Thus, we verify that the random effects estimator is equivalent to the generative model (1). Specifically, if u(x) = 1 for all x X, we use ( X, P,ψ) for simplicity. Due to the separability of ψ, we have the following coreset definition. Definitions 2.2 and 2.3, the regression objectives of OLSE and GLSE can be decomposed into Thus, we can apply the above definition to define coresets for OLSE and GLSE. Now we are ready to describe the FL framework in the language of a query space. We first prove Theorem C.1 and propose the corresponding algorithm that constructs an Next, we prove Theorem C.2 and propose the corresponding algorithm that constructs an accurate Caratheodory's Theorem, there must exist at most To accelerate the running time, Jubran et al. [ By the Caratheodory's Theorem, there must exist at most In this section, we complete the proofs for GLSE.

This material is organized as follows. Section 1 presents more numerical Results.

We give proofs for the theorems and propositions in the order they appeared in the main paper. We prove the more detailed claim. Proposition 1 is a part of the following proposition. The followings are equivalent 1. There exist α,β such that α > β 0 and Z satisfies ( α,β, g)-w.l.c. 2. nullZ, g null > 0 .

We introduce a novel class of noise distributions for approximating marginals via perturb-and-MAP .

Table 4: The detailed sampling hyperparameters for Chameleon.

The Π denotes the set of measure couplings whose marginals are P and Q, respectively. "blank" augmentation, which is an important property for proving the main theorem. Let d be a metric on X and 0 X . Therefore, we prove that the equality case must hold by contradiction. But practically the augmentations are specified to be different from the elements in the sets.

We introduce a novel class of noise distributions for approximating marginals via perturb-and-MAP .

Table 4: The detailed sampling hyperparameters for Chameleon.

In this paper, we study the inexact Moreau envelope Lagrangian (iMELa) method for solving smooth non-convex optimization problems over a simple polytope with additional convex inequality constraints. By incorporating a proximal term into the traditional Lagrangian function, the iMELa method approximately solves a convex optimization subproblem over the polyhedral set at each main iteration. Under the assumption of a local error bound condition for subsets of the feasible set defined by subsets of the constraints, we establish that the iMELa method can find an $\epsilon$-Karush-Kuhn-Tucker point with $\tilde O(\epsilon^{-2})$ gradient oracle complexity.