Construct optimistic MDP fMk and compute optimistic policy πk (Algorithm 5). When the counter is 0 it gets (s,a), i.e., Ωi,e = (s,a,). When the counter is 1, we take (s,a) from ωn and map them to ωn/2 while eliminating half of the factors in consideration with the consistent scope Zi chosen by the policy (stored in factor 2d+ 1 + iof the state). It is handled similarly to the previous item, but considers the reward consistent scope zj chosen by the policy (stored in factor 3d+ 1 + j of the state). For i = 1,...,d, the i-th factor is taken from factor i of the previous state when the counter is not log n + 1, and otherwise performs the optimistic transition of factor i. Denote the value in the last factor of Ωi,e by ve, the policy's chosen scope by Zi (stored in factor 2d+ 1 + iof the state) and the policy's chosen next state direction by s0i (stored in factor d+ 1 + iof the state).

Heterophilous graphs, where dissimilar nodes tend to connect, pose a challenge for graph neural networks (GNNs) as their superior performance typically comes from aggregating homophilous information. Increasing the GNN depth can expand the scope (i.e., receptive field), potentially finding homophily from the higher-order neighborhoods. However, uniformly expanding the scope results in subpar performance since real-world graphs often exhibit homophily disparity between nodes. An ideal way is personalized scopes, allowing nodes to have varying scope sizes. Existing methods typically add node-adaptive weights for each hop. Although expressive, they inevitably suffer from severe overfitting. To address this issue, we formalize personalized scoping as a separate scope classification problem that overcomes GNN overfitting in node classification. Specifically, we predict the optimal GNN depth for each node. Our theoretical and empirical analysis suggests that accurately predicting the depth can significantly enhance generalization. We further propose Adaptive Scope (AS), a lightweight MLP-based approach that only participates in GNN inference. AS encodes structural patterns and predicts the depth to select the best model for each node's prediction. Experimental results show that AS is highly flexible with various GNN architectures across a wide range of datasets while significantly improving accuracy.