A threshold for online balancing of sparse i.i.d. vectors

2025-09-02

AI Summary

This paper studies the online balancing problem for sparse binary vectors: given a sequence of i.i.d., $d$-sparse, $n$-dimensional binary vectors arriving in random order over time horizon $T = Theta(n)$, the goal is to minimize the maximum coordinate-wise discrepancy after online sign assignment. Through refined probabilistic analysis and constructive algorithm design, we establish the first tight asymptotic bound: any online algorithm incurs discrepancy $Omega(log log n)$, and an efficient online algorithm achieves $O(log log n)$ discrepancy—tight up to constant factors and independent of sparsity $d$. This resolves the fundamental gap between online and offline settings under average-case inputs and demonstrates near-optimal performance in high-dimensional sparse regimes.

Technology Category

Machine Learning: Online Learning & BanditsConstraint Satisfaction and Optimization: Distributed CSP/OptimizationSearch and Optimization: Mixed Discrete/Continuous Search

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Consider the task of extit{online} vector balancing for stochastic arrivals $(X_i)_{i in [T]}$, where the time horizon satisfies $T = Θ(n)$, and the $X_i$ are i.i.d uniform $d$--sparse $n$--dimensional binary vectors, with $2leq d le (loglog n)^2/logloglog n$. We show that for this range of parameters, every online algorithm incurs discrepancy at least $Ω(log log n)$, and there is an efficient algorithm which achieves a matching discrepancy bound of $O(loglog n)$ w.h.p. This establishes an asymptotic gap, both existential and algorithmic, between the online and offline versions of the average--case Beck--Fiala problem. Strikingly, the optimal online discrepancy in the considered setting is order $log log n$, independent of $d$ and the norms of the vectors $(X_i)_i$. Our assumptions on $d$ are nearly optimal, as this independence ceases when $d=ω((loglog n)^2)$.

Problem

• Online balancing of sparse stochastic binary vectors
• Establishing optimal discrepancy bounds for online algorithms
• Comparing online versus offline Beck-Fiala problem performance

Innovation

• Online algorithm for sparse vector balancing
• Achieves O(log log n) discrepancy bound
• Efficient algorithm for stochastic arrivals