The Optimizer's Windfall: Bounding Hindsight Gain under Random Restrictions
Abstract
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.
Related
- Failure Is the Mother of Success: Recovering Supervision with Grounding Hindsight
- Brauer--Manin obstruction for Weil restrictions
- Dual mixing and quasi-mixing of inner function restrictions
- On partitions with bounded multiplicity restrictions on even parts and with the largest part being odd
- Restrictions of Infinite Circuits
BibTeX
@misc{bei2026optimizers,
title = {{The Optimizer's Windfall: Bounding Hindsight Gain under Random Restrictions}},
author = {Xiaohui Bei and Wenhao Wu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03468},
eprint = {2611.03468},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03468},
}
RIS
TY - UNPB
TI - The Optimizer's Windfall: Bounding Hindsight Gain under Random Restrictions
AU - Xiaohui Bei
AU - Wenhao Wu
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03468
UR - https://arxiv.org/abs/2611.03468
AB - 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=(A_e)_e among r distinct variables and independent random weights ζ_i, the gain is at most O_r(|A|_F M^(r−1)/2σ_max), where |A|_F^2 is the sum of squared coefficients, M=∑_i𝔼ζ_i^2, and σ_max^2=max_iVar(ζ_i); only second moments are needed. Under uniform retention with probability p, we sharpen this to O_r(p^r−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 n^r, and this sample size is optimal. (ii) The same approach estimates MaxCut and the distance to bipartiteness within ±ε n^2 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 n^r, improving the previous O(ε^−6log(1/ε)) bound.
ER -