This paper provides a finite-time analysis of linear stochastic approximation (LSA) algorithms with fixed step size, a core method in statistics and machine learning. LSA is used to compute approximate solutions of a $d$-dimensional linear system $\bar{\mathbf{A}} \theta = \bar{\mathbf{b}}$ for which $(\bar{\mathbf{A}}, \bar{\mathbf{b}})$ can only be estimated by (asymptotically) unbiased observations $\{(\mathbf{A}(Z_n),\mathbf{b}(Z_n))\}_{n \in \mathbb{N}}$. We consider here the case where $\{Z_n\}_{n \in \mathbb{N}}$ is an i.i.d. sequence or a uniformly geometrically ergodic Markov chain. We derive $p$-th moment and high-probability deviation bounds for the iterates defined by LSA and its Polyak-Ruppert-averaged version. Our finite-time instance-dependent bounds for the averaged LSA iterates are sharp in the sense that the leading term we obtain coincides with the local asymptotic minimax limit. Moreover, the remainder terms of our bounds admit a tight dependence on the mixing time $t_{\operatorname{mix}}$ of the underlying chain and the norm of the noise variables. We emphasize that our result requires the SA step size to scale only with logarithm of the problem dimension $d$.

翻译：本文对固定步长下的线性随机逼近（LSA）算法进行了有限时间分析，该算法是统计学和机器学习中的核心方法。LSA用于计算 $d$ 维线性系统 $\bar{\mathbf{A}} \theta = \bar{\mathbf{b}}$ 的近似解，其中 $(\bar{\mathbf{A}}, \bar{\mathbf{b}})$ 只能通过（渐近）无偏观测 $\{(\mathbf{A}(Z_n),\mathbf{b}(Z_n))\}_{n \in \mathbb{N}}$ 进行估计。我们考虑 $\{Z_n\}_{n \in \mathbb{N}}$ 为独立同分布序列或一致几何遍历马尔可夫链的情形。我们推导了LSA迭代及其Polyak-Ruppert平均版本的 $p$ 阶矩和高概率偏差界。对于平均LSA迭代，我们的有限时间实例相关界是紧的，因为所得主导项与局部渐近极小极大极限一致。此外，我们的边界余项与底层链的混合时间 $t_{\operatorname{mix}}$ 及噪声变量范数具有紧密依赖关系。我们强调，该结果要求SA步长仅随问题维度 $d$ 的对数缩放。