We derive upper bounds for random design linear regression with dependent ($\beta$-mixing) data absent any realizability assumptions. In contrast to the strictly realizable martingale noise regime, no sharp instance-optimal non-asymptotics are available in the literature. Up to constant factors, our analysis correctly recovers the variance term predicted by the Central Limit Theorem -- the noise level of the problem -- and thus exhibits graceful degradation as we introduce misspecification. Past a burn-in, our result is sharp in the moderate deviations regime, and in particular does not inflate the leading order term by mixing time factors.

翻译：我们针对带有相依（$\beta$-混合）数据的随机设计线性回归问题，推导了上界，且无需任何可实现性假设。与严格可实现的鞅噪声情形相比，文献中尚无最优实例依赖的非渐近结果。在忽略常数因子的情况下，我们的分析正确恢复了中心极限定理所预测的方差项——即该问题的噪声水平——从而在引入模型误设定时表现出优雅的性能退化。经过初始阶段后，我们的结果在中等偏差区间内是尖锐的，特别地，主导阶项不会被混合时间因子放大。