Sharper Bounds for Uniformly Stable Algorithms with Stationary Mixing Process

Shi Fu, Yunwen Lei, Qiong Cao, Xinmei Tian, Dacheng Tao

Keywords: Algorithmic Stability, Non-I.I.D. Learning, Generalization Error, Learning Theory

Abstract: Generalization analysis of learning algorithms often builds on a critical assumption that training examples are independently and identically distributed, which is often violated in practical problems such as time series prediction. In this paper, we use algorithmic stability to study the generalization performance of learning algorithms with $\psi$-mixing data, where the dependency between observations weakens over time. We show uniformly stable algorithms guarantee high-probability generalization bounds of the order $O(1/\sqrt{n})$ (within a logarithmic factor), where $n$ is the sample size. We apply our general result to specific algorithms including regularization schemes, stochastic gradient descent and localized iterative regularization, and develop excess population risk bounds for learning with $\psi$-mixing data. Our analysis builds on a novel moment bound for weakly-dependent random variables on a $\varphi$-mixing sequence and a novel error decomposition of generalization error.

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TL;DR: We develop stability and generalization bounds for learning with mixing sequences.