Stability and Deviation Optimal Risk Bounds with Convergence Rate O(1/n)
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The sharpest known high probability generalization bounds for uniformly stable algorithms (Feldman, Vondrák, 2018, 2019), (Bousquet, Klochkov, Zhivotovskiy, 2020) contain a generally inevitable sampling error term of order Θ(1/√(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 Θ(1/√(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 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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