Stability and Deviation Optimal Risk Bounds with Convergence Rate $O(1/n)$

TL;DR

This paper improves high probability risk bounds for stable algorithms to an optimal rate of O(1/n) under the Bernstein condition, resolving a longstanding open problem in stochastic convex optimization.

Contribution

It demonstrates that the O(1/√n) sampling error can be eliminated under the Bernstein condition, achieving near-optimal risk bounds for empirical risk minimization and gradient descent.

Findings

High probability excess risk bounds of O(log n/n] are achievable.

The results apply to any empirical risk minimization method under the Bernstein condition.

O(1/n) bounds are obtained without smoothness assumptions for gradient descent.

Abstract

The sharpest known high probability generalization bounds for uniformly stable algorithms (Feldman, Vondr\'{a}k, 2018, 2019), (Bousquet, Klochkov, Zhivotovskiy, 2020) contain a generally inevitable sampling error term of order $\Theta(1/\sqrt{n})$. When applied to excess risk bounds, this leads to suboptimal results in several standard stochastic convex optimization problems. We show that if the so-called Bernstein condition is satisfied, the term $\Theta(1/\sqrt{n})$ can be avoided, and high probability excess risk bounds of order up to $O(1/n)$ are possible via uniform stability. Using this result, we show a high probability excess risk bound with the rate $O(\log n/n)$ for strongly convex and Lipschitz losses valid for \emph{any} empirical risk minimization method. This resolves a question of Shalev-Shwartz, Shamir, Srebro, and Sridharan (2009). We discuss how $O(\log n/n)$ high probability excess risk bounds are possible for projected gradient descent in the case of strongly convex and Lipschitz losses without the usual smoothness assumption.