The sharpest known high probability generalization bounds for uniformly stable algorithms (Feldman, Vondrak, NeurIPS 2018, COLT, 2019), (Bousquet, Klochkov, Zhivotovskiy, COLT, 2020) contain a generally inevitable sampling error term of order $\Theta(1/\sqrt{n})$. When applied to excess risk bounds, this leads to suboptimal results in several standard stochastic convex optimization problems. We show that if the so-called Bernstein condition is satisfied, the term $\Theta(1/\sqrt{n})$ can be avoided, and high probability excess risk bounds of order up to $O(1/n)$ are possible via uniform stability. Using this result, we show a high probability excess risk bound with the rate $O(\log n/n)$ for strongly convex and Lipschitz losses valid for \emph{any} empirical risk minimization method.

The sharpest known high probability excess risk bounds are up to $O\left( 1/n \right)$ for empirical risk minimization and projected gradient descent via algorithmic stability (Klochkov \& Zhivotovskiy, 2021). In this paper, we show that high probability excess risk bounds of order up to $O\left( 1/n^2 \right)$ are possible. We discuss how high probability excess risk bounds reach $O\left( 1/n^2 \right)$ under strongly convexity, smoothness and Lipschitz continuity assumptions for empirical risk minimization, projected gradient descent and stochastic gradient descent. Besides, to the best of our knowledge, our high probability results on the generalization gap measured by gradients for nonconvex problems are also the sharpest.

The sharpest known high probability generalization bounds for uniformly stable algorithms (Feldman, Vondrak, NeurIPS 2018, COLT, 2019), (Bousquet, Klochkov, Zhivotovskiy, COLT, 2020) contain a generally inevitable sampling error term of order \Theta(1/\sqrt{n}) . When applied to excess risk bounds, this leads to suboptimal results in several standard stochastic convex optimization problems. We show that if the so-called Bernstein condition is satisfied, the term \Theta(1/\sqrt{n}) can be avoided, and high probability excess risk bounds of order up to O(1/n) are possible via uniform stability. Using this result, we show a high probability excess risk bound with the rate O(\log n/n) for strongly convex and Lipschitz losses valid for \emph{any} empirical risk minimization method. We discuss how O(\log n/n) high probability excess risk bounds are possible for projected gradient descent in the case of strongly convex and Lipschitz losses without the usual smoothness assumption.