On the tightness of information-theoretic bounds on generalization error of learning algorithms
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A recent line of works, initiated by Russo and Xu, has shown that the generalization error of a learning algorithm can be upper bounded by information measures. In most of the relevant works, the convergence rate of the expected generalization error is in the form of O(√(λ/n)) where λ is some information-theoretic quantities such as the mutual information or conditional mutual information between the data and the learned hypothesis. However, such a learning rate is typically considered to be “slow", compared to a “fast rate" of O(λ/n) in many learning scenarios. In this work, we first show that the square root does not necessarily imply a slow rate, and a fast rate result can still be obtained using this bound under appropriate assumptions. Furthermore, we identify the critical conditions needed for the fast rate generalization error, which we call the (η,c)-central condition. Under this condition, we give information-theoretic bounds on the generalization error and excess risk, with a fast convergence rate for specific learning algorithms such as empirical risk minimization and its regularized version. Finally, several analytical examples are given to show the effectiveness of the bounds.
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