A unified variance-reduced accelerated gradient method for convex optimization
This work addresses optimization efficiency for machine learning and data science applications, offering a unified algorithm with theoretical guarantees, though it appears incremental in advancing existing variance-reduction and acceleration techniques.
The authors tackled the problem of finite-sum convex optimization by proposing Varag, a variance-reduced accelerated gradient method that achieves unified optimal convergence rates for smooth convex problems, regardless of strong convexity, and provides optimal linear convergence under strong convexity or error bound conditions.
We propose a novel randomized incremental gradient algorithm, namely, VAriance-Reduced Accelerated Gradient (Varag), for finite-sum optimization. Equipped with a unified step-size policy that adjusts itself to the value of the condition number, Varag exhibits the unified optimal rates of convergence for solving smooth convex finite-sum problems directly regardless of their strong convexity. Moreover, Varag is the first accelerated randomized incremental gradient method that benefits from the strong convexity of the data-fidelity term to achieve the optimal linear convergence. It also establishes an optimal linear rate of convergence for solving a wide class of problems only satisfying a certain error bound condition rather than strong convexity. Varag can also be extended to solve stochastic finite-sum problems.