All experiments were conducted on a Linux server with a Tesla P40 GPU. Our Contrastive Graph Poisson Network (CGPN) was implemented via PyTorch 1.4.0 [1]. We adopted the Adam optimizer [2] for training. The number of GAT layers was set as two. Other hyperparameters were adjusted based on the corresponding datasets. Tables 1, 2, 3, and 4 provide the details of the important hyperparameters.

Graph Neural Networks (GNNs) have achieved remarkable performance in the task of semi-supervised node classification. However, most existing GNN models require sufficient labeled data for effective network training. Their performance can be seriously degraded when labels are extremely limited. To address this issue, we propose a new framework termed Contrastive Graph Poisson Networks (CGPN) for node classification under extremely limited labeled data. Specifically, our CGPN derives from variational inference; integrates a newly designed Graph Poisson Network (GPN) to effectively propagate the limited labels to the entire graph and a normal GNN, such as Graph Attention Network, that flexibly guides the propagation of GPN; applies a contrastive objective to further exploit the supervision information from the learning process of GPN and GNN models. Essentially, our CGPN can enhance the learning performance of GNNs under extremely limited labels by contrastively propagating the limited labels to the entire graph. We conducted extensive experiments on different types of datasets to demonstrate the superiority of CGPN.

Calculation of T Given data D, disaggregate Y into M equal-size bins, and the m-th bin is denoted as Bm. Let m = |Bm| denote the number of samples in Bm. For distribution p 2 (V A Y) conditioned on y in Bm, pV,A|ym, pV|ym and pA|ym are denoted as the joint distribution of (V,A), marginal distribution of V and A, respectively. As detailed in Section 5.1 of [33] and Algorithm 4 of [32], Um could be calculated through U-statistic. Specifically, in [33], they consider designing kernel as ij(av)= I(Ai = a,Vi = v) I(Ai = a)I(Vi = v), for i and j-th sample in Dt.

We study fair machine learning (ML) under predictive uncertainty to enable reliable and trustworthy decision-making. The seminal work of "equalized coverage" proposed an uncertainty-aware fairness notion. However, it does not guarantee equal coverage rates across more fine-grained groups (e.g., low-income females) conditioning on the true label and is biased in the assessment of uncertainty. To tackle these limitations, we propose a new uncertainty-aware fairness - Equal Opportunity of Coverage (EOC) - that aims to achieve two properties: (1) coverage rates for different groups with similar outcomes are close, and (2) the coverage rate for the entire population remains at a predetermined level. Further, the prediction intervals should be narrow to be informative. We propose Binned Fair Quantile Regression (BFQR), a distribution-free post-processing method to improve EOC with reasonable width for any trained ML models. It first calibrates a hold-out set to bound deviation from EOC, then leverages conformal prediction to maintain EOC on a test set, meanwhile optimizing prediction interval width. Experimental results demonstrate the effectiveness of our method in improving EOC.

Define an agent's current state information as s = (x,y,θ,vx,vy), which includes the x,y positions in the coordinate space, and the yaw angle θ, and the velocities in the X and Y directions. The inverse kinematics can be used to calculate the actions for behavior cloning purpose. With the Bicycle action space, we propose a model to approximate the vehicle dynamics with the goal of minimizing the discrepancy between the predicted vehicle states and the recorded vehicle states. More specifically, define the vehicle's coordinates as x,y in the global coordinate system, and the predicted coordinates as ˆx,ˆy, the goal is to minimize (x ˆx)2 + (y ˆy)2. Define the current vehicle's state information as s, which includes the coordinates of the vehicle in the global coordinate system (x,y), the vehicle's yaw angle θ, the vehicle's speed in the x and y direction vx,vy.