric subregularity assumption without the need to adapt the algorithm parameters. Choosing suitable monotone operators allows for incorporating deviations into other algorithms, such as Chambolle--Pock and Krasnoselsky--Mann iterations. We propose a novel inertial primal--dual algorithm by selecting the deviations along a momentum direction and deciding their size using the norm condition. Numerical experiments demonstrate our convergence claims and show that even this simple choice of deviation vector can improve the performance, compared, e.g., to the standard Chambolle--Pock algorithm.
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How the literature uses it
2 arXiv papers cite this in 15 sentences
Papers citing it · 10 sentences without a confident label
Builds on it 1 paper
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Our proposed method is an instance of <ref> from [ref] , where the deviations are chosen based on ideas from the extrapolation step of Anderson acceleration with the goal of improving local performance while benefiting from the global convergence properties of <ref>.
DWIFOB: A Dynamically Weighted Inertial Forward-Backward Algorithm for Monotone Inclusions · Forward–backward splitting with deviations · 0.91
Uses its method or tool 1 paper
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However, by utilizing a similar approach as in [ref] , the extra evaluations can be efficiently done by reusing some of the previous computations.
DWIFOB: A Dynamically Weighted Inertial Forward-Backward Algorithm for Monotone Inclusions · Efficient evaluation of the $M$-induced norm · 0.97
Cites it as background 1 paper
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The FB algorithm with deviations is an extension of the standard FB algorithm and was introduced recently in [ref] .
DWIFOB: A Dynamically Weighted Inertial Forward-Backward Algorithm for Monotone Inclusions · Forward–backward splitting with deviations · 0.94
What it cites · 5 sentences without a confident label
Compares with 1 paper
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Moreover, our convergence result is slightly stronger than [ref] .
Accelerated Forward-Backward Optimization using Deep Learning · Introduction · 1.00
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