The Optimizer's Windfall: Bounding Hindsight Gain under Random Restrictions

Suppose an optimization problem is randomly restricted, independently on each variable, and re-optimized after the restriction is revealed. For multilinear objectives, every fixed solution is unbiased after rescaling, but re-optimizing introduces an upward bias. We bound this excess, the hindsight gain, over products of arbitrary compact local domains in Euclidean unit balls. For interactions A=(Aₑ)ₑ among r distinct variables and independent random weights ζᵢ, the gain is at most Oᵣ(\|A\|F M(r-1)/2σₘₐₓ), where \|A\|F² is the sum of squared coefficients, M=∑ᵢEζᵢ², and σₘₐₓ²=maxᵢVar(ζᵢ); only second moments are needed. Under uniform retention with probability p, we sharpen this to Oᵣ(pr-1/2√1-p\,n(r-1)/2\|A\|F), which is tight for every fixed r and p∈(0,1). Our proof combines a comparison principle along a single support direction with tensor projection, vector contraction, and repeated thinning. Applications: (i) For fixed arity and alphabet size, solving the subproblem induced by O(ε⁻²) sampled variables estimates a dense Max-rCSP optimum within ±ε nʳ, and this sample size is optimal. (ii) The same approach estimates MaxCut and the distance to bipartiteness within ±ε n² using O(ε⁻⁴) nonadaptive queries and O(ε⁻²) examined vertices, both optimal. (iii) We answer a question of Yufei Zhao: vertex percolation at rate p preserves the max-cut fraction whenever p\,e(G)/v(G)→∞, and this condition is sharp. (iv) For bounded r-local quantum Hamiltonians of fixed local dimension, O(ε⁻³) uniformly sampled particles suffice to estimate the ground-state energy within ±ε nʳ, improving the previous O(ε⁻⁶log(1/ε)) bound.