On Empirical Risk Minimization with Dependent and Heavy-Tailed Data

Part of Advances in Neural Information Processing Systems 34 pre-proceedings (NeurIPS 2021)

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Abhishek Roy, Krishnakumar Balasubramanian, Murat A. Erdogdu


In this work, we establish risk bounds for Empirical Risk Minimization (ERM) with both dependent and heavy-tailed data-generating processes. We do so by extending the seminal works~\cite{pmlr-v35-mendelson14, mendelson2018learning} on the analysis of ERM with heavy-tailed but independent and identically distributed observations, to the strictly stationary exponentially $\beta$-mixing case. We allow for the interaction between the noise and inputs to be even polynomially heavy-tailed, which covers a significantly large class of heavy-tailed models beyond what is analyzed in the learning theory literature. We illustrate our theoretical results by obtaining rates of convergence for high-dimensional linear regression with dependent and heavy-tailed data.