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The sharpest known high probability generalization bounds for uniformlystable algorithms (Feldman, Vondr\'{a}k, 2018, 2019), (Bousquet, Klochkov,Zhivotovskiy, 2020) contain a generally inevitable sampling error term of order$\Theta(1/\sqrt{n})$. When applied to excess risk bounds, this leads tosuboptimal 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 oforder up to $O(1/n)$ are possible via uniform stability. Using this result, weshow a high probability excess risk bound with the rate $O(\log n/n)$ forstrongly convex and Lipschitz losses valid for \emph{any} empirical riskminimization method. This resolves a question of Shalev-Shwartz, Shamir,Srebro, and Sridharan (2009). We discuss how $O(\log n/n)$ high probabilityexcess risk bounds are possible for projected gradient descent in the case ofstrongly convex and Lipschitz losses without the usual smoothness assumption.

Stability and Deviation Optimal Risk Bounds with Convergence Rate $O(1/n)$