Optimization of Graph Neural Networks: Implicit Acceleration by Skip Connections and More Depth

Graph Neural Networks (GNNs) have been studied from the lens of expressive power and generalization. However, their optimization properties are less well understood. We take the first step towards analyzing GNN training by studying the gradient dynamics of GNNs. First, we analyze linearized GNNs and prove that despite the non-convexity of training, convergence to a global minimum at a linear rate is guaranteed under mild assumptions that we validate on real-world graphs. Second, we study what may affect the GNNs' training speed. Our results show that the training of GNNs is implicitly accelerated by skip connections, more depth, and/or a good label distribution. Empirical results confirm that our theoretical results for linearized GNNs align with the training behavior of nonlinear GNNs. Our results provide the first theoretical support for the success of GNNs with skip connections in terms of optimization, and suggest that deep GNNs with skip connections would be promising in practice.

Counting Substructures with Higher-Order Graph Neural Networks: Possibility and Impossibility Results

While massage passing based Graph Neural Networks (GNNs) have become increasingly popular architectures for learning with graphs, recent works have revealed important shortcomings in their expressive power. In response, several higher-order GNNs have been proposed, which substantially increase the expressive power, but at a large computational cost. Motivated by this gap, we introduce and analyze a new recursive pooling technique of local neighborhoods that allows different tradeoffs of computational cost and expressive power. First, we show that this model can count subgraphs of size $k$, and thereby overcomes a known limitation of low-order GNNs. Second, we prove that, in several cases, the proposed algorithm can greatly reduce computational complexity compared to the existing higher-order $k$-GNN and Local Relational Pooling (LRP) networks. We also provide a (near) matching information-theoretic lower bound for graph representations that can provably count subgraphs, and discuss time complexity lower bounds as well.

Contrastive Learning with Hard Negative Samples

We consider the question: how can you sample good negative examples for contrastive learning? We argue that, as with metric learning, learning contrastive representations benefits from hard negative samples (i.e., points that are difficult to distinguish from an anchor point). The key challenge toward using hard negatives is that contrastive methods must remain unsupervised, making it infeasible to adopt existing negative sampling strategies that use label information. In response, we develop a new class of unsupervised methods for selecting hard negative samples where the user can control the amount of hardness. A limiting case of this sampling results in a representation that tightly clusters each class, and pushes different classes as far apart as possible. The proposed method improves downstream performance across multiple modalities, requires only few additional lines of code to implement, and introduces no computational overhead.

Graph Adversarial Networks: Protecting Information against Adversarial Attacks

We study the problem of protecting information when learning with graph-structured data. While the advent of Graph Neural Networks (GNNs) has greatly improved node and graph representational learning in many applications, the neighborhood aggregation paradigm exposes additional vulnerabilities to attackers seeking to extract node-level information about sensitive attributes. To counter this, we propose a minimax game between the desired GNN encoder and the worst-case attacker. The resulting adversarial training creates a strong defense against inference attacks, while only suffering a small loss in task performance. We analyze the effectiveness of our framework against a worst-case adversary, and characterize the trade-off between predictive accuracy and adversarial defense. Experiments across multiple datasets from recommender systems, knowledge graphs and quantum chemistry demonstrate that the proposed approach provides a robust defense across various graph structures and tasks, while producing competitive GNN encoders. Our code is available at https://github.com/liaopeiyuan/GAL.

How Neural Networks Extrapolate: From Feedforward to Graph Neural Networks

We study how neural networks trained by gradient descent extrapolate, i.e., what they learn outside the support of the training distribution. Previous works report mixed empirical results when extrapolating with neural networks: while multilayer perceptrons (MLPs) do not extrapolate well in certain simple tasks, Graph Neural Network (GNN), a structured network with MLP modules, has shown some success in more complex tasks. Working towards a theoretical explanation, we identify conditions under which MLPs and GNNs extrapolate well. First, we quantify the observation that ReLU MLPs quickly converge to linear functions along any direction from the origin, which implies that ReLU MLPs do not extrapolate most non-linear functions. But, they can provably learn a linear target function when the training distribution is sufficiently "diverse". Second, in connection to analyzing successes and limitations of GNNs, these results suggest a hypothesis for which we provide theoretical and empirical evidence: the success of GNNs in extrapolating algorithmic tasks to new data (e.g., larger graphs or edge weights) relies on encoding task-specific non-linearities in the architecture or features.

Testing Determinantal Point Processes

Determinantal point processes (DPPs) are popular probabilistic models of diversity. In this paper, we investigate DPPs from a new perspective: property testing of distributions. Given sample access to an unknown distribution $q$ over the subsets of a ground set, we aim to distinguish whether $q$ is a DPP distribution, or $\epsilon$-far from all DPP distributions in $\ell_1$-distance. In this work, we propose the first algorithm for testing DPPs. Furthermore, we establish a matching lower bound on the sample complexity of DPP testing. This lower bound also extends to showing a new hardness result for the problem of testing the more general class of log-submodular distributions.

Estimating Generalization under Distribution Shifts via Domain-Invariant Representations

When machine learning models are deployed on a test distribution different from the training distribution, they can perform poorly, but overestimate their performance. In this work, we aim to better estimate a model's performance under distribution shift, without supervision. To do so, we use a set of domain-invariant predictors as a proxy for the unknown, true target labels. Since the error of the resulting risk estimate depends on the target risk of the proxy model, we study generalization of domain-invariant representations and show that the complexity of the latent representation has a significant influence on the target risk. Empirically, our approach (1) enables self-tuning of domain adaptation models, and (2) accurately estimates the target error of given models under distribution shift. Other applications include model selection, deciding early stopping and error detection.

Debiased Contrastive Learning

A prominent technique for self-supervised representation learning has been to contrast semantically similar and dissimilar pairs of samples. Without access to labels, dissimilar (negative) points are typically taken to be randomly sampled datapoints, implicitly accepting that these points may, in reality, actually have the same label. Perhaps unsurprisingly, we observe that sampling negative examples from truly different labels improves performance, in a synthetic setting where labels are available. Motivated by this observation, we develop a debiased contrastive objective that corrects for the sampling of same-label datapoints, even without knowledge of the true labels. Empirically, the proposed objective consistently outperforms the state-of-the-art for representation learning in vision, language, and reinforcement learning benchmarks. Theoretically, we establish generalization bounds for the downstream classification task.