This paper provides statistical sample complexity bounds for score-matching and its applications in causal discovery. We demonstrate that accurate estimation of the score function is achievable by training a standard deep ReLU neural network using stochastic gradient descent. We establish bounds on the error rate of recovering causal relationships using the score-matching-based causal discovery method of Rolland et al. [2022], assuming a sufficiently good estimation of the score function. Finally, we analyze the upper bound of score-matching estimation within the scorebased generative modeling, which has been applied for causal discovery but is also of independent interest within the domain of generative models.

This paper provides statistical sample complexity bounds for score-matching and its applications in causal discovery. We demonstrate that accurate estimation of the score function is achievable by training a standard deep ReLU neural network using stochastic gradient descent. We establish bounds on the error rate of recovering causal relationships using the score-matching-based causal discovery method of Rolland et al. [2022], assuming a sufficiently good estimation of the score function. Finally, we analyze the upper bound of score-matching estimation within the scorebased generative modeling, which has been applied for causal discovery but is also of independent interest within the domain of generative models.

A.1 Proof of Monotonicity and Submodularity In Equation (3a), we stated the objective of the knapsack cover to be Remark 1. f+M is monotonically increasing. A.2 Knapsack Cover To find a solution to problem 3, we use the greedy algorithm proposed by Badanidiyuru and Vondrák [2], which deals with submodular maximization subject to a system of lknapsack constraints and with pmatroid constraints. We present an adapted version of the algorithm in Algorithm 2 where l = 1. Theparameter allows us to 16 trade-off solution time and solution quality. In this work, we set = 0.2.