Decoupling via Affine Spectral-Independence: Beck-Fiala and Komlós Bounds Beyond Banaszczyk

Abstract: The Beck-Fiala Conjecture [Discrete Appl. Math, 1981] asserts that any set system of $n$ elements with degree $k$ has combinatorial discrepancy $O(\sqrt{k})$. A substantial generalization is the Koml\'os Conjecture, which states that any $m \times n$ matrix with unit length columns has discrepancy $O(1)$. In this work, we resolve the Beck-Fiala Conjecture for $k \geq \log^2 n$. We also give an $\widetilde{O}(\sqrt{k} + \sqrt{\log n})$ bound for $k \leq \log^2 n$, where $\widetilde{O}(\cdot)$ hides $\mathsf{poly}(\log \log n)$ factors. These bounds improve upon the $O(\sqrt{k \log n})$ bound due to Banaszczyk [Random Struct. Algor., 1998]. For the Komlos problem, we give an $\widetilde{O}(\log^{1/4} n)$ bound, improving upon the previous $O(\sqrt{\log n})$ bound [Random Struct. Algor., 1998]. All of our results also admit efficient polynomial-time algorithms. To obtain these results, we consider a new notion of affine spectral-independence in designing random walks. In particular, our algorithms obtain the desired colorings via a discrete Brownian motion, guided by a semidefinite program (SDP). Besides standard constraints used in prior works, we add some extra affine spectral-independence constraints, which effectively decouple the evolution of discrepancy across different rows, and allow us to better control how many rows accumulate large discrepancy at any point during the process. This technique of ``decoupling via affine spectral-independence'' is quite general and may be of independent interest.