From our theoretical analysis and empirical evaluation, we conclude that the model selection procedure needs tobe guided by careful considerations regardingtheobserveddata,thefactorsusedforcorrection,andthestructureofthe data-generatingprocess.

Inparticular,forany function in a classC of bounded VC dimension, we guarantee a low test error rate and a low rejection ratewith respect toP. Our algorithm is efficient given an Empirical Risk Minimizer (ERM) forC.

Graph Contrastive Learning (GCL) has shown superior performance in representation learning in graph-structured data. Despite their success, most existing GCL methods rely on prefabricated graph augmentation and homophily assumptions. Thus, they fail to generalize well to heterophilic graphs where connected nodes may have different class labels and dissimilar features.

These assumptions are not strong and can be satisfied in most of environments includes MuJoCo, Atarigamesandsoon. Let f be an Lebesgue integrable function, P and Q are two probability distributions, |f| C,then EP(x)f(x) EQ(x)f(x) CDTV(P,Q) (5) Proof. Suppose there are two actions a1, a2 under state s, and let Q1(s,a1) = u, Q1(s,a2) = v. In this way, we can derive the upper bound of Ea π2Q1(s,a) Ea π1Q1(s,a)asabove. Since both sides of the above equation have the same minimum (here the minima are given by Qk = Q), we can replace the objective in Problem 2 with the upper bound in Eq. (10) and solve therelaxedoptimizationproblem.