Practical Schemes for Finding Near-Stationary Points of Convex Finite-Sums
This work addresses a gap in convex optimization for researchers and practitioners, offering practical and simple schemes that could enable future advancements, though it is incremental as it builds on existing methods like OGM-G and SVRG.
The paper tackles the problem of finding near-stationary points in convex finite-sum optimization, a less studied area compared to function value minimization, and presents new algorithms including a memory-saving variant of OGM-G, an accelerated SVRG variant for fast gradient norm and function value rates, and an adaptively regularized variant with near-optimal complexities.
In convex optimization, the problem of finding near-stationary points has not been adequately studied yet, unlike other optimality measures such as the function value. Even in the deterministic case, the optimal method (OGM-G, due to Kim and Fessler (2021)) has just been discovered recently. In this work, we conduct a systematic study of algorithmic techniques for finding near-stationary points of convex finite-sums. Our main contributions are several algorithmic discoveries: (1) we discover a memory-saving variant of OGM-G based on the performance estimation problem approach (Drori and Teboulle, 2014); (2) we design a new accelerated SVRG variant that can simultaneously achieve fast rates for minimizing both the gradient norm and function value; (3) we propose an adaptively regularized accelerated SVRG variant, which does not require the knowledge of some unknown initial constants and achieves near-optimal complexities. We put an emphasis on the simplicity and practicality of the new schemes, which could facilitate future work.