Generalization Bounds for Uniformly Stable Algorithms

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 a $γ$-uniformly stable learning algorithm on $n$ samples is known to be within $O((γ+1/n) \sqrt{n \log(1/δ)})$ of the empirical error with probability at least $1-δ$. Unfortunately, this bound does not lead to meaningful generalization bounds in many common settings where $γ\geq 1/\sqrt{n}$. At the same time the bound is known to be tight only when $γ= O(1/n)$. We substantially improve generalization bounds for uniformly stable algorithms without making any additional assumptions. First, we show that the bound in this setting is $O(\sqrt{(γ+ 1/n) \log(1/δ)})$ 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 estimation error. The best previous bound on the second moment 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.