The methodology involves transforming an optimization problem to improve the convergence rate of iterative descent methods. A symmetric, positive definite matrix is used to precondition the gradient, altering the search direction. This adjustment aims to align the search more closely with the optimal solution, accelerating the iterative process. For instance, when minimizing a poorly conditioned quadratic function, this technique can significantly reduce the number of iterations required to reach a desired level of accuracy compared to standard gradient descent.
This approach is valuable in various fields, including machine learning, image processing, and structural engineering, where large-scale optimization problems are prevalent. By modifying the curvature of the objective function, the preconditioning step reduces the eccentricity of the level sets, resulting in a more stable and efficient descent. Historically, this technique has evolved from basic steepest descent to more sophisticated methods that dynamically adapt the preconditioning matrix during the optimization process, further enhancing performance.
