The Power of Two-Choices in Regulating Interval Partitions Ohad N. - - PowerPoint PPT Presentation

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The Power of Two-Choices in Regulating Interval Partitions Ohad N. - - PowerPoint PPT Presentation

Feb 03, 2023 263 likes •1.67k views

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition The Power of Two-Choices in Regulating Interval Partitions Ohad N. Feldheim (Stanford) Joint work with Ori

slide-1
SLIDE 1

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

The Power of Two-Choices in Regulating Interval Partitions

Ohad N. Feldheim (Stanford) Joint work with Ori Gurel-Gurevich (HUJI) September 2016

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SLIDE 2

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balls and bins model

Consider an online process in which N balls are randomly assigned,

  • ne by one to M bins.
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SLIDE 3

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balls and bins model

Consider an online process in which N balls are randomly assigned,

  • ne by one to M bins.
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SLIDE 4

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balls and bins model

Consider an online process in which N balls are randomly assigned,

  • ne by one to M bins.
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SLIDE 5

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balls and bins model

Consider an online process in which N balls are randomly assigned,

  • ne by one to M bins.
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SLIDE 6

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balls and bins model

Consider an online process in which N balls are randomly assigned,

  • ne by one to M bins.
slide-7
SLIDE 7

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balls and bins model

Consider an online process in which N balls are randomly assigned,

  • ne by one to M bins.
slide-8
SLIDE 8

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balls and bins model

Consider an online process in which N balls are randomly assigned,

  • ne by one to M bins.
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SLIDE 9

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balls and bins model

Consider an online process in which N balls are randomly assigned,

  • ne by one to M bins.
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SLIDE 10

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balls and bins model

Consider an online process in which N balls are randomly assigned,

  • ne by one to M bins.
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SLIDE 11

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balls and bins model

Consider an online process in which N balls are randomly assigned,

  • ne by one to M bins.
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SLIDE 12

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balls and bins model

Consider an online process in which N balls are randomly assigned,

  • ne by one to M bins.
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SLIDE 13

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balls and bins model

Consider an online process in which N balls are randomly assigned,

  • ne by one to M bins.
  • After M balls, highest occupancy is a.a.s. (1 + o(1))

log M log log M

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SLIDE 14

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balls and bins model

Consider an online process in which N balls are randomly assigned,

  • ne by one to M bins.
  • After M balls, highest occupancy is a.a.s. (1 + o(1))

log M log log M

  • After N ≫ M balls, highest occupancy is a.a.s.

N M + Θ

  • N log M

M

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SLIDE 15

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balls and bins model

Consider an online process in which N balls are randomly assigned,

  • ne by one to M bins.
  • After M balls, highest occupancy is a.a.s. (1 + o(1))

log M log log M

  • After N ≫ M balls, highest occupancy is a.a.s.

N M + Θ

  • N log M

M

  • typical deviation from expectation is Θ
  • N

M

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SLIDE 16

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balls and bins model

Consider an online process in which N balls are randomly assigned,

  • ne by one to M bins.
  • After M balls, highest occupancy is a.a.s. (1 + o(1))

log M log log M

  • After N ≫ M balls, highest occupancy is a.a.s.

N M + Θ

  • N log M

M

  • typical deviation from expectation is Θ
  • N

M

  • Load balancing is an effort to reduce these quantities.

(possible with control over the distribution of the balls.)

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SLIDE 17

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Power of two choices

Azar, Broder, Karlin, and Upfal (1994): Very little choice is sufficient for rather balanced allocation - choice between two random cells per ball. Greedy strategy: choose the least occupied cell.

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SLIDE 18

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Power of two choices

Azar, Broder, Karlin, and Upfal (1994): Very little choice is sufficient for rather balanced allocation - choice between two random cells per ball.

? ?

Greedy strategy: choose the least occupied cell.

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SLIDE 19

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Power of two choices

Azar, Broder, Karlin, and Upfal (1994): Very little choice is sufficient for rather balanced allocation - choice between two random cells per ball.

? ?

Greedy strategy: choose the least occupied cell.

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SLIDE 20

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Power of two choices

Azar, Broder, Karlin, and Upfal (1994): Very little choice is sufficient for rather balanced allocation - choice between two random cells per ball.

? ?

Greedy strategy: choose the least occupied cell.

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SLIDE 21

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Power of two choices

Azar, Broder, Karlin, and Upfal (1994): Very little choice is sufficient for rather balanced allocation - choice between two random cells per ball.

??

Greedy strategy: choose the least occupied cell.

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SLIDE 22

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Power of two choices

Azar, Broder, Karlin, and Upfal (1994): Very little choice is sufficient for rather balanced allocation - choice between two random cells per ball.

? ?

Greedy strategy: choose the least occupied cell.

slide-23
SLIDE 23

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Power of two choices

Azar, Broder, Karlin, and Upfal (1994): Very little choice is sufficient for rather balanced allocation - choice between two random cells per ball.

? ?

Greedy strategy: choose the least occupied cell.

slide-24
SLIDE 24

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Power of two choices

Azar, Broder, Karlin, and Upfal (1994): Very little choice is sufficient for rather balanced allocation - choice between two random cells per ball.

? ?

Greedy strategy: choose the least occupied cell.

slide-25
SLIDE 25

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Power of two choices

Azar, Broder, Karlin, and Upfal (1994): Very little choice is sufficient for rather balanced allocation - choice between two random cells per ball.

? ?

Greedy strategy: choose the least occupied cell.

slide-26
SLIDE 26

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Power of two choices

Azar, Broder, Karlin, and Upfal (1994): Very little choice is sufficient for rather balanced allocation - choice between two random cells per ball.

? ?

Greedy strategy: choose the least occupied cell.

slide-27
SLIDE 27

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Power of two choices

Azar, Broder, Karlin, and Upfal (1994): Very little choice is sufficient for rather balanced allocation - choice between two random cells per ball. Greedy strategy: choose the least occupied cell.

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SLIDE 28

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Power of two choices

Azar, Broder, Karlin, and Upfal (1994): Very little choice is sufficient for rather balanced allocation - choice between two random cells per ball. Greedy strategy: choose the least occupied cell.

Balls no-choice 2-choices no-choice 2-choices

  • max. dev.
  • max. dev.
  • typ. dev.
  • typ. dev.

M

log M log log M log log M 2

O(1) O(1) N ≫ M

Θ

  • N log M

M

  • Θ(log M)

Θ

  • N

M

  • O(1)
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SLIDE 29

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Power of two choices - remarks

This observation had many applications

  • Server load-balancing
  • Distributed shared memory
  • Efficient on-line hashing
  • Low-congestion circuit routing
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SLIDE 30

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Power of two choices - remarks

This observation had many applications

  • Server load-balancing
  • Distributed shared memory
  • Efficient on-line hashing
  • Low-congestion circuit routing

It is interesting to note that

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SLIDE 31

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Power of two choices - remarks

This observation had many applications

  • Server load-balancing
  • Distributed shared memory
  • Efficient on-line hashing
  • Low-congestion circuit routing

It is interesting to note that

  • More choice does not significantly reduce the maximum load.
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SLIDE 32

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Power of two choices - remarks

This observation had many applications

  • Server load-balancing
  • Distributed shared memory
  • Efficient on-line hashing
  • Low-congestion circuit routing

It is interesting to note that

  • More choice does not significantly reduce the maximum load.
  • If balls keep appearing and dying at rate 1 the phenomenon

persists (Luczak & McDiarmid ’05)

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SLIDE 33

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Power of two choices - remarks

This observation had many applications

  • Server load-balancing
  • Distributed shared memory
  • Efficient on-line hashing
  • Low-congestion circuit routing

It is interesting to note that

  • More choice does not significantly reduce the maximum load.
  • If balls keep appearing and dying at rate 1 the phenomenon

persists (Luczak & McDiarmid ’05)

  • However if one can’t keep track of the number of balls per bin

(due to having M 1−ǫ bits of memory), then no asymptotic improvement over no-choice is possible (Alon, Gurel-Gurevich, Lubetzky ’09)

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SLIDE 34

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and One-retry

One-retry: A related intermediate setup. The chooser is only

  • ffered a chance to re-roll the target bin once per ball.
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SLIDE 35

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and One-retry

One-retry: A related intermediate setup. The chooser is only

  • ffered a chance to re-roll the target bin once per ball.
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SLIDE 36

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and One-retry

One-retry: A related intermediate setup. The chooser is only

  • ffered a chance to re-roll the target bin once per ball.
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SLIDE 37

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and One-retry

One-retry: A related intermediate setup. The chooser is only

  • ffered a chance to re-roll the target bin once per ball.
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SLIDE 38

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and One-retry

One-retry: A related intermediate setup. The chooser is only

  • ffered a chance to re-roll the target bin once per ball.
slide-39
SLIDE 39

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and One-retry

One-retry: A related intermediate setup. The chooser is only

  • ffered a chance to re-roll the target bin once per ball.
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SLIDE 40

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and One-retry

One-retry: A related intermediate setup. The chooser is only

  • ffered a chance to re-roll the target bin once per ball.
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SLIDE 41

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and One-retry

One-retry: A related intermediate setup. The chooser is only

  • ffered a chance to re-roll the target bin once per ball.
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SLIDE 42

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and One-retry

One-retry: A related intermediate setup. The chooser is only

  • ffered a chance to re-roll the target bin once per ball.
slide-43
SLIDE 43

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and One-retry

One-retry: A related intermediate setup. The chooser is only

  • ffered a chance to re-roll the target bin once per ball.
slide-44
SLIDE 44

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and One-retry

One-retry: A related intermediate setup. The chooser is only

  • ffered a chance to re-roll the target bin once per ball.

Equivalent to being oblivious to one of the two bins in the two choices setup.

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SLIDE 45

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and One-retry

One-retry: A related intermediate setup. The chooser is only

  • ffered a chance to re-roll the target bin once per ball.

Equivalent to being oblivious to one of the two bins in the two choices setup. Asymptotic discrepancy like two-choices when N ≫ M.

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SLIDE 46

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and One-retry

One-retry: A related intermediate setup. The chooser is only

  • ffered a chance to re-roll the target bin once per ball.

Equivalent to being oblivious to one of the two bins in the two choices setup. Asymptotic discrepancy like two-choices when N ≫ M. When N = M, the discrepancy is

  • log M/ log log M.
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SLIDE 47

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Interval partition

Consider an online process in which an interval is partitioned into smaller and smaller subintervals by selecting new partition points uniformly at random.

slide-48
SLIDE 48

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Interval partition

Consider an online process in which an interval is partitioned into smaller and smaller subintervals by selecting new partition points uniformly at random.

slide-49
SLIDE 49

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Interval partition

Consider an online process in which an interval is partitioned into smaller and smaller subintervals by selecting new partition points uniformly at random.

slide-50
SLIDE 50

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Interval partition

Consider an online process in which an interval is partitioned into smaller and smaller subintervals by selecting new partition points uniformly at random.

slide-51
SLIDE 51

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Interval partition

Consider an online process in which an interval is partitioned into smaller and smaller subintervals by selecting new partition points uniformly at random.

slide-52
SLIDE 52

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Interval partition

Consider an online process in which an interval is partitioned into smaller and smaller subintervals by selecting new partition points uniformly at random.

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SLIDE 53

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Interval partition

Consider an online process in which an interval is partitioned into smaller and smaller subintervals by selecting new partition points uniformly at random.

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SLIDE 54

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Interval partition

Consider an online process in which an interval is partitioned into smaller and smaller subintervals by selecting new partition points uniformly at random. We view the points at time n as each having 1/n mass, call this µn

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SLIDE 55

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Interval partition

Consider an online process in which an interval is partitioned into smaller and smaller subintervals by selecting new partition points uniformly at random. We view the points at time n as each having 1/n mass, call this µn Convergence to uniform measure: limn→∞ µn T.V. = U[0, 1]

slide-56
SLIDE 56

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Interval partition

Consider an online process in which an interval is partitioned into smaller and smaller subintervals by selecting new partition points uniformly at random. We view the points at time n as each having 1/n mass, call this µn Convergence to uniform measure: limn→∞ µn T.V. = U[0, 1] Three natural ways to measure discrepancy/rate of convergence:

slide-57
SLIDE 57

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Interval partition

Consider an online process in which an interval is partitioned into smaller and smaller subintervals by selecting new partition points uniformly at random. We view the points at time n as each having 1/n mass, call this µn Convergence to uniform measure: limn→∞ µn T.V. = U[0, 1] Three natural ways to measure discrepancy/rate of convergence: Geometric:

  • Interval variation - maxa,b |µn((a, b)) − µ((a, b))|
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SLIDE 58

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Interval partition

Consider an online process in which an interval is partitioned into smaller and smaller subintervals by selecting new partition points uniformly at random. We view the points at time n as each having 1/n mass, call this µn Convergence to uniform measure: limn→∞ µn T.V. = U[0, 1] Three natural ways to measure discrepancy/rate of convergence: Geometric:

  • Interval variation - maxa,b |µn((a, b)) − µ((a, b))|
  • Largest/smallest interval - maxa,b |(µn((a, b)) = 0) − 1/n|
slide-59
SLIDE 59

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Interval partition

Consider an online process in which an interval is partitioned into smaller and smaller subintervals by selecting new partition points uniformly at random. We view the points at time n as each having 1/n mass, call this µn Convergence to uniform measure: limn→∞ µn T.V. = U[0, 1] Three natural ways to measure discrepancy/rate of convergence: Geometric:

  • Interval variation - maxa,b |µn((a, b)) − µ((a, b))|
  • Largest/smallest interval - maxa,b |(µn((a, b)) = 0) − 1/n|

Non-geometric:

  • Empirical normalized interval distribution.
slide-60
SLIDE 60

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and interval partitions

Benjamini: Can two choices regulate interval partitions?

slide-61
SLIDE 61

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and interval partitions

Benjamini: Can two choices regulate interval partitions? In particular what if we partition the largest interval? What if we choose the point furthest from neighbour?

? ?

slide-62
SLIDE 62

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and interval partitions

Benjamini: Can two choices regulate interval partitions? In particular what if we partition the largest interval? What if we choose the point furthest from neighbour?

? ?

slide-63
SLIDE 63

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and interval partitions

Benjamini: Can two choices regulate interval partitions? In particular what if we partition the largest interval? What if we choose the point furthest from neighbour?

? ?

slide-64
SLIDE 64

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and interval partitions

Benjamini: Can two choices regulate interval partitions? In particular what if we partition the largest interval? What if we choose the point furthest from neighbour?

  • Cf. Kakutani process - uniform partition of the largest interval.
slide-65
SLIDE 65

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and interval partitions

Benjamini: Can two choices regulate interval partitions? In particular what if we partition the largest interval? What if we choose the point furthest from neighbour?

  • Cf. Kakutani process - uniform partition of the largest interval.
slide-66
SLIDE 66

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and interval partitions

Benjamini: Can two choices regulate interval partitions? In particular what if we partition the largest interval? What if we choose the point furthest from neighbour?

  • Cf. Kakutani process - uniform partition of the largest interval.
slide-67
SLIDE 67

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and interval partitions

Benjamini: Can two choices regulate interval partitions? In particular what if we partition the largest interval? What if we choose the point furthest from neighbour?

  • Cf. Kakutani process - uniform partition of the largest interval.
slide-68
SLIDE 68

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and interval partitions

Benjamini: Can two choices regulate interval partitions? In particular what if we partition the largest interval? What if we choose the point furthest from neighbour?

  • Cf. Kakutani process - uniform partition of the largest interval.

Empirical normalized interval dist. Uniform interval partition → Exp(1) Kakutani interval partition → U(0, 2) (Pyke 80’) Max-2 interval partition → ???

slide-69
SLIDE 69

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Two-choices and interval partitions

Benjamini: Can two choices regulate interval partitions? In particular what if we partition the largest interval? What if we choose the point furthest from neighbour?

  • Cf. Kakutani process - uniform partition of the largest interval.

Empirical normalized interval dist. Uniform interval partition → Exp(1) Kakutani interval partition → U(0, 2) (Pyke 80’) Max-2 interval partition → ??? However, even Kakutani process offers merely a factor 2 improvement over the uniform process in terms of interval variation (Pyke-Zwet 2004).

slide-70
SLIDE 70

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Convergence of 2-Max interval partition process

Studying 2-Max is a rather difficult task: Maillard & Paquette ’14: 2-Max converges to some limit distribution.

slide-71
SLIDE 71

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Convergence of 2-Max interval partition process

Studying 2-Max is a rather difficult task: Maillard & Paquette ’14: 2-Max converges to some limit distribution. Junge ’15: 2-Max converges to U[0, 1].

slide-72
SLIDE 72

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Convergence of 2-Max interval partition process

Studying 2-Max is a rather difficult task: Maillard & Paquette ’14: 2-Max converges to some limit distribution. Junge ’15: 2-Max converges to U[0, 1]. Both experimental and heuristic arguments suggest that 2-Max

  • ffers no improvement in interval variation when compared with

uniform.

slide-73
SLIDE 73

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Convergence of 2-Max interval partition process

Studying 2-Max is a rather difficult task: Maillard & Paquette ’14: 2-Max converges to some limit distribution. Junge ’15: 2-Max converges to U[0, 1]. Both experimental and heuristic arguments suggest that 2-Max

  • ffers no improvement in interval variation when compared with

uniform. We believe that no local algorithm can obtain significant improvement.

slide-74
SLIDE 74

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Convergence of 2-Max interval partition process

Studying 2-Max is a rather difficult task: Maillard & Paquette ’14: 2-Max converges to some limit distribution. Junge ’15: 2-Max converges to U[0, 1]. Both experimental and heuristic arguments suggest that 2-Max

  • ffers no improvement in interval variation when compared with

uniform. We believe that no local algorithm can obtain significant

  • improvement. In a sense, corresponds to Alon, Gurel-Gurevich,

Lubetzky.

slide-75
SLIDE 75

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Global strategy that regulates interval variation

Our main result is a global one-retry strategy which reduces discrepancy in interval partitions significantly.

slide-76
SLIDE 76

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Global strategy that regulates interval variation

Our main result is a global one-retry strategy which reduces discrepancy in interval partitions significantly. Power of two-choices in regulating interval discrepancy (F. & Gurel-Gurevich 2016+) In a power of one-retry process on U[0, 1], the chooser can obtain lim

n→∞ P

  • max

a,b |µn((a, b)) − µ((a, b))| < C log3 N

N

  • = 1.
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SLIDE 77

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Global strategy that regulates interval variation

Our main result is a global one-retry strategy which reduces discrepancy in interval partitions significantly. Power of two-choices in regulating interval discrepancy (F. & Gurel-Gurevich 2016+) In a power of one-retry process on U[0, 1], the chooser can obtain lim

n→∞ P

  • max

a,b |µn((a, b)) − µ((a, b))| < C log3 N

N

  • = 1.
  • Cf. lower bound of C log N

N

, no choice estimate

  • C log N

N

.

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SLIDE 78

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Global strategy that regulates interval variation

Our main result is a global one-retry strategy which reduces discrepancy in interval partitions significantly. Power of two-choices in regulating interval discrepancy (F. & Gurel-Gurevich 2016+) In a power of one-retry process on U[0, 1], the chooser can obtain lim

n→∞ P

  • max

a,b |µn((a, b)) − µ((a, b))| < C log3 N

N

  • = 1.
  • Cf. lower bound of C log N

N

, no choice estimate

  • C log N

N

.

  • There exists a single universal strategy which obtains this for

all N.

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SLIDE 79

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Global strategy that regulates interval variation

The result is obtained through a discrete counterpart which also may be of interest.

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SLIDE 80

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Global strategy that regulates interval variation

The result is obtained through a discrete counterpart which also may be of interest. Discrete counterpart For N balls on U([M]), a probabilistic retry strategy obtains P

  • max

a<b∈[M] |µn([a, b]) − µ([a, b])| > ∆ log3 M

  • ≤ Ce−c∆.
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SLIDE 81

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Statistical implication

Consider a researcher who is interested in gathering cardiovascular data on a population.

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SLIDE 82

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Statistical implication

Consider a researcher who is interested in gathering cardiovascular data on a population.

  • It is well known that this data is correlated with the height of

the sampled person.

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SLIDE 83

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Statistical implication

Consider a researcher who is interested in gathering cardiovascular data on a population.

  • It is well known that this data is correlated with the height of

the sampled person.

  • Height test is cheap, cardiovascular test - expansive.
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SLIDE 84

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Statistical implication

Consider a researcher who is interested in gathering cardiovascular data on a population.

  • It is well known that this data is correlated with the height of

the sampled person.

  • Height test is cheap, cardiovascular test - expansive.
  • Height distribution in the population is well known.
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SLIDE 85

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Statistical implication

Consider a researcher who is interested in gathering cardiovascular data on a population.

  • It is well known that this data is correlated with the height of

the sampled person.

  • Height test is cheap, cardiovascular test - expansive.
  • Height distribution in the population is well known.

One by one volunteers suggest themselves to be tested, and it is desirable to obtain an overall sample which matches the empirical distribution of height.

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SLIDE 86

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Statistical implication

Consider a researcher who is interested in gathering cardiovascular data on a population.

  • It is well known that this data is correlated with the height of

the sampled person.

  • Height test is cheap, cardiovascular test - expansive.
  • Height distribution in the population is well known.

One by one volunteers suggest themselves to be tested, and it is desirable to obtain an overall sample which matches the empirical distribution of height. Our result implies that by rejecting at most one of every two candidates this could be done.

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SLIDE 87

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Stochastic point of view on the power of one-retry

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SLIDE 88

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

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SLIDE 89

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Large family of one-retry distributions

What kind of distributions could be realized using one retry?

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SLIDE 90

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Large family of one-retry distributions

What kind of distributions could be realized using one retry? Observation I Every distribution with “density”

1 2M ≤ g(x) ≤ 3 2M on [M] could

be realized by a (probabilistic) one-retry strategy.

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SLIDE 91

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Large family of one-retry distributions

What kind of distributions could be realized using one retry? Observation I Every distribution with “density”

1 2M ≤ g(x) ≤ 3 2M on [M] could

be realized by a (probabilistic) one-retry strategy. In general any distribution with Radon-Nikodim Derivative w.r.t the base distribution between 0.5 and 1.5 could be realized

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SLIDE 92

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Large family of one-retry distributions

What kind of distributions could be realized using one retry? Observation I Every distribution with “density”

1 2M ≤ g(x) ≤ 3 2M on [M] could

be realized by a (probabilistic) one-retry strategy. In general any distribution with Radon-Nikodim Derivative w.r.t the base distribution between 0.5 and 1.5 could be realized

  • Proof. Write f (x) = 3

2 − Mg(x), and retry x with probability f (x).

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SLIDE 93

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Large family of one-retry distributions

What kind of distributions could be realized using one retry? Observation I Every distribution with “density”

1 2M ≤ g(x) ≤ 3 2M on [M] could

be realized by a (probabilistic) one-retry strategy. In general any distribution with Radon-Nikodim Derivative w.r.t the base distribution between 0.5 and 1.5 could be realized

  • Proof. Write f (x) = 3

2 − Mg(x), and retry x with probability f (x).

The probability of a random bin to be re-rolled is M

i=1 f (x) M = 1 2.

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SLIDE 94

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Large family of one-retry distributions

What kind of distributions could be realized using one retry? Observation I Every distribution with “density”

1 2M ≤ g(x) ≤ 3 2M on [M] could

be realized by a (probabilistic) one-retry strategy. In general any distribution with Radon-Nikodim Derivative w.r.t the base distribution between 0.5 and 1.5 could be realized

  • Proof. Write f (x) = 3

2 − Mg(x), and retry x with probability f (x).

The probability of a random bin to be re-rolled is M

i=1 f (x) M = 1 2.

Hence the probability that x is chosen is now 1 − f (x) M + 1 2M = g(x) .

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SLIDE 95

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Self regulating point process

What kind of distributions do we wish to realize? .

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SLIDE 96

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Self regulating point process

What kind of distributions do we wish to realize?

  • First - how to recover original balls and bins result with N ≫ M.

.

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SLIDE 97

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Self regulating point process

What kind of distributions do we wish to realize?

  • First - how to recover original balls and bins result with N ≫ M.

Consider a point process Xt with changing causal intensity λ(t), defined by λ(t) = 1 + θ Xt ≤ t λ(t) = 1 − θ Xt > t .

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SLIDE 98

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Self regulating point process

What kind of distributions do we wish to realize?

  • First - how to recover original balls and bins result with N ≫ M.

Consider a point process Xt with changing causal intensity λ(t), defined by λ(t) = 1 + θ Xt ≤ t λ(t) = 1 − θ Xt > t Proposition For such a process P

  • |Xt − t| > ∆

θ

  • < Cθ−2e−∆/3

.

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SLIDE 99

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Self regulating point process

What kind of distributions do we wish to realize?

  • First - how to recover original balls and bins result with N ≫ M.

Consider a point process Xt with changing causal intensity λ(t), defined by λ(t) = 1 + θ Xt ≤ t λ(t) = 1 − θ Xt > t Proposition For such a process P

  • |Xt − t| > ∆

θ

  • < Cθ−2e−∆/3

i.e. for θ < 1, such a process has typical fluctuation O(− log θ

θ

).

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SLIDE 100

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Recovering the N ≫ M balls and bins result

Now let us consider M such self regulating processes.

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SLIDE 101

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Recovering the N ≫ M balls and bins result

Now let us consider M such self regulating processes.

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SLIDE 102

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Recovering the N ≫ M balls and bins result

Now let us consider M such self regulating processes.

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SLIDE 103

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Recovering the N ≫ M balls and bins result

Now let us consider M such self regulating processes.

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SLIDE 104

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Recovering the N ≫ M balls and bins result

Now let us consider M such self regulating processes.

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SLIDE 105

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Recovering the N ≫ M balls and bins result

Now let us consider M such self regulating processes.

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SLIDE 106

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Recovering the N ≫ M balls and bins result

Now let us consider M such self regulating processes.

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SLIDE 107

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Recovering the N ≫ M balls and bins result

Now let us consider M such self regulating processes.

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SLIDE 108

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Recovering the N ≫ M balls and bins result

Now let us consider M such self regulating processes.

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SLIDE 109

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Recovering the N ≫ M balls and bins result

Now let us consider M such self regulating processes.

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SLIDE 110

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Recovering the N ≫ M balls and bins result

Now let us consider M such self regulating processes.

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SLIDE 111

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Recovering the N ≫ M balls and bins result

Now let us consider M such self regulating processes.

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SLIDE 112

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Recovering the N ≫ M balls and bins result

Now let us consider M such self regulating processes. Proposition (repeat) For such a process P

  • |Xt − t| > ∆

θ

  • < Cθ−2e−∆/3

Among M such processes, for fixed θ, the extremal fluctuation is O(log M).

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SLIDE 113

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Recovering the N ≫ M balls and bins result

The plan: realize a self regulating process with θ = 1/5 at every bin.

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SLIDE 114

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Recovering the N ≫ M balls and bins result

The plan: realize a self regulating process with θ = 1/5 at every bin. Can we do it?

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SLIDE 115

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Recovering the N ≫ M balls and bins result

The plan: realize a self regulating process with θ = 1/5 at every bin. Can we do it? Yes - px, the probability of the next ball falling into a bin x is bounded by 1 2M ≤ 1 − θ M(1 + θ) ≤ px ≤ 1 + θ M(1 − θ) ≤ 3 2M .

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SLIDE 116

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Recovering the N ≫ M balls and bins result

The plan: realize a self regulating process with θ = 1/5 at every bin. Can we do it? Yes - px, the probability of the next ball falling into a bin x is bounded by 1 2M ≤ 1 − θ M(1 + θ) ≤ px ≤ 1 + θ M(1 − θ) ≤ 3 2M . Suppose we use this strategy until time N/M. We expect N balls, Maximal load of N/M + Θ(log M).

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SLIDE 117

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Recovering the N ≫ M balls and bins result

The plan: realize a self regulating process with θ = 1/5 at every bin. Can we do it? Yes - px, the probability of the next ball falling into a bin x is bounded by 1 2M ≤ 1 − θ M(1 + θ) ≤ px ≤ 1 + θ M(1 − θ) ≤ 3 2M . Suppose we use this strategy until time N/M. We expect N balls, Maximal load of N/M + Θ(log M). All that is left is to show that the same holds after N balls were

  • distributed. - i.e. show concentration of the stopping time.
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Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

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SLIDE 119

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Proving the main Theorem

We will illustrate the proof for discrepancy of log4 N/N, in a continuous setting. The proof in the paper for discrepancy of log3 N/N is similar but requires working with expectations rather than probabilities.

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SLIDE 120

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balancing poisson point process

Consider two point processes X 0

t , X 1 t with changing causal

intensities λ0(t), λ1(t) (which may depend on other variables), which satisfy λ1

t , λ0 t < 2

∀t λ0

t − λ1 t ≥ θ

X 0

t ≤ X 1 t

λ0

t − λ1 t ≤ −θ

X 0

t > X 1 t

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SLIDE 121

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balancing poisson point process

Consider two point processes X 0

t , X 1 t with changing causal

intensities λ0(t), λ1(t) (which may depend on other variables), which satisfy λ1

t , λ0 t < 2

∀t λ0

t − λ1 t ≥ θ

X 0

t ≤ X 1 t

λ0

t − λ1 t ≤ −θ

X 0

t > X 1 t

Proposition For such a process P

  • |X 1

t − X 0 t | > ∆ θ

  • < Cθ−2e−∆/2
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SLIDE 122

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balancing poisson point process

We now define an intensity µt on [0, 1] which could be realized by a one-retry strategy.

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SLIDE 123

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balancing poisson point process

We now define an intensity µt on [0, 1] which could be realized by a one-retry strategy. Each (a, b) ⊂ [0, 1] has (a, b) = ⌊log2 N⌋

i=0

Ii + R where Ii are diadic and |R| < 1/N.

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SLIDE 124

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Balancing poisson point process

We now define an intensity µt on [0, 1] which could be realized by a one-retry strategy. Each (a, b) ⊂ [0, 1] has (a, b) = ⌊log2 N⌋

i=0

Ii + R where Ii are diadic and |R| < 1/N. For every diadic interval I write Ileft, Iright for its left and right halves.

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SLIDE 125

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Hierarchy of drifts

We build a hierarchy of log2 N drifts of strength C/ log2 N, to control diadic discrepancies.

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SLIDE 126

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Hierarchy of drifts

We build a hierarchy of log2 N drifts of strength C/ log2 N, to control diadic discrepancies. The intensity of every point x is 1+

drift hierarchy

C log2 N

  • x∈I

(−1)1{x∈Ileft}(−1)1{µt(Ileft≤Iright)}+

rate regulating term

c(−1)1{µt([0,1])>t}

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SLIDE 127

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Hierarchy of drifts

We build a hierarchy of log2 N drifts of strength C/ log2 N, to control diadic discrepancies. The intensity of every point x is 1+

drift hierarchy

C log2 N

  • x∈I

(−1)1{x∈Ileft}(−1)1{µt(Ileft≤Iright)}+

rate regulating term

c(−1)1{µt([0,1])>t}

7:9 5:4 1:2 2:3 2:0

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SLIDE 128

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Hierarchy of drifts

We build a hierarchy of log2 N drifts of strength C/ log2 N, to control diadic discrepancies. The intensity of every point x is 1+

drift hierarchy

C log2 N

  • x∈I

(−1)1{x∈Ileft}(−1)1{µt(Ileft≤Iright)}+

rate regulating term

c(−1)1{µt([0,1])>t}

  • +

+ + + +

  • +
  • +
  • +
  • +

+

  • +

+

  • +
  • +

+

  • +
  • +
  • +

+

  • +
  • +
  • +
  • +
  • +

+

  • +
  • +
  • +
  • +
  • +
  • +
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SLIDE 129

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Hierarchy of drifts

We build a hierarchy of log2 N drifts of strength C/ log2 N, to control diadic discrepancies. The intensity of every point x is 1+

drift hierarchy

C log2 N

  • x∈I

(−1)1{x∈Ileft}(−1)1{µt(Ileft≤Iright)}+

rate regulating term

c(−1)1{µt([0,1])>t} This drift could be made between 1 − 1

5 and 1 + 1 5 so that it could

be realized by a one-retry strategy as before.

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SLIDE 130

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Hierarchy of drifts - Ct.

Write D(I) = #{balls in Ileft} − #{balls in Iright}. By the balancing processes lemma, D(I) is typically (log2 n)2.

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SLIDE 131

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Hierarchy of drifts - Ct.

Write D(I) = #{balls in Ileft} − #{balls in Iright}. By the balancing processes lemma, D(I) is typically (log2 n)2. We now bound the discrepancy of a diadic interval I of size

1 2j+1 by j−1

  • i=0

1 2j−i D(I i) Where I i

n is an interval containing I of size 2−i.

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SLIDE 132

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Hierarchy of drifts - Ct.

Write D(I) = #{balls in Ileft} − #{balls in Iright}. By the balancing processes lemma, D(I) is typically (log2 n)2. We now bound the discrepancy of a diadic interval I of size

1 2j+1 by j−1

  • i=0

1 2j−i D(I i) Where I i

n is an interval containing I of size 2−i. This is also

typically of size (log2 n)2.

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SLIDE 133

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Hierarchy of drifts - Ct.

Write D(I) = #{balls in Ileft} − #{balls in Iright}. By the balancing processes lemma, D(I) is typically (log2 n)2. We now bound the discrepancy of a diadic interval I of size

1 2j+1 by j−1

  • i=0

1 2j−i D(I i) Where I i

n is an interval containing I of size 2−i. This is also

typically of size (log2 n)2. Since any interval with diadic endpoint could be decomposed into at most 2 log2 n diadic intervals - the theorem follows.

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SLIDE 134

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Future directions

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SLIDE 135

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Future directions

  • Other spaces:

Benjamini expressed particular interest in whether similar methods could improve discrepancy bounds on a sphere, where the known bounds are far from tight.

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SLIDE 136

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Future directions

  • Other spaces:

Benjamini expressed particular interest in whether similar methods could improve discrepancy bounds on a sphere, where the known bounds are far from tight.

  • Other measures of discrepancy.
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SLIDE 137

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Future directions

  • Other spaces:

Benjamini expressed particular interest in whether similar methods could improve discrepancy bounds on a sphere, where the known bounds are far from tight.

  • Other measures of discrepancy.
  • Reducing the power of the log.
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SLIDE 138

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition

Future directions

  • Other spaces:

Benjamini expressed particular interest in whether similar methods could improve discrepancy bounds on a sphere, where the known bounds are far from tight.

  • Other measures of discrepancy.
  • Reducing the power of the log.
  • Simpler algorithm?
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SLIDE 139

Two-choices for Balls and Bins Interval Partition Results and Method Method - Balls & Bins Method - Interval Partition