High probability generalization bounds for uniformly stable algorithms with nearly optimal rate

Algorithmic stability is a classical approach to understanding and analysis of the generalization error of learning algorithms. A notable weakness of most stability-based generalization bounds is that they hold only in expectation. Generalization with high probability has been established in a landmark paper of Bousquet and Elisseeff (2002) albeit at the expense of an additional √n factor in the bound. Specifically, their bound on the estimation error of any γ-uniformly stable learning algorithm on n samples and range in [0,1] is O(γ √n log(1/δ) + √log(1/δ)/n) with probability ≥ 1-δ. The √n overhead makes the bound vacuous in the common settings where γ ≥ 1/√n. A stronger bound was recently proved by the authors (Feldman and Vondrak, 2018) that reduces the overhead to at most O(n1/4). Still, both of these results give optimal generalization bounds only when γ = O(1/n). We prove a nearly tight bound of O(γ log(n)log(n/δ) + √log(1/δ)/n) on the estimation error of any γ-uniformly stable algorithm. It implies that for algorithms that are uniformly stable with γ = O(1/√n), estimation error is essentially the same as the sampling error. Our result leads to the first high-probability generalization bounds for multi-pass 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.