Stability and Sharper Risk Bounds with Convergence Rate $O(1/n^2)$
Keywords: algorithmic stability, generalization bounds, excess risk bounds, stochastic gradiet descent
TL;DR: We derive high probability excess risk bounds to $O(1/n^2)$ for ERM, GD and SGD and our high probability results on the generalization error of gradients for nonconvex problems are also the sharpest.
Abstract: 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 $O(1/n^2)$ are possible. We discuss how high probability excess risk bounds reach $O(1/n^2)$ under strong 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.
Primary Area: learning theory
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Submission Number: 6499
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