Optimal Online Discrepancy Minimization in Linear Time

2026-07-05

AI Summary

This work addresses the problem of minimizing prefix discrepancy for online sequences of unit-Euclidean-norm vectors, where each incoming vector must be assigned a ±1 sign to minimize the ℓ∞ norm of all prefix sums. The paper proposes a linear-time online algorithm that combines Gaussian process coupling with a randomized signing strategy, representing each prefix sum as the sum of three coupled standard Gaussian vectors. Running in O(dT) time, the method achieves— with high probability—the optimal prefix discrepancy bound $\max_t \|\sum_{i=1}^t \varepsilon_i v_i\|_\infty = O(\sqrt{\log T})$. This result marks the first algorithm to attain this optimal theoretical guarantee while reducing computational complexity from exponential to linear in the input size, thereby overcoming a longstanding efficiency bottleneck in prior approaches.

Abstract

We provide an online algorithm with the following guarantee: for any fixed sequence of vectors $v_1,\dots,v_T \in \mathbf{R}^d$ with $\|v_i\|_2\le 1$, the algorithm assigns each arriving vector $v_t$ a random sign $\varepsilon_t$ such that every prefix sum $\sum_{i=1}^t \varepsilon_i v_i $ can be written as the sum of three coupled standard Gaussian vectors. Our algorithm runs in $O(dT)$ time and achieves the optimal prefix discrepancy bound \[ \max_{1 \le t \le T}\left\| \sum_{i=1}^t \varepsilon_i v_i \right\|_\infty = O\left( \sqrt{\log T} \right), \] with high probability. This recovers the optimal bound of Kulkarni, Reis, and Rothvoss, whose algorithm runs in time exponential in $T$ and $d$. The algorithm and main proof were discovered in a GPT-5.5 Pro Extended conversation prompted by the author.

Problem

• online discrepancy minimization
• prefix discrepancy
• linear time algorithm
• vector balancing
• Gaussian coupling

Innovation

• online discrepancy minimization
• linear time algorithm
• prefix discrepancy
• Gaussian coupling
• randomized signing