Adaptive Restart of the Optimized Gradient Method for Convex Optimization

Abstract

First-order methods with momentum such as Nesterov's fast gradient method are very useful for convex optimization problems, but can exhibit undesirable oscillations yielding slow convergence rates for some applications. An adaptive restarting scheme can improve the convergence rate of the fast gradient method, when the parameter of a strongly convex cost function is unknown or when the iterates of the algorithm enter a locally strongly convex region. Recently, we introduced the optimized gradient method, a first-order algorithm that has an inexpensive per-iteration computational cost similar to that of the fast gradient method, yet has a worst-case cost function rate that is twice faster than that of the fast gradient method and that is optimal for large-dimensional smooth convex problems. Building upon the success of accelerating the fast gradient method using adaptive restart, this paper investigates similar heuristic acceleration of the optimized gradient method. We first derive a new first-order method that resembles the optimized gradient method for strongly convex quadratic problems with known function parameters, yielding a linear convergence rate that is faster than that of the analogous version of the fast gradient method. We then provide a heuristic analysis and numerical experiments that illustrate that adaptive restart can accelerate the convergence of the optimized gradient method. Numerical results also illustrate that adaptive restart is helpful for a proximal version of the optimized gradient method for nonsmooth composite convex functions.

References in corpus (2)

• Convex Optimization for Big Data
• On the convergence analysis of the optimized gradient method

Cited by in corpus (6)

• Acceleration Methods
• Efficient approximation of Jacobian matrices involving a non-uniform fast Fourier transform (NUFFT)
• Practical Perspectives on Symplectic Accelerated Optimization
• Additive Schwarz methods for semilinear elliptic problems with convex energy functionals: Convergence rate independent of nonlinearity
• A piecewise conservative method for unconstrained convex optimization
• Fast Empirical Scenarios