Remark. The general upper bound proven above is unfortunately not tight. Consider the (non-sparse) minimum bisection problem with d = n/2. Under log 2σ the same general assumptions for the weights, it can be shown that a tighter bound holds for β^ ≤ √ 1 lim E[log Z(β, X)] + βµ log m √N log m β2σ2 ^ ^
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Sources: Posterior Agreement for Large Parameter Rich Optimization Problems, Posterior Agreement for Large Parameter Rich Optimization Problems