Exact Optimal Accelerated Complexity for Fixed-Point Iterations

@InProceedings{pmlr-v162-park22c,
  title = 	 {Exact Optimal Accelerated Complexity for Fixed-Point Iterations},
  author =       {Park, Jisun and Ryu, Ernest K},
  booktitle = 	 {Proceedings of the 39th International Conference on Machine Learning},
  pages = 	 {17420--17457},
  year = 	 {2022},
  editor = 	 {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan},
  volume = 	 {162},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {17--23 Jul},
  publisher =    {PMLR},
  pdf = 	 {https://proceedings.mlr.press/v162/park22c/park22c.pdf},
  url = 	 {https://proceedings.mlr.press/v162/park22c.html},
  abstract = 	 {Despite the broad use of fixed-point iterations throughout applied mathematics, the optimal convergence rate of general fixed-point problems with nonexpansive nonlinear operators has not been established. This work presents an acceleration mechanism for fixed-point iterations with nonexpansive operators, contractive operators, and nonexpansive operators satisfying a Hölder-type growth condition. We then provide matching complexity lower bounds to establish the exact optimality of the acceleration mechanisms in the nonexpansive and contractive setups. Finally, we provide experiments with CT imaging, optimal transport, and decentralized optimization to demonstrate the practical effectiveness of the acceleration mechanism.}
}