Generalization Bounds for Uniformly Stable Algorithms
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Uniform stability of a learning algorithm is a classical notion of algorithmic stability introduced to derive high-probability bounds on the generalization error (Bousquet and Elisseeff, 2002). Specifically, for a loss function with range bounded in [0,1], the generalization error of γ-uniformly stable learning algorithm on n samples is known to be at most O((γ +1/n) √(n (1/δ))) with probability at least 1-δ. Unfortunately, this bound does not lead to meaningful generalization bounds in many common settings where γ≥ 1/√(n). At the same time the bound is known to be tight only when γ = O(1/n). Here we prove substantially stronger generalization bounds for uniformly stable algorithms without any additional assumptions. First, we show that the generalization error in this setting is at most O(√((γ + 1/n) (1/δ))) with probability at least 1-δ. In addition, we prove a tight bound of O(γ^2 + 1/n) on the second moment of the generalization error. The best previous bound on the second moment of the generalization error is O(γ + 1/n). Our proofs are based on new analysis techniques and our results imply substantially stronger generalization guarantees for several well-studied algorithms.
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