The problem of finding near-stationary points in convex optimization has not been adequately studied yet, unlike other optimality measures such as minimizing 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 the algorithmic techniques in 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 both minimizing 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 developments.
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