Stability and Sharper Risk Bounds with Convergence Rate $\tilde{O}(1/n^2)$

Yong Liu (Tsinghua University, Tsinghua University) · Bowei Zhu (Renmin University of China) · Shaojie Li (Renmin University of China) · Mingyang Yi (Renmin University of China)

algorithmic stability convergence rates excess risk boundsgeneralization gapsgradient-based generalizationhigh-probability boundslipschitz continuitynonconvex settingsoptimization theorypolyak-lojasiewicz conditionsmoothness statistical learning theorystrongly-convex learnerstheoretical analysis

Prior work (Klochkov \& Zhivotovskiy, 2021) establishes at most $O\left(\log (n)/n\right)$ excess risk bounds via algorithmic stability for strongly-convex learners with high probability. We show that under the similar common assumptions — Polyak-Lojasiewicz condition, smoothness, and Lipschitz continous for losses — rates of $O\left(\log^2(n)/n^2\right)$ are at most achievable. To our knowledge, our analysis also provides the tightest high-probability bounds for gradient-based generalization gaps in nonconvex settings.