Graph Neural Networks (GNNs) extend basic Neural Networks (NNs) by using graph structures based on the relational inductive bias (homophily assumption). While GNNs have been commonly believed to outperform NNs in real-world tasks, recent work has identified a non-trivial set of datasets where their performance compared to NNs is not satisfactory. Heterophily has been considered as the main cause of this empirical observation and numerous works have been put forward to address it. In this paper, we first revisit the widely used homophily metrics and point out that their consideration of only graph-label consistency is a shortcoming. Then, we study heterophily from the perspective of post-aggregation node similarity and define new homophily metrics, which are verified to be advantageous compared to existing ones. Based on this investigation, we prove that some harmful cases of heterophily can be effectively addressed by local diversification operation. Then, we propose the Adaptive Channel Mixing (ACM), a framework to adaptively exploit aggregation, diversification and identity channels node-wisely to extract richer localized information for diverse node heterophily situations. ACM is more powerful than the commonly used uni-channel framework for node classification tasks on heterophilic graphs and is easy to be implemented in baseline GNN layers. When evaluated on 10 benchmark node classification tasks, ACM-augmented baselines consistently achieve significant performance gain, exceeding state-of-theart GNNs on most tasks without incurring significant computational burden.

We develop a framework for analyzing the training and learning rate dynamics on a large class of high-dimensional optimization problems, which we call the high line, trained using one-pass stochastic gradient descent (SGD) with adaptive learning rates. We give exact expressions for the risk and learning rate curves in terms of a deterministic solution to a system of ODEs. We then investigate in detail two adaptive learning rates -- an idealized exact line search and AdaGrad-Norm -- on the least squares problem. When the data covariance matrix has strictly positive eigenvalues, this idealized exact line search strategy can exhibit arbitrarily slower convergence when compared to the optimal fixed learning rate with SGD. Moreover we exactly characterize the limiting learning rate (as time goes to infinity) for line search in the setting where the data covariance has only two distinct eigenvalues. For noiseless targets, we further demonstrate that the AdaGrad-Norm learning rate converges to a deterministic constant inversely proportional to the average eigenvalue of the data covariance matrix, and identify a phase transition when the covariance density of eigenvalues follows a power law distribution.

We evaluate Okapi using three datasets - iWildCam, PovertyMap, and CivilComments - taken from the WILDS 2.0 benchmark [63]. These datasets were chosen specifically due to the poor performance reported by [63] for semi-supervised and domain adaptation methods across the board, in relation to the ERM baselines. For PovertyMap in particular, ERM was found to vastly outperform any competing methods utilising the unlabelled data and/or domain labels. The task is multiclass species classification of animals in camera trap images. The dataset contains 1022K images of animals annotated with the domain, s, that identifies the camera trap that captured it.

If there is a bingo on mode-k, the m-th row of the mode-k expansion of P is a constant multiple of the (m 1)-th row, where mis a number determined by the bingo position. When a row is a constant multiple of another row, the rank of the matrix is reduced by a maximum of one, which means Rank(P(k)) Ik 1. In the same way, if there are bk bingos, then bk rows are constant multiple of the other rows, which means Rank(P(k)) Ik bk. For any positive tensor P, rank(P) = 1 if and only if its all many-body θparameters are 0. Proof. First, we show that rank(P) = 1 implies all many-body θ-parameters are 0. From the assumption of rank(P) = 1, the m-th row of the mode-k expansion of P have to be a constant multiple of the (m 1)-th row for all m= {2,...,Ik}and k [d].

We investigate the parameterized complexity of Bayesian Network Structure Learn-1 ing (BNSL), a classical problem that has received significant attention in empirical2 but also purely theoretical studies. We follow up on previous works that have3 analyzed the complexity of BNSL w.r.t. the so-called superstructure of the input.4 While known results imply that BNSL is unlikely to be fixed-parameter tractable5 even when parameterized by the size of a vertex cover in the superstructure, here we6 show that a different kind of parameterization--notably by the size of a feedback7 edge set--yields fixed-parameter tractability. We proceed by showing that this8 result can be strengthened to a localized version of the feedback edge set, and9 provide corresponding lower bounds that complement previous results to provide a10 complexity classification of BNSL w.r.t.

We provide an extended version of the Experimental Setup from Section 5 below. Linear Model This domain involves learning a linear model when the underlying mapping between features and predictions is cubic. Concretely, the aim is to choose the top B =1 out of N = 50 resources using a linear model. The fact that the features can be seen as 1-dimensional allows us to visualize the learned models (as seen in Figure 4). Predict: Given a feature xn U[0,1], use a linear model to predict the utility ˆyof choosing resource n, where the true utility is given by yn = 10x3n 6.5xn.