AI 中文总结
该研究针对大规模线性统计模型的普通最小二乘估计问题，提出顺序预条件共轭梯度法（SPCG）。它通过构建子问题、应用PCG并热启动，降低计算成本。建立收敛理论，数值实验显示SPCG比其他方法用更少迭代次数和时间达目标精度。
AI 中文摘要
我们针对大规模线性统计模型中的普通最小二乘估计问题提出了一种随机迭代方法，即顺序预条件共轭梯度法（SPCG）。SPCG构建一系列草图大小不断增加的最小二乘子问题，以内层求解器应用PCG，并从前一个解热启动每个子问题。然后对全规模问题进行最终细化阶段。由于大多数迭代在较小子问题上进行，显著降低了总体计算成本。我们建立了收敛理论，证明SPCG达到OLS预测精度，并推导了每个子问题的迭代界和复杂度估计。数值实验表明，SPCG比全数据PCG和迭代双草图（IDS）以更少的迭代次数和更少的CPU时间达到目标预测精度。
英文摘要
We propose a randomized iterative method for the ordinary least-squares estimation problem in large-scale linear statistical models, namely the Sequential Preconditioned Conjugate Gradient Method (SPCG). SPCG constructs a sequence of sketched least-squares subproblems with increasing sketch sizes, applies PCG as the inner solver, and warm-starts each subproblem from the previous solution. A final refinement stage is then performed on the full-scale problem. Since most iterations are carried out on smaller subproblems, the overall computational cost is significantly reduced. We establish the convergence theory, prove that SPCG attains OLS prediction accuracy, and derive per-subproblem iteration bounds and complexity estimates. Numerical experiments show that SPCG reaches the target prediction accuracy with fewer iterations and less CPU time than full-data PCG and Iterative Double Sketching (IDS).
