aiqbench analysis
This paper establishes nearly quadratic lower bounds for randomized algorithms solving linear optimization and uniform sampling over convex bodies, effectively closing the gap between theoretical limits and existing upper bounds in high-dimensional spaces. By proving that volume estimation shares this complexity constraint, the work provides fundamental insights into the computational hardness of core machine learning tasks that rely on these geometric primitives.
A Nearly Quadratic Lower Bound for Linear Optimization over Convex Bodies in the Membership Oracle Model
arXiv:2609.30215v1 Announce Type: cross Abstract: We prove nearly quadratic lower bounds for randomized algorithms for linear optimization and uniform sampling over convex bodies in the membership oracle model. For linear optimization, this matches the known nearly quadratic upper bound up to a polylog factor in the dimension. For uniform sampling, this improves on the previous linear lower bound. Our construction also implies the same lower bound for volume estimation.
Abstract:We prove nearly quadratic lower bounds for randomized algorithms for linear optimization and uniform sampling over convex bodies in the membership oracle model. For linear optimization, this matches the known nearly quadratic upper bound up to a polylog factor in the dimension. For uniform sampling, this improves on the previous linear lower bound. Our construction also implies the same lower bound for volume estimation.
Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG); Functional Analysis (math.FA); Optimization and Control (math.OC)
arXiv:2609.30215 [cs.DS]
(or arXiv:2609.30215v1 [cs.DS] for this version)
https://doi.org/10.48550/arXiv.2609.30215 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 170 words
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