WHEN DOES PRECONDITIONING HELP OR HURT GEN-ERALIZATION?

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 implicit bias of optimizers affects the comparison of generalization properties. We provide an exact asymptotic biasvariance decomposition of the generalization error of preconditioned ridgeless regression in the overparameterized regime, and consider the inverse population Fisher information matrix (used in NGD) as a particular example. We determine the optimal preconditioner 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 with the features, 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 several approaches to manage the bias-variance tradeoff, and the potential 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 preconditioning can lead to more efficient decrease in the population risk. Lastly, we empirically compare the generalization error of first-and second-order optimizers in neural network experiments, and observe robust trends matching our theoretical analysis.

1. INTRODUCTION

We study the generalization property of an estimator θ obtained by minimizing the empirical risk (or the training error) L(f θ ) via a preconditioned gradient update with preconditioner P : θ t+1 = θ t -ηP (t)∇ θt L(f θt ), t = 0, 1, . . . (1.1) Setting P = I recovers gradient descent (GD). Choices of P which exploit second-order information include the inverse Fisher information matrix, which gives the natural gradient descent (NGD) (Amari, 1998); the inverse Hessian, which leads to Newton's method (LeCun et al., 2012) ; and diagonal matrices estimated from past gradients, which include various adaptive gradient methods (Duchi et al., 2011; Kingma & Ba, 2014) . These preconditioners often alleviate the effect of pathological curvature and speed up optimization, but their impact on generalization has been under debate: Wilson et al. (2017) reported that in neural network optimization, adaptive or secondorder methods generalize worse compared to gradient descent (GD), whereas other empirical studies showed that second-order methods achieve comparable, if not better generalization (Xu et al., 2020) . The generalization property of optimizers relates to the discussion of implicit bias (Gunasekar et al., 2018a), i.e. preconditioning may lead to a different converged solution (with potentially the same training loss), as illustrated in Figure 1 . While many explanations have been proposed, our starting point is the well-known observation that GD often implicitly regularizes the parameter 2 norm. For instance in overparameterized least squares regression, GD and many first-order methods find the minimum 2 norm solution from zero initialization (without explicit regularization), but preconditioned updates may not. This being said, while the minimum 2 norm solution can generalize well

