A Fast Anderson-Chebyshev Acceleration for Nonlinear Optimization [article]

Zhize Li, Jian Li
2020 arXiv   pre-print
Anderson acceleration (or Anderson mixing) is an efficient acceleration method for fixed point iterations x_t+1=G(x_t), e.g., gradient descent can be viewed as iteratively applying the operation G(x) x-α∇ f(x). It is known that Anderson acceleration is quite efficient in practice and can be viewed as an extension of Krylov subspace methods for nonlinear problems. In this paper, we show that Anderson acceleration with Chebyshev polynomial can achieve the optimal convergence rate O(√(κ)ln1/ϵ),
more » ... ch improves the previous result O(κln1/ϵ) provided by (Toth and Kelley, 2015) for quadratic functions. Moreover, we provide a convergence analysis for minimizing general nonlinear problems. Besides, if the hyperparameters (e.g., the Lipschitz smooth parameter L) are not available, we propose a guessing algorithm for guessing them dynamically and also prove a similar convergence rate. Finally, the experimental results demonstrate that the proposed Anderson-Chebyshev acceleration method converges significantly faster than other algorithms, e.g., vanilla gradient descent (GD), Nesterov's Accelerated GD. Also, these algorithms combined with the proposed guessing algorithm (guessing the hyperparameters dynamically) achieve much better performance.
arXiv:1809.02341v4 fatcat:5kghvryk2re2rhiehbllsywt3y