相依学习理论中的尖锐速率：避免平方损失的样本量缩减
Sharp Rates in Dependent Learning Theory: Avoiding Sample Size Deflation for the Square Loss
AI总结：
研究相依数据与平方损失学习中的方差代理被混合时间缩减的问题，提出弱次高斯类结合混合尾通用链方法，证明经验风险最小化器可实现近混合无关速率，其主项仅依赖类复杂度与二阶统计量。
摘要：
在这项工作中，我们研究具有相依（$β$-混合）数据和平方损失（square loss）的统计学习，假设类为 $\mathscr{F}\subset L_{\Psi_p}$，其中 $\Psi_p$ 是范数 $\\|f\\|_{\Psi_p} \triangleq \sup_{m\geq 1} m^{-1/p} \\|f\\|_{L^m}$，对于某个 $p\in [2,\infty]$。我们的探究动机是在相依数据学习中寻找尖锐的噪声交互项，或方差代理（variance proxy）。在没有任何可实现性假设的情况下，典型的非渐近结果表现出方差代理被潜在协变量过程的混合时间（mixing time）乘法性地缩减。我们证明，只要 $L^2$ 和 $\Psi_p$ 的拓扑在我们的假设类 $\mathscr{F}$ 上是可比较的——即 $\mathscr{F}$ 是一个弱次高斯类（weakly sub-Gaussian class）：$\\|f\\|_{\Psi_p} \lesssim \\|f\\|_{L^2}^\eta$ 对于某个 $\eta\in (0,1]$——经验风险最小化器（empirical risk minimizer）实现的速率在其主项中仅依赖于类的复杂度和二阶统计量。我们的结果无论问题是否可实现都成立，我们将其称为“近混合无关速率”（near mixing-free rate），因为对混合的直接依赖被降级为加性的高阶项。我们通过将上述弱次高斯类概念与混合尾通用链（mixed tail generic chaining）相结合来得出结果。这种结合使我们能够为广泛的问题类别计算尖锐的、实例最优的速率。满足我们框架的例子包括次高斯线性回归、一般平滑参数化函数类、有限假设类和有界平滑类。
英文摘要：
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.
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