Online Beck--Fiala Down to Logarithmic Sparsity

2026-07-15

AI Summary

This work investigates prefix discrepancy minimization in the online Beck–Fiala setting under low sparsity, specifically when $d \approx \log T$. We propose an efficient online algorithm based on a tightly supported Metropolis fixed-point walk, integrating recent techniques from the online Komlós problem with human-in-the-loop proof strategies. Our algorithm achieves a nearly optimal prefix discrepancy of $O(\sqrt{d})$ for $d \geq \log^{1+o(1)} T$, matching the known lower bound. This result extends the applicability of classical Beck–Fiala bounds into this sparse regime and resolves an open problem in online vector balancing within Spencer's framework.

Abstract

The Beck--Fiala conjecture asserts that every matrix $A\in\{0,1\}^{n\times T}$ with at most $d$ nonzero entries in each column has discrepancy $O(\sqrt d)$. A major breakthrough result of Bansal and Jiang recently established the validity of the conjecture for $d \ge \log(T)^2$. The present article extends the validity of the classical \textit{offline} Beck--Fiala conjecture to $d \ge \log(T)^{1+o(1)}$; moreover, the main thrust of the result is that it is actually obtained by an efficient \textit{online} algorithm that minimizes prefix discrepancy. The result is also essentially optimal, since online prefix discrepancy is known to scale as $ω(\sqrt{d})$ for $d =o(\log T)$. As an immediate corollary, the open question of online vector balancing in the Spencer setting is also resolved. The algorithm is based on a compactly supported Metropolis fixed-point walk, constructed by combining ideas from several recent works on the online Komlós problem. The proof was generated in conversation with ChatGPT 5.6 Pro; the authors provided high-level guidance in several rounds of prompting, followed by manual checking and rewriting of the proof.

Problem

• online discrepancy
• Beck–Fiala conjecture
• prefix discrepancy
• logarithmic sparsity
• vector balancing

Innovation

• online discrepancy minimization
• Beck–Fiala conjecture
• prefix discrepancy
• Metropolis walk
• vector balancing