Online Beck--Fiala Down to Logarithmic Sparsity
For researchers in discrepancy theory and online algorithms, this provides the first optimal online algorithm for a broad regime, but the proof was largely generated by ChatGPT, raising concerns about novelty.
The authors prove the Beck-Fiala conjecture for sparsity down to logarithmic factors, achieving an online algorithm with optimal prefix discrepancy. The result resolves the open question of online vector balancing in the Spencer setting.
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.