Stochastic gradient descent (SGD) exhibits strong algorithmic regularization effects in practice and plays an important role in the generalization of modern machine learning. However, prior research has revealed instances where the generalization performance of SGD is worse than ridge regression due to uneven optimization along different dimensions. Preconditioning offers a natural solution to this issue by rebalancing optimization across different directions. Yet, the extent to which preconditioning can enhance the generalization performance of SGD and whether it can bridge the existing gap with ridge regression remains uncertain. In this paper, we study the generalization performance of SGD with preconditioning for the least squared problem. We make a comprehensive comparison between preconditioned SGD and (standard \& preconditioned) ridge regression. Our study makes several key contributions toward understanding and improving SGD with preconditioning. First, we establish excess risk bounds (generalization performance) for preconditioned SGD and ridge regression under an arbitrary preconditions matrix. Second, leveraging the excessive risk characterization of preconditioned SGD and ridge regression, we show that (through construction) there exists a simple preconditioned matrix that can outperform (standard \& preconditioned) ridge regression. Finally, we show that our proposed preconditioning matrix is straightforward enough to allow robust estimation from finite samples while maintaining a theoretical advantage over ridge regression. Our empirical results align with our theoretical findings, collectively showcasing the enhanced regularization effect of preconditioned SGD.

通过对预处理的随机梯度下降（SGD）和岭回归的综合比较研究，我们建立了预处理的SGD和岭回归的过度风险界限，并证明了存在一个简单的预处理矩阵能够优于标准的和预处理的岭回归，从而展示了预处理的SGD的增强正则化效果。

使用预处理改进最小二乘问题的隐式正则化 SGD