On the Convergence Analysis of the Optimized Gradient Method.

Area of Science:

• Optimization Theory
• Numerical Analysis
• Machine Learning

Background:

• Unconstrained minimization of smooth convex functions with Lipschitz continuous gradients is a fundamental problem.
• Existing methods like Nesterov's fast gradient method have established convergence bounds.
• The recently proposed optimized gradient method offers improved worst-case convergence bounds.

Purpose of the Study:

• To fully investigate the convergence properties of the optimized gradient method.
• To derive an analytic convergence bound for the primary sequence generated by the optimized gradient method.
• To complement existing theoretical understanding of optimal first-order methods.

Main Methods:

• Analysis of the optimized gradient method's convergence for smooth convex functions.
• Derivation of analytic convergence bounds for the primary sequence.
• Identification of worst-case functions for the method.

Main Results:

• An analytic convergence bound for the primary sequence of the optimized gradient method is established.
• The optimized gradient method demonstrates optimal complexity for cost function decrease among first-order methods.
• Two types of worst-case functions (piecewise affine-quadratic and quadratic) are identified for the method.

Conclusions:

• The derived convergence bound completes the theoretical analysis of the optimized gradient method.
• The findings solidify the optimized gradient method's status as an optimal first-order method for smooth convex minimization.
• This research contributes to the understanding of efficient algorithms for convex optimization problems.