We initiate a study of supervised learning from many independent sequences ("trajectories") of non-independent covariates, reflecting tasks in sequence modeling, control, and reinforcement learning. Conceptually, our multi-trajectory setup sits between two traditional settings in statistical learning theory: learning from independent examples and learning from a single auto-correlated sequence. Our conditions for efficient learning generalize the former setting--trajectories must be non-degenerate in ways that extend standard requirements for independent examples. Notably, we do not require that trajectories be ergodic, long, nor strictly stable. For linear least-squares regression, given $n$-dimensional examples produced by $m$ trajectories, each of length $T$, we observe a notable change in statistical efficiency as the number of trajectories increases from a few (namely $m \lesssim n$) to many (namely $m \gtrsim n$). Specifically, we establish that the worst-case error rate of this problem is $\Theta(n / m T)$ whenever $m \gtrsim n$. Meanwhile, when $m \lesssim n$, we establish a (sharp) lower bound of $\Omega(n^2 / m^2 T)$ on the worst-case error rate, realized by a simple, marginally unstable linear dynamical system. A key upshot is that, in domains where trajectories regularly reset, the error rate eventually behaves as if all of the examples were independent, drawn from their marginals. As a corollary of our analysis, we also improve guarantees for the linear system identification problem.

翻译：我们启动了一项关于从多个独立序列（“轨迹”）中监督学习的研究，这些序列包含非独立的协变量，反映了序列建模、控制和强化学习中的任务。从概念上讲，我们的多轨迹设置介于统计学习理论中的两种传统设置之间：从独立样本中学习和从单条自相关序列中学习。我们高效学习的条件推广了前一种设置——轨迹必须在某些方面非退化，以扩展独立样本的标准要求。值得注意的是，我们并不要求轨迹是遍历的、长时间稳定的或严格稳定的。对于线性最小二乘回归，给定由$m$条轨迹产生的$n$维样本（每条轨迹长度为$T$），我们观察到当轨迹数量从少量（即$m \lesssim n$）增加到大量（即$m \gtrsim n$）时，统计效率发生了显著变化。具体而言，我们证明该问题的最坏情况误差率为$\Theta(n / m T)$，只要$m \gtrsim n$。同时，当$m \lesssim n$时，我们建立了最坏情况误差率的（精确）下界$\Omega(n^2 / m^2 T)$，该下界由一个简单、边缘不稳定的线性动力系统实现。一个关键结论是，在轨迹定期重置的领域中，误差率最终表现得如同所有样本独立且来自其边际分布。作为我们分析的推论，我们还改进了线性系统辨识问题的保证。