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

In this work, we study statistical learning with dependent (β-mixing) data and square loss in a hypothesis class mathscrFsubset LPsip where Psip is the norm |f|Psip triangleq supm≥ 1 m-1/p |f|Lm for some p∈ [2,∞]. 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 L2 and Psip are comparable on our hypothesis class mathscrF -- that is, mathscrF is a weakly sub-Gaussian class: |f|Psip lesssim |f|L2η for some η∈ (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 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.