Constructive l2-Discrepancy Minimization with Additive Deviations

2025-08-29

AI Summary

This paper addresses the signed series problem under the ℓ₂ norm: given $n$ unit ℓ₂-norm vectors $v_1,dots,v_n in mathbb{R}^d$, can one construct signs $varepsilon_1,dots,varepsilon_n in {pm 1}$ such that the ℓ₂ norms of all prefix sums $sum_{j=1}^i varepsilon_j v_j$ are uniformly bounded? To meet path-control requirements with additive bias, we propose a novel random-walk framework integrating linear and spectral orthogonality constraints, coupled with covariance matrix design, sub-Gaussian chaos analysis, and an improved Hanson–Wright-type concentration inequality. Our method yields, for the first time in polynomial time, a signing achieving maximum prefix-sum norm $O(sqrt{d} + log^2 n)$. Crucially, when $d geq log^2 n$, we constructively attain the tighter bound $O(sqrt{d} + log n)$, thereby resolving two long-standing open conjectures in discrepancy theory.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

The emph{signed series} problem in the $ell_2$ norm asks, given set of vectors $v_1,ldots,v_nin mathbf{R}^d$ having at most unit $ell_2$ norm, does there always exist a series $(varepsilon_i)_{iin [n]}$ of $pm 1$ signs such that for all $iin [n]$, $max_{iin [n]} |sum_{j=1}^i varepsilon_i v_i|_2 = O(sqrt{d})$. A result of Banaszczyk [2012, emph{Rand. Struct. Alg.}] states that there exist signs $varepsilon_iin {-1,1},; iin [n]$ such that $max_{iin [n]} |sum_{j=1}^i varepsilon_i v_i|_2 = O(sqrt{d+log n})$. The best constructive bound known so far is of $O(sqrt{dlog n})$, by Bansal and Garg [2017, emph{STOC.}, 2019, emph{SIAM J. Comput.}]. We give a polynomial-time randomized algorithm to find signs $x(i) in {-1,1},; iin [n]$ such that [ max_{iin [n]} |sum_{j=1}^i x(i)v_i|_2 = O(sqrt{d + log^2 n}) = O(sqrt{d}+log n).] By the constructive reduction of Harvey and Samadi [emph{COLT}, 2014], this also yields a constructive bound of $O(sqrt{d}+log n)$ for the Steinitz problem in the $ell_2$-norm. Thus, our result settles both conjectures when $d geq log^2n$. Our algorithm is based on the framework on Bansal and Garg, together with a new analysis involving $(i)$ additional linear and spectral orthogonality constraints during the construction of the covariance matrix of the random walk steps, which allow us to control the quadratic variation in the linear as well as the quadratic components of the discrepancy increment vector, alongwith $(ii)$ a ``Freedman-like" version of the Hanson-Wright concentration inequality, for filtration-dependent sums of subgaussian chaoses.

Problem

• Minimizing additive deviations in constructive l2-discrepancy
• Finding optimal sign sequences for vector series
• Improving bounds for signed series and Steinitz problems

Innovation

• Randomized algorithm for sign selection
• Orthogonality constraints in covariance matrix
• Freedman-like Hanson-Wright concentration inequality