We analyze graph smoothing with mean aggregation, where each node successively receives the average of the features of its neighbors. Indeed, it has quickly been observed that Graph Neural Networks (GNNs), which generally follow some variant of Message-Passing (MP) with repeated aggregation, may be subject to the oversmoothing phenomenon: by performing too many rounds of MP, the node features tend to converge to a non-informative limit. In the case of mean aggregation, for connected graphs, the node features become constant across the whole graph. At the other end of the spectrum, it is intuitively obvious that some MP rounds are necessary, but existing analyses do not exhibit both phenomena at once: beneficial "finite" smoothing and oversmoothing in the limit. In this paper, we consider simplified linear GNNs, and rigorously analyze two examples for which a finite number of mean aggregation steps provably improves the learning performance, before oversmoothing kicks in. We consider a latent space random graph model, where node features are partial observations of the latent variables and the graph contains pairwise relationships between them. We show that graph smoothing restores some of the lost information, up to a certain point, by two phenomena: graph smoothing shrinks non-principal directions in the data faster than principal ones, which is useful for regression, and shrinks nodes within communities faster than they collapse together, which improves classification.

In all environments, we use a Transformer architecture with four layers and four self-attention heads. The total input vocabulary of the model is V (N + M +2) to account for states, actions, rewards, and rewards-to-go, but the output linear layer produces logits only over a vocabulary of size V; output tokens can be interpreted unambiguously because their offset is uniquely determined by that of the previous input. The dimension of each token embedding is 128. Dropout is applied at the end of each block with probability 0.1. We follow the learning rate scheduling of (Radford et al., 2018), increasing linearly from 0 to 2.5 10 4 over the course of 2000 updates.

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.