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# Sharp Rates in Dependent Learning Theory: Avoiding Sample Size Deflation for the Square Loss

*Proceedings of the 41st International Conference on Machine Learning*, PMLR 235:62779-62802, 2024.

#### Abstract

In this work, we study statistical learning with dependent data and square loss in a hypothesis class with tail decay in Orlicz space: $\mathscr{F}\subset L_{\Psi_p}$. Our inquiry is motivated by the search for a sharp noise interaction term, or variance proxy, in learning with dependent (e.g. $\beta$-mixing) data. Typical non-asymptotic results exhibit variance proxies that are deflated

*multiplicatively*in the mixing time of the underlying covariates process. We show that whenever the topologies of $L^2$ and $\Psi_p$ are comparable on our hypothesis class $\mathscr{F}$, the empirical risk minimizer achieves a rate that only depends on the complexity of the class and second order statistics in its leading term. We refer to this as a*near mixing-free rate*, since direct dependence on mixing is relegated to an additive higher order term. Our approach, reliant on mixed tail generic chaining, allows us to obtain sharp, instance-optimal rates. Examples that satisfy our framework include for instance sub-Gaussian linear regression and bounded smoothness classes.