Non-ergodic Convergence Analysis of Heavy-Ball Algorithms
This work provides incremental theoretical advancements for optimization algorithms, benefiting researchers in machine learning and optimization by offering more robust convergence guarantees.
The paper improves convergence complexity results for the Heavy-ball method in convex optimization, achieving the first non-ergodic O(1/k) rate for coercive functions and linear convergence under relaxed strongly convex conditions with weaker assumptions.
In this paper, we revisit the convergence of the Heavy-ball method, and present improved convergence complexity results in the convex setting. We provide the first non-ergodic O(1/k) rate result of the Heavy-ball algorithm with constant step size for coercive objective functions. For objective functions satisfying a relaxed strongly convex condition, the linear convergence is established under weaker assumptions on the step size and inertial parameter than made in the existing literature. We extend our results to multi-block version of the algorithm with both the cyclic and stochastic update rules. In addition, our results can also be extended to decentralized optimization, where the ergodic analysis is not applicable.