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Workshop: OPT2020: Optimization for Machine Learning

Contributed Video: When Does Preconditioning Help or Hurt Generalization?, Denny Wu

Denny Wu

Abstract: While second order optimizers such as natural gradient descent (NGD) often speed up optimization, their effect on generalization has been called into question. This work presents a more nuanced view on how the \textit{implicit bias} of optimizers affects the comparison of generalization properties. We provide an exact 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 determine the optimal $\boldsymbol{P}$ for both the bias and variance, and find that the relative generalization performance of different optimizers depends on label noise and ``shape'' of the signal (true parameters): when the labels are noisy, the model is misspecified, or the signal is misaligned, NGD can achieve lower risk; conversely, GD generalizes better under clean labels, a well-specified model, or aligned signal. Based on this analysis, we discuss approaches to manage the bias-variance tradeoff, and the benefit of interpolating between first- and second-order updates. 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. Lastly, we empirically compare the generalization error of first- and second-order optimizers in neural network, and observe robust trends matching our theoretical analysis.