Generalization Bounds for Uniformly Stable Algorithms

TL;DR

This paper improves the theoretical understanding of the generalization error bounds for uniformly stable algorithms, providing tighter bounds that hold under broader conditions without extra assumptions.

Contribution

The authors derive significantly tighter generalization bounds for uniformly stable algorithms, including a new bound on the second moment of the estimation error, using novel analysis techniques.

Findings

Bound on generalization error is improved to O(√((γ + 1/n) log(1/δ)))

Established a tight bound of O(γ^2 + 1/n) on the second moment of the estimation error

Results imply stronger guarantees for several well-studied algorithms.

Abstract

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 $\gamma$-uniformly stable learning algorithm on $n$ samples is known to be within $O((\gamma +1/n) \sqrt{n \log(1/\delta)})$ of the empirical error with probability at least $1-\delta$. Unfortunately, this bound does not lead to meaningful generalization bounds in many common settings where $\gamma \geq 1/\sqrt{n}$. At the same time the bound is known to be tight only when $\gamma = O(1/n)$. We substantially improve generalization bounds for uniformly stable algorithms without making any additional assumptions. First, we show that the bound in this setting is $O(\sqrt{(\gamma + 1/n) \log(1/\delta)})$ with probability at least $1-\delta$. In addition, we prove a tight bound of $O(\gamma^2 + 1/n)$ on the second moment of the estimation error. The best previous bound on the second moment is $O(\gamma + 1/n)$. Our proofs are based on new analysis techniques and our results imply substantially stronger generalization guarantees for several well-studied algorithms.