The widely studiedGeneralized Min-Sum-Set-Cover(GMSSC) problem serves as a formal model for the setting above. GMSSC is NP-hard and the standard application ofno-regretonline learning algorithms iscomputationally inefficient, because they operate in the space of rankings. In this work, we show how to achievelowregret for GMSSC inpolynomial-time.

Traditional solutions prioritizing easy or hard samples lack the flexibility to handle such a variety simultaneously. Recent work has proposed a more reasonable data selection principle by examining the data's impact on the model's generalization loss.

To resolve the first challenge, we propose to sample anestimate ofthefirst-order andsecond-order derivativestoapproximate theoptimality andKKT conditions.

Real-world problems such as ad allocation and matching havebeen extensively studied under the lens of combinatorial optimization. In several applications, uncertainty in the input appears naturally and this has led to the study of online stochastic optimization models for such problems.

Now we look into the term[(A+I)X]TV,:, which is the aggregated feature vectors within neighborhood N1 for nodes in the training set. Note that [(A + I)X]TS,: is a circulant matrix, therefore its inverse exists. Now consider an arbitrary training datapoint(v,yv) TV, and a perturbation added to the neighborhood N(v) of node v, such that the number of nodes with a randomly selected class labelyp Y 6=yv isδ1lessthanexpectedin N(v). Now we move on to discuss the GCN layer formulated asf(X;A,W) = AXW without self loops. We regardcs,i as the coefficient ofs at frequency componenti and regard the coefficients at all frequencies components{cs,i} as the spectrum of signalswith respect to graphG.