The appendix consists of proofs (Section A), additional theoretical results (Section B), and numerical experiments (Section C). When g is strictly increasing, the mapping x7 g(x) is sign preserving. Specifically, if x 0, then g(x) g(0) = 0. Conversely, ifg(x) 0 = g(0), then applying g 1 to both sides givesx 0. When g is strictly decreasing, the mappingx7 g(x) is sign reversing. See Section B.2 for constructive examples. Based on the definition of classification lossL(,), the function Risk() relies only on the sign pattern of the tensor.

See Section B.2 for constructive examples.Proof of Proposition 2. Based on (3) in Proposition 2, we have Risk( Z) Risk( Θ) = E null |sgnZ sgn Θ|| Θ|null . We divide the proof into two cases: α > 0 and α = . The inequality (6) now becomes Risk( Z) Risk( Θ) t null MAE(sgn Θ, sgnZ) C snull, for all 0 t < ρ(π, N) . Consider the same setup as in Theorem 2. Fix The conclusion (10) then directly follows by applying Remark A.1 to (11). 3 Proof of Theorem 2. To simplify the notation, we denote ρ = ρ(π, N). It follows from Kosorok (2007, Theorem 9.22) that the Proof of Theorem 3. By definition of ˆ Θ, we have MAE( ˆ Θ, Θ) = E null null null null null 1 2H + 1 null Assumption A.1, we establish the estimation accuracy guarantee for the large-margin estimators H log H. (29) In particualr, setting H null (1 + |N|) To apply Theorem A.1, we choose the pair ( L Here, we describe the details of the example set-up.

We propose an approach to modelling large-scale multi-agent dynamical systems allowing interactions among more than just pairs of agents using the theory of mean field games and the notion of hypergraphons, which are obtained as limits of large hypergraphs. To the best of our knowledge, ours is the first work on mean field games on hypergraphs. Together with an extension to a multi-layer setup, we obtain limiting descriptions for large systems of non-linear, weakly-interacting dynamical agents. On the theoretical side, we prove the well-foundedness of the resulting hypergraphon mean field game, showing both existence and approximate Nash properties. On the applied side, we extend numerical and learning algorithms to compute the hypergraphon mean field equilibria. To verify our approach empirically, we consider a social rumor spreading model, where we give agents intrinsic motivation to spread rumors to unaware agents, and an epidemics control problem.

Let V {1,..., n} be a set of n items that could represent for example, people in a social network, genes in a biological network or researchers in academic networks. Models of interaction among the n items could be conveniently represented in the form of a graph or a hypergraph, G(V, E), where the items form the nodes of the graph and the hyperedge set E represents the interactions among the items. Network datasets that capture such complex interactions between a set of objects are becoming increasingly prevalent in several scientific fields. Developing realistic generative models for such networks is a challenging problem that has been an active subject of research across diverse fields spanning from statistics, physics, computer science; see Kolaczyk (2009); Goldenberg et al. (2010); Battiston et al. (2020) for comprehensive overview. A majority of the existing work has focussed on the case of modeling pairwise interactions.