While second order optimizers such as natural gradient descent (NGD) often speed up optimization, their effect on generalization remains controversial. For instance, it has been pointed out that gradient descent (GD), in contrast to second-order optimizers, converges to solutions with small Euclidean norm in many overparameterized models, leading to favorable generalization properties. In this work, we question the common belief that first-order optimizers generalize better. We provide a precise asymptotic bias-variance decomposition of the generalization error of overparameterized ridgeless regression under a general class of preconditioner $\boldsymbol{P}$, and consider the inverse population Fisher information matrix (used in NGD) as a particular example. We characterize the optimal $\boldsymbol{P}$ for the bias and variance, and find that the relative generalization performance of different optimizers depends on the label noise and the "shape" of the signal (true parameters). Specifically, when the labels are noisy, the model is misspecified, or the signal is misaligned with the features, NGD can generalize better than GD. Conversely, in the setting with clean labels, a well-specified model, and well-aligned signal, GD achieves better generalization. Based on this analysis, we consider several approaches to manage the bias-variance tradeoff, and find that interpolating between GD and NGD may generalize better than either algorithm. We then extend our analysis to regression in the reproducing kernel Hilbert space and demonstrate that preconditioned GD can decrease the population risk faster than GD. In our empirical comparisons of first- and second-order optimization of neural networks, we observe robust trends matching our theoretical analysis.

此研究探讨一些最优化方法（包括用于神经网络的一阶梯度下降和二阶自然梯度下降）的隐式偏差如何影响其泛化性能，并提出了管理偏差方差的几种方法及在回归问题中的应用。

预处理何时有助于或损害泛化能力？