Supplementary material for: Convergence Rate Analysis of MAP Coordinate Minimization Algorithms
نویسندگان
چکیده
Proof. ‖∇F (δ)‖∞ ≤ guarantees that μ = μ(δ) are -consistent in the sense that |μi(xi) − μc(xi)| ≤ for all c, i ∈ c and xi. Algorithm 1 maps any such -consistent μ to locally consistent marginals μ̃ such that |μi(xi)− μ̃i(xi)| ≤ 3 Nmax, |μc(xc)− μ̃c(xc)| ≤ 2 N max, (3) for all i, xi, c, and xc, where Nmax = max{maxiNi,maxcNc}. In other words, ‖μ− μ̃‖∞ ≤ K . This can be easily derived from the update in Algorithm 1 and the fact that |μi(xi)− μc(xi)| ≤ . Next, it can be shown that F (δ) = Pτ (μ(δ)). And it follows that P ∗ τ ≤ F (δ) ≤ Pτ (μ), where the first inequality follows from weak duality. Thus we have: P ∗ τ ≤ Pτ (μ) = μ · θ + 1 τ H(μ) = (μ̃+ μ− μ̃) · θ + 1 τ H(μ̃) + 1 τ (H(μ)−H(μ̃)) (4)
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