Sharp Rates in Dependent Learning Theory: Avoiding Sample Size Deflation for the Square Loss
This provides a theoretical foundation for learning with dependent data, offering near mixing-free rates that are broadly applicable across various function classes, though it is incremental in refining existing bounds.
The paper tackles the problem of statistical learning with dependent data, showing that under a weakly sub-Gaussian condition, the empirical risk minimizer achieves a rate that depends primarily on class complexity and second-order statistics, with mixing effects relegated to higher-order terms, avoiding multiplicative sample size deflation.
In this work, we study statistical learning with dependent ($β$-mixing) data and square loss in a hypothesis class $\mathscr{F}\subset L_{Ψ_p}$ where $Ψ_p$ is the norm $\|f\|_{Ψ_p} \triangleq \sup_{m\geq 1} m^{-1/p} \|f\|_{L^m} $ for some $p\in [2,\infty]$. Our inquiry is motivated by the search for a sharp noise interaction term, or variance proxy, in learning with dependent data. Absent any realizability assumption, typical non-asymptotic results exhibit variance proxies that are deflated multiplicatively by the mixing time of the underlying covariates process. We show that whenever the topologies of $L^2$ and $Ψ_p$ are comparable on our hypothesis class $\mathscr{F}$ -- that is, $\mathscr{F}$ is a weakly sub-Gaussian class: $\|f\|_{Ψ_p} \lesssim \|f\|_{L^2}^η$ for some $η\in (0,1]$ -- the empirical risk minimizer achieves a rate that only depends on the complexity of the class and second order statistics in its leading term. Our result holds whether the problem is realizable or not and we refer to this as a \emph{near mixing-free rate}, since direct dependence on mixing is relegated to an additive higher order term. We arrive at our result by combining the above notion of a weakly sub-Gaussian class with mixed tail generic chaining. This combination allows us to compute sharp, instance-optimal rates for a wide range of problems. Examples that satisfy our framework include sub-Gaussian linear regression, more general smoothly parameterized function classes, finite hypothesis classes, and bounded smoothness classes.