The power of online thinning in reducing discrepancy

Dwivedi, Raaz; Feldheim, Ohad N.; Gurel-Gurevich, Ori; Ramdas, Aaditya

doi:10.1007/s00440-018-0860-y

Cited by 22 publications

(16 citation statements)

References 24 publications

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“…In contrast, the proof approach of [JKS19] completely breaks down for the Tusnády's problem even in two dimensions and does not give any better lower bounds in terms of d. We recently learned that results similar to Theorems 1.1 and 1.3 were also obtained by Dwivedi et al [DFGGR19], in the context of understanding the power of online thinning in reducing discrepancy.…”

Section: Our Discrepancy Boundsmentioning

confidence: 97%

Online vector balancing and geometric discrepancy

Bansal

Jiang

Singla

et al. 2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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We consider an online vector balancing question where T vectors, chosen from an arbitrary distribution over [−1, 1] n , arrive one-byone 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 remains always nearly balanced. As random coloring incurs Ω(T 1/2 ) discrepancy, while the worst-case offline bounds are Θ( n log(T /n)) for vector balancing and 1 for interval balancing, a natural question is whether one can (nearly) match the offline bounds in the online setting for these problems. One must utilize the stochasticity as in the worst-case scenario it is known that discrepancy is Ω(T 1/2 ) for any online algorithm.In a special case of online vector balancing, Bansal and Spencer [BS19] recently show an O( √ n logT ) bound when each coordinate is independently chosen. When there are dependencies among the coordinates, as in the interval discrepancy problem, the problem becomes much more challenging, as evidenced by a recent work of Jiang, Kulkarni, and Singla [JKS19] that gives a non-trivial O(T 1/log log T ) bound for online interval discrepancy. Although this beats random coloring, it is still far from the offline bound.In this work, we introduce a new framework that allows us to handle online vector balancing even when the input distribution has dependencies across coordinates. In particular, this lets us obtain a poly(n, logT ) bound for online vector balancing under arbitrary input distributions, and a polylog(T ) bound for online interval discrepancy. Our framework is powerful enough to capture other well-studied geometric discrepancy problems; e.g., we obtain a poly(log d (T )) bound for the online d-dimensional Tusnády's problem. All our bounds are tight up to polynomial factors.A key new technical ingredient in our work is an anticoncentration inequality for sums of pairwise uncorrelated random variables, which might also be of independent interest.

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Section: Our Discrepancy Boundsmentioning

confidence: 97%

Online vector balancing and geometric discrepancy

Bansal

Jiang

Singla

et al. 2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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“…The name "two-thinning" is due to yet another point of view on this setting. According to this view, an infinite sequence of allocations has been drawn independently and uniformly at random, and the overseer is allowed to thin it on-line (i.e., delete some of the allocations depending only on the past), as long as at most one of every two consecutive entries is deleted (for a more thorough discussion of the model see joint work with Ramdas and Dwivedi [6], where the model was introduced).…”

Section: Discussionmentioning

confidence: 99%

“…This notion was recently formulated and studied by Peres, Talwar and Wieder [9], viewing it as having two-choices with probability β and no-choice with probability (1 − β), independently for every ball. Once errors of this nature are introduced to the model, two-choices and one-retry are equivalent up to a parameter change, and in lightly loaded case of ρn balls allocated into n bins, both offer no improvement over having no-choice at all (see [6] for more details).…”

Section: Discussionmentioning

confidence: 99%

The power of thinning in balanced allocation

Feldheim

Gurel-Gurevich

2021

Electron. Commun. Probab.

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Balls are sequentially allocated into n bins as follows: for each ball, an independent, uniformly random bin is generated. An overseer may then choose to either allocate the ball to this bin, or else the ball is allocated to a new independent uniformly random bin. The goal of the overseer is to reduce the load of the most heavily loaded bin after Θ(n) balls have been allocated. We provide an asymptotically optimal strategy yielding a maximum load of (1 + o(1)) 8 log n log log n balls.

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“…This is a significant improvement over One-Choice, but also the total number of samples is (1+o(1))•m, which is an improvement over Two-Choice. Similar threshold processes have been studied in queuing [9], [20,Section 5] and discrepancy theory [8]. For values of m sufficiently larger than n, [11] and [18] prove some lower and upper bounds for a more general class of adaptive thinning protocols (here, adaptive means that the choice of the threshold may depend on the load configuration).…”

Section: Introductionmentioning

confidence: 86%

Balanced Allocations: Caching and Packing, Twinning and Thinning

Los,

Sauerwald,

Sylvester

2021

Preprint

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We consider the sequential allocation of m balls (jobs) into n bins (servers) by allowing each ball to choose from some bins sampled uniformly at random. The goal is to maintain a small gap between the maximum load and the average load.In this paper, we present a general framework that allows us to analyze various allocation processes that slightly prefer allocating into underloaded, as opposed to overloaded bins. Our analysis covers several natural instances of processes, including:• The Caching process (a.k.a. memory protocol) as studied by Mitzenmacher, Prabhakar and Shah (2002): At each round we only take one bin sample, but we also have access to a cache in which the most recently used bin is stored. We place the ball into the least loaded of the two.• The Packing process: At each round we only take one bin sample. If the load is below some threshold (e.g., the average load), then we place as many balls until the threshold is reached; otherwise, we place only one ball.• The Twinning process: At each round, we only take one bin sample. If the load is below some threshold, then we place two balls; otherwise, we place only one ball.• The Thinning process as recently studied by Feldheim and Gurel-Gurevich (2021):At each round, we first take one bin sample. If its load is below some threshold, we place one ball; otherwise, we place one ball into a second bin sample.As we demonstrate, our general framework implies for all these processes a gap of O(log n) between the maximum load and average load, even when an arbitrary number of balls m n are allocated (heavily loaded case). Our analysis is inspired by a previous work of Peres, Talwar and Wieder (2010) for the (1 + β)-process, however here we rely on the interplay between different potential functions to prove stabilization.

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The power of online thinning in reducing discrepancy

Cited by 22 publications

References 24 publications

Online vector balancing and geometric discrepancy

Online vector balancing and geometric discrepancy

The power of thinning in balanced allocation

Balanced Allocations: Caching and Packing, Twinning and Thinning

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