High probability generalization bounds for uniformly stable algorithms with nearly optimal rate
[edit]
Proceedings of the ThirtySecond Conference on Learning Theory, PMLR 99:12701279, 2019.
Abstract
Algorithmic stability is a classical approach to understanding and analysis of the generalization error of learning algorithms. A notable weakness of most stabilitybased generalization bounds is that they hold only in expectation. Generalization with high probability has been established in a landmark paper of Bousquet and Elisseeff (2001) albeit at the expense of an additional $\sqrt{n}$ factor in the bound. Specifically, their bound on the estimation error of any $\gamma$uniformly stable learning algorithm on $n$ samples and range in $[0,1]$ is $O(\gamma \sqrt{n \log(1/\delta)} + \sqrt{\log(1/\delta)/n})$ with probability $\geq 1\delta$. The $\sqrt{n}$ overhead makes the bound vacuous in the common settings where $\gamma \geq 1/\sqrt{n}$. A stronger bound was recently proved by the authors (Feldman and Vondrak, 2018) that reduces the overhead to at most $O(n^{1/4})$. Still, both of these results give optimal generalization bounds only when $\gamma = O(1/n)$. We prove a nearly tight bound of $O(\gamma \log(n)\log(n/\delta) + \sqrt{\log(1/\delta)/n})$ on the estimation error of any $\gamma$uniformly stable algorithm. It implies that for algorithms that are uniformly stable with $\gamma = O(1/\sqrt{n})$, estimation error is essentially the same as the sampling error. Our result leads to the first highprobability generalization bounds for multipass stochastic gradient descent and regularized ERM for stochastic convex problems with nearly optimal rate — resolving open problems in prior work. Our proof technique is new and we introduce several analysis tools that might find additional applications.
Related Material


