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\title{Tuning Algorithmic and Architectural Hyperparameters in Graph-Based Semi-Supervised Learning with Provable Guarantees}

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\author[1]{Ally Yalei Du}
\author[1]{Eric Huang}
\author[2]{Dravyansh Sharma}
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    Carnegie Mellon University
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    Toyota Technological Institute at Chicago
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\begin{abstract}
Graph-based semi-supervised learning is a powerful paradigm in machine learning for modeling and exploiting the underlying graph structure that captures the relationship between labeled and unlabeled data. A large number of classical as well as modern deep learning based algorithms have been proposed for this problem, often having tunable hyperparameters. 
We initiate a formal study of tuning algorithm hyperparameters from parameterized algorithm families for this problem.
We obtain novel $O(\log n)$ pseudo-dimension upper bounds for hyperparameter selection in three classical label propagation-based algorithm families, where $n$ is the number of nodes, implying bounds on the amount of data needed for learning provably good parameters.
We further provide matching $\Omega(\log n)$ pseudo-dimension lower bounds, thus asymptotically characterizing the learning-theoretic complexity of the parameter tuning problem.
We extend our study to selecting architectural hyperparameters in modern graph neural networks. We bound the Rademacher complexity for tuning the self-loop weighting in recently proposed Simplified Graph Convolution (SGC) networks.
We further propose a tunable architecture that interpolates graph convolutional neural networks (GCN) and graph attention networks (GAT) in every layer, and provide Rademacher complexity bounds for tuning the interpolation coefficient. 
\end{abstract}

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\section{Conclusion}
We study the problem of hyperparameter tuning in graph-based semi-supervised learning for both classical label-propagation based techniques as well as modern deep learning algorithms. For the former, we obtain tight learning guarantees by bounding the pseudo-dimension of the relevant loss function classes. For the latter, we study a novel interpolation of convolutional and attention based graph neural network architectures and provide data-dependent bounds on the complexity of tuning the  hyperparameter the interpolates the two architectures. We obtain a sharper generalization error bound for tuning the hyperparameter in the simplified graph convolutional networks proposed in prior work. Our experiments indicate that we can achieve consistently good empirical performance across datasets by tuning the interpolation parameter.

An interesting direction for further investigation involves improving computational efficiency. \citet{sharma2023efficiently} introduced techniques for approximating loss functions to reduce the cost of tuning graph kernel hyperparameters. It would be worthwhile to explore whether their methods can be adapted to our setting to alleviate computational burdens.

Another natural extension is the tuning of multiple hyperparameters. Although this increases analytical complexity, we anticipate that our techniques remain applicable. For GNN-based algorithms, Rademacher Complexity may still provide a suitable capacity measure. Our approach would aim to bound the variation in predicted values as hyperparameters change and then apply a covering argument. For label propagation methods, we would instead analyze how the scoring matrix evolves with respect to hyperparameter shifts. These directions offer a compelling foundation for extending our techniques to more complex tuning scenarios.

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\title{Supplementary Material}
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\appendix
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