We present a weighted-majority classification approach over subtrees of a fixed tree, which provably achieves excess-risk of the same order as the best tree-pruning. Furthermore, the computational efficiency of pruning is maintained at both training and testing time despite having to aggregate over an exponential number of subtrees. We believe this is the first subtree aggregation approach with such guarantees.

Note that computing bR value can be done in constant time ifWp and Wn values are given. We stress that this result holds for any loss functionℓ satisfying ℓ(v,y) > ℓ(y,y) 0, with v =y. We performed additional experiments to empirically investigate the difference between uPU and nnPU risk estimators in regards to overfitting. In Table 11 we report the training risks (measured 19 asPUriskasdataisPU)andtesting risks(measured asPNriskasdataisPN)using zero-one loss ℓ0/1(v,y)=(1 sign(vy))/2onanumberofdatasets. From the results we can see that the training risk issignificantly smaller than the test risk in the uPU setting as compared to the nnPU setting, confirming that uPU suffers more from overfittingthannnPU. Table11: TrainingandtestingriskofPUET. Figure 4shows that the normalized risk reduction importance makes manymore pixels more important.

Graph Anomaly Detection (GAD) aims to identify uncommon, deviated, or suspicious objects within graph-structured data. Existing methods generally focus on a single graph object type (node, edge, graph, etc.) and often overlook the inherent connections among different object types of graph anomalies. For instance, a money laundering transaction might involve an abnormal account and the broader community it interacts with. To address this, we present UniGAD, the first unified framework for detecting anomalies at node, edge, and graph levels jointly. Specifically, we develop the Maximum Rayleigh Quotient Subgraph Sampler (MRQSampler) that unifies multi-level formats by transferring objects at each level into graph-level tasks on subgraphs.

Most open online courses on programming makeuse ofautomated grading systems tosupport programming assignments and give real-time feedback. These systems usually rely on test results toquantify theprograms' functional correctness.

In order to perform a wide search over the space of the trees, it is critical to solve this projection problem fast.

Despite recent advances in automated machine learning, model selection is still acomplexandcomputationally intensiveprocess.

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