## Abstract

The fast iterative shrinkage-thresholding algorithm (FISTA) is a widely used procedure for minimizing the sum of two convex functions, such that one has a L-Lipschitz continuous gradient and the other is possible nonsmooth. While FISTA's theoretical rate of convergence (RoC) is pro-1 portional to _{αkt}_{2k} , and it is related to (i) its extragradient rule / inertial sequence, which depends on sequence t_{k}, and (ii) the stepsize αk, which estimates L, its worst-case complexity results in O(k^{−}^{2}) since, originally, (i) by construction t_{k} ≥ ^{k}^{+1}_{2} , and (ii) the condition αk ≥ αk_{+1} was imposed. Attempts to improve FISTA's RoC include alternative inertial sequences, and intertwining the selection of the inertial sequence and the step-size. In this paper, we show that if a bounded and non-decreasing step-size sequence (α_{k} ≤ αk+1, decoupled from the inertial sequence) can be generated via some adaptive scheme, then FISTA can achieve a RoC proportional to k^{−}^{3} for the indexes where the step-size exhibits an approximate linear growth, with the default O(k^{−}^{2}) behavior when the step-size's bound is reached. Furthermore, such exceptional step-size sequence can be easily generated, and it indeed boots FISTA's practical performance.

Original language | English |
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Title of host publication | EUSIPCO 2019 - 27th European Signal Processing Conference |

Publisher | European Signal Processing Conference, EUSIPCO |

ISBN (Electronic) | 9789082797039 |

DOIs | |

State | Published - Sep 2019 |

Event | 27th European Signal Processing Conference, EUSIPCO 2019 - A Coruna, Spain Duration: 2 Sep 2019 → 6 Sep 2019 |

### Publication series

Name | European Signal Processing Conference |
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Volume | 2019-September |

ISSN (Print) | 2219-5491 |

### Conference

Conference | 27th European Signal Processing Conference, EUSIPCO 2019 |
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Country/Territory | Spain |

City | A Coruna |

Period | 2/09/19 → 6/09/19 |

## Keywords

- Convolutional sparse representations
- FISTA
- Step-size

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