Convergence Acceleration via Chebyshev Step: Plausible Interpretation of Deep-Unfolded Gradient Descent
This work provides theoretical insights into deep unfolding for researchers in optimization and machine learning, though it is incremental as it builds on existing iterative algorithms.
The study tackled the problem of understanding convergence acceleration in deep-unfolded gradient descent by proposing an interpretation using Chebyshev steps, showing that this leads to an asymptotically optimal convergence rate without momentum terms. It also introduced Chebyshev-PSOR, which demonstrated significantly faster convergence in numerical experiments for methods like Jacobi and proximal gradient methods.
Deep unfolding is a promising deep-learning technique, whose network architecture is based on expanding the recursive structure of existing iterative algorithms. Although convergence acceleration is a remarkable advantage of deep unfolding, its theoretical aspects have not been revealed yet. The first half of this study details the theoretical analysis of the convergence acceleration in deep-unfolded gradient descent (DUGD) whose trainable parameters are step sizes. We propose a plausible interpretation of the learned step-size parameters in DUGD by introducing the principle of Chebyshev steps derived from Chebyshev polynomials. The use of Chebyshev steps in gradient descent (GD) enables us to bound the spectral radius of a matrix governing the convergence speed of GD, leading to a tight upper bound on the convergence rate. The convergence rate of GD using Chebyshev steps is shown to be asymptotically optimal, although it has no momentum terms. We also show that Chebyshev steps numerically explain the learned step-size parameters in DUGD well. In the second half of the study, %we apply the theory of Chebyshev steps and Chebyshev-periodical successive over-relaxation (Chebyshev-PSOR) is proposed for accelerating linear/nonlinear fixed-point iterations. Theoretical analysis and numerical experiments indicate that Chebyshev-PSOR exhibits significantly faster convergence for various examples such as Jacobi method and proximal gradient methods.