Online Beck--Fiala Down to Logarithmic Sparsity
TL;DR
The main thrust of the result is that it is actually obtained by an efficient \textit{online} algorithm that minimizes prefix discrepancy, and is also essentially optimal, since online prefix discrepancy is known to scale as $\omega(\sqrt{d})$ for $d =o(\log T)$.
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 $\omega(\sqrt{d})$ 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 Koml\'os 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.