We analyze the bit complexity of efficient algorithms for fundamental optimization problems, such as linear regression, $p$-norm regression, and linear programming (LP). State-of-the-art algorithms are iterative, and in terms of the number of arithmetic operations, they match the current time complexity of multiplying two $n$-by-$n$ matrices (up to polylogarithmic factors). However, previous work has typically assumed infinite precision arithmetic, and due to complicated inverse maintenance techniques, the actual running times of these algorithms are unknown. To settle the running time and bit complexity of these algorithms, we demonstrate that a core common subroutine, known as \emph{inverse maintenance}, is backward-stable. Additionally, we show that iterative approaches for solving constrained weighted regression problems can be accomplished with bounded-error pre-conditioners. Specifically, we prove that linear programs can be solved approximately in matrix multiplication time multiplied by polylog factors that depend on the condition number $\kappa$ of the matrix and the inner and outer radius of the LP problem. $p$-norm regression can be solved approximately in matrix multiplication time multiplied by polylog factors in $\kappa$. Lastly, linear regression can be solved approximately in input-sparsity time multiplied by polylog factors in $\kappa$. Furthermore, we present results for achieving lower than matrix multiplication time for $p$-norm regression by utilizing faster solvers for sparse linear systems.

翻译：我们分析了高效算法在基本优化问题（如线性回归、$p$-范数回归和线性规划（LP））中的比特复杂度。当前最先进的算法是迭代式的，就算术运算次数而言，它们与两个$n \times n$矩阵相乘的当前时间复杂度相匹配（最多相差多对数因子）。然而，先前的研究通常假设无限精度算术，且由于复杂的逆矩阵维护技术，这些算法的实际运行时间仍是未知的。为解决这些算法的运行时间和比特复杂度问题，我们证明了一个名为逆矩阵维护的核心公共子程序具有后向稳定性。此外，我们展示了解决约束加权回归问题的迭代方法可以通过有界误差预条件器实现。具体而言，我们证明线性规划可以在矩阵乘法时间乘以依赖于矩阵条件数$\kappa$及LP问题内半径和外半径的多对数因子的范围内近似求解。$p$-范数回归可以在矩阵乘法时间乘以$\kappa$的多对数因子的范围内近似求解。最后，线性回归可以在输入稀疏时间乘以$\kappa$的多对数因子的范围内近似求解。此外，我们通过利用稀疏线性系统的更快速求解器，展示了在$p$-范数回归中实现低于矩阵乘法时间的结果。