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

• URL: http://arxiv.org/abs/2103.12024v1
• Date: Mon, 22 Mar 2021 17:28:40 GMT
• Title: Stability and Deviation Optimal Risk Bounds with Convergence Rate $O(1/n)$
• Authors: Yegor Klochkov and Nikita Zhivotovskiy
• Abstract summary: We show a high probability excess risk bound with the rate $O(log n/n)$ for strongly convex and Lipschitz losses valid for emphany 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.
• Score: 4.1499725848998965
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The sharpest known high probability generalization bounds for uniformly stable 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 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. This resolves a question of Shalev-Shwartz, Shamir, Srebro, and Sridharan (2009). 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.

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