The proof of infeasibility condition (Theorem 3.2) is provided in Section B. Explanations on conditions derived in Theorem 3.2 are included in Section C. The proof of properties of the proposed model (r)LogSpecT (Proposition 3.4 The truncated Hausdorff distance based proof details of Theorem 4.1 and Corollary 4.4 are Details of L-ADMM and its convergence analysis are in Section F. Additional experiments and discussions on synthetic data are included in Section G. ( m 1) Again, from Farkas' lemma, this implies that the following linear system does not have a solution: Example 3.1 we know δ = 2|h Since the constraint set S is a cone, it follows that for all γ > 0, γ S = S . Opt(C, α) = α Opt(C, 1), which completes the proof. The proof will be conducted by constructing a feasible solution for rLogSpecT. Since the LogSpecT is a convex problem and Slater's condition holds, the KKT conditions We show that it is feasible for rLogSpecT. R, its epigraph is defined as epi f: = {( x, y) | y f ( x) }. Before presenting the proof, we first introduce the following lemma.

First, note that we shall use the same MDPs defined in Appendix D.1 as follows

As such, in games where the marginalization of CCE coincides with the game's Nash equilibria (like

The following proof is from Daniely and V ardi [15], and we give it here for completeness. By Lemma A.1, there exists a DNF formula We construct such an affine layer in Lemma A.2. At least one of the k size-n slices in z contains 0 more than once. We define the outputs of our affine layer as follows. Pr [z represents a hyperedge ] = n (n 1) ... (n k + 1) null 1 n null Pr null z Z null 1 2 log(n) .

Gaussian, and the weight matrix is non-degenerate.

Let k (,) be the ν = 2 .5 Matérn kernel.