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Line search (or backtracking) procedures have been widely employed intofirst-order methods for solving convex optimization problems, especially thosewith unknown problem parameters (e.g., Lipschitz constant). In this paper, weshow that line search is superfluous in attaining the optimal rate ofconvergence for solving a convex optimization problem whose parameters are notgiven a priori. In particular, we present a novel accelerated gradient descenttype algorithm called auto-conditioned fast gradient method (AC-FGM) that canachieve an optimal $\mathcal{O}(1/k^2)$ rate of convergence for smooth convexoptimization without requiring the estimate of a global Lipschitz constant orthe employment of line search procedures. We then extend AC-FGM to solve convexoptimization problems with H\"{o}lder continuous gradients and show that itautomatically achieves the optimal rates of convergence uniformly for allproblem classes with the desired accuracy of the solution as the only input.Finally, we report some encouraging numerical results that demonstrate theadvantages of AC-FGM over the previously developed parameter-free methods forconvex optimization.

A simple uniformly optimal method without line search for convex optimization