In this document we provide supplementary materials that we are not able to fit into the main manuscriptduetothepagelimit. The dimensions of the hidden features of the three-layer GCN are set toF, F/2, and F respectively. The dataset is separated into ten parts. We generate ten validation accuracy curves when setting each of parts as the validation one. The ten curves are then averaged.

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.

Traditionally, each component of task-oriented dialogue systems is trained independently with different supervision.

Dn) 1 as n, which means the most posterior mass falls in the neighbourhood of true parameter. Remarkonthenotation: ν() is similar toν() defined in Section 2.1 of the main text. Thenotationsweusedinthis proof are the same as in the proof of Theorem 2.1. Theorem 2.2 implies that a faithful prediction interval can be constructed for the sparse neural network learned by the proposed algorithms. In practice, for a normal regression problem with noise N(0,σ2), to construct the prediction interval for a test pointx0, the terms σ2 and Σ = γ µ(β,x0)TH 1 γ µ(β,x0) in Theorem 2.2 need to be estimated from data.

Self-play is a technique for machine learning in multi-agent systems where a learning algorithm learns by interacting with copies of itself.

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We study first-order optimization algorithms for computing the barycenter of Gaussian distributions with respect to the optimal transport metric.