Optimal score estimation via empirical Bayes smoothing

We study the problem of estimating the score function of an unknown probability distribution ρ^* from n independent and identically distributed observations in d dimensions. Assuming that ρ^* is subgaussian and has a Lipschitz-continuous score function s^*, we establish the optimal rate of Θ̃(n^-2/d+4) for this estimation problem under the loss function ŝ - s^*^2_L^2(ρ^*) that is commonly used in the score matching literature, highlighting the curse of dimensionality where sample complexity for accurate score estimation grows exponentially with the dimension d. Leveraging key insights in empirical Bayes theory as well as a new convergence rate of smoothed empirical distribution in Hellinger distance, we show that a regularized score estimator based on a Gaussian kernel attains this rate, shown optimal by a matching minimax lower bound. We also discuss the implication of our theory on the sample complexity of score-based generative models.