# Online vector balancing and geometric discrepancy

@article{Bansal2019OnlineVB, title={Online vector balancing and geometric discrepancy}, author={Nikhil Bansal and Haotian Jiang and Sahil Singla and Makrand Sinha}, journal={Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing}, year={2019} }

We consider an online vector balancing question where T vectors, chosen from an arbitrary distribution over [−1,1] n , arrive one-by-one and must be immediately given a ± sign. The goal is to keep the discrepancy—the ℓ∞-norm of any signed prefix-sum—as small as possible. A concrete example of this question is the online interval discrepancy problem where T points are sampled one-by-one uniformly in the unit interval [0,1], and the goal is to immediately color them ± such that every sub-interval…

## 22 Citations

### Online Discrepancy with Recourse for Vectors and Graphs

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The core ideas are to dynamically maintain an expander-decomposition with low recourse (using a very simple approach), and then to show that, as the expanders change over time, a natural local-search algorithm converges quickly (i.e., with high recourse) to a low-discrepancy solution.

### Prefix Discrepancy, Smoothed Analysis, and Combinatorial Vector Balancing

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A well-known result of Banaszczyk in discrepancy theory concerns the preﬁx discrepancy problem (also known as the signed series problem): given a sequence of T unit vectors in R d , ﬁnd ± signs for…

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### GraB: Finding Provably Better Data Permutations than Random Reshuffling

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An online Gradient Balancing algorithm (GraB) is proposed that can outperform random reshuffling in terms of both training and validation performance, and even outperform state-of-the-art greedy ordering while reducing memory usage over 100 × .

### Resolving Matrix Spencer Conjecture Up to Poly-logarithmic Rank

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A simple proof of the matrix Spencer conjecture up to poly-logarithmic rank is given, which implies a log n − Ω(log log n ) qubit lower bound for quantum random access codes encoding n classical bits with advantage ≫ 1 / √ n .

### A Gaussian fixed point random walk

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A discrete random walk on the real line which takes steps 0,±1 (and one with steps in {±1, 2}) where at least 96% of the signs are ±1 in expectation, and which has N (0, 1) as a stationary distribution is designed.

### Flow time scheduling and prefix Beck-Fiala

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A general reduction is given that allows us to transfer discrepancy bounds in the prefix Beck-Fiala (bounded ℓ1-norm) setting to bounds on the flow time of an optimal schedule, improving upon the previous best guarantees of O(logn) and O( logn logP).

### Algorithms and Barriers in the Symmetric Binary Perceptron Model

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It is shown that at high enough densities the SBP exhibits the multi Overlap Gap Property (m-OGP), an intricate geometrical property known to be a rigorous barrier for large classes of algorithms.

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This work shows that in the online setting, the two desired properties (envy-freeness and efficiency) are in direct contention, in that any algorithm achieving additive counterfactual envy- freeness up to a factor of LT necessarily suffers a efficiency loss of at least 1 / LT.

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