Constructive Discrepancy Minimization for Convex Sets Thomas - - PowerPoint PPT Presentation

Constructive Discrepancy Minimization for Convex Sets

Thomas Rothvoss

UW Seattle

Discrepancy theory

◮ Set system S = {S1, . . . , Sm}, Si ⊆ [n]

i S

Discrepancy theory

◮ Set system S = {S1, . . . , Sm}, Si ⊆ [n] ◮ Coloring χ : [n] → {−1, +1}

i S

b b

−1

b

+1 −1

Discrepancy theory

◮ Set system S = {S1, . . . , Sm}, Si ⊆ [n] ◮ Coloring χ : [n] → {−1, +1} ◮ Discrepancy

disc(S) = min

χ:[n]→{±1} max S∈S

• i∈S

χ(i)

• .

i S

b b

−1

b

+1 −1

Discrepancy theory

◮ Set system S = {S1, . . . , Sm}, Si ⊆ [n] ◮ Coloring χ : [n] → {−1, +1} ◮ Discrepancy

disc(S) = min

χ:[n]→{±1} max S∈S

• i∈S

χ(i)

• .

i S

b b

−1

b

+1 −1 Known results:

◮ n sets, n elements: disc(S) = O(√n) [Spencer ’85]

Discrepancy theory

◮ Set system S = {S1, . . . , Sm}, Si ⊆ [n] ◮ Coloring χ : [n] → {−1, +1} ◮ Discrepancy

disc(S) = min

χ:[n]→{±1} max S∈S

• i∈S

χ(i)

• .

i S

b b

−1

b

+1 −1 Known results:

◮ n sets, n elements: disc(S) = O(√n) [Spencer ’85] ◮ Every element in ≤ t sets: disc(S) < 2t [Beck & Fiala ’81]

Discrepancy theory

◮ Set system S = {S1, . . . , Sm}, Si ⊆ [n] ◮ Coloring χ : [n] → {−1, +1} ◮ Discrepancy

disc(S) = min

χ:[n]→{±1} max S∈S

• i∈S

χ(i)

• .

i S

b b

−1

b

+1 −1 Known results:

◮ n sets, n elements: disc(S) = O(√n) [Spencer ’85] ◮ Every element in ≤ t sets: disc(S) < 2t [Beck & Fiala ’81]

Main method: Find a partial coloring χ : [n] → {0, ±1}

◮ low discrepancy maxS∈S |χ(S)| ◮ |supp(χ)| ≥ Ω(n)

Discrepancy theory (2)

Lemma (Spencer)

For m set on n ≤ m elements there is a partial coloring of discrepancy O(

• n log 2m

n ). ◮ Run argument log n times ◮ Total discrepancy is

√n +

• n/2 +
• n/22 + . . . + 1 = O(√n)

Thm of Spencer-Gluskin-Giannopolous

(−1, −1) (1, −1) (−1, 1) (1, 1)

Thm of Spencer-Gluskin-Giannopolous

Goal: For K :=

• x ∈ Rn : |

j∈Si xj| ≤ 100√n ∀i ∈ [n]

• K

Thm of Spencer-Gluskin-Giannopolous

Goal: For K :=

• x ∈ Rn : |

j∈Si xj| ≤ 100√n ∀i ∈ [n]

• find a point x ∈ {−1, 1}n ∩ K

K x

Thm of Spencer-Gluskin-Giannopolous

Goal: For K :=

• x ∈ Rn : |

j∈Si xj| ≤ 100√n ∀i ∈ [n]

• find a point x ∈ {−1, 0, 1}n ∩ K with |supp(x)| ≥ Ω(n).

K x

Thm of Spencer-Gluskin-Giannopolous

Goal: For K :=

• x ∈ Rn : |

j∈Si xj| ≤ 100√n ∀i ∈ [n]

• find a point x ∈ {−1, 0, 1}n ∩ K with |supp(x)| ≥ Ω(n).

K

≥ 100 ≥ 100

◮ K is intersection of n strips of width 100

Thm of Spencer-Gluskin-Giannopolous

Goal: For K :=

• x ∈ Rn : |

j∈Si xj| ≤ 100√n ∀i ∈ [n]

• find a point x ∈ {−1, 0, 1}n ∩ K with |supp(x)| ≥ Ω(n).

K

≥ 100 ≥ 100

◮ K is intersection of n strips of width 100 ◮ Gaussian measure

γn(K) ≥ (γn(width 100 strip))n ≥ e−n/100

Thm of Spencer-Gluskin-Giannopolous

Goal: For K :=

• x ∈ Rn : |

j∈Si xj| ≤ 100√n ∀i ∈ [n]

• find a point x ∈ {−1, 0, 1}n ∩ K with |supp(x)| ≥ Ω(n).

K x

◮ K is intersection of n strips of width 100 ◮ Gaussian measure

γn(K) ≥ (γn(width 100 strip))n ≥ e−n/100

◮ Counting argument: any such K admits partial coloring

Gaussian measure

Lemma (Sidak-Kathri ’67)

For K convex and symmetric and strip S, γn(K ∩ S) ≥ γn(K) · γn(S) K S

Gaussian measure

Lemma (Sidak-Kathri ’67)

For K convex and symmetric and strip S, γn(K ∩ S) ≥ γn(K) · γn(S) S K

Partial coloring approaches

◮ Spencer ’85, Gluskin ’87, Giannopolous ’97:

◮ (+) very general ◮ (−) non-constructive

Partial coloring approaches

◮ Spencer ’85, Gluskin ’87, Giannopolous ’97:

◮ (+) very general ◮ (−) non-constructive

◮ Bansal ’10: SDP-based random walk for Spencer’s Thm

◮ (+) poly-time ◮ (−) custom-tailored to Spencers setting

Partial coloring approaches

◮ Spencer ’85, Gluskin ’87, Giannopolous ’97:

◮ (+) very general ◮ (−) non-constructive

◮ Bansal ’10: SDP-based random walk for Spencer’s Thm

◮ (+) poly-time ◮ (−) custom-tailored to Spencers setting

◮ Lovett-Meka ’12:

◮ (+) poly-time ◮ (+) simple and elegant ◮ (+/−) Works for any K = {x : |x, vi| ≤ λi∀i ∈ [m]}

with m

i=1 e−λ2

i /16 ≤ n

16

Our main result

Theorem (R. 2014)

For a convex symmetric set K ⊆ Rn with γn(K) ≥ e−δn,

• ne can find a y ∈ K ∩ [−1, 1]n with |{i : yi = ±1}| ≥ εn in

poly-time. K y∗ [−1, 1]n

Our main result

Theorem (R. 2014)

For a convex symmetric set K ⊆ Rn with γn(K) ≥ e−δn,

• ne can find a y ∈ K ∩ [−1, 1]n with |{i : yi = ±1}| ≥ εn in

poly-time. Algorithm: (1) take a random x∗ ∼ γn K x∗ [−1, 1]n

Our main result

Theorem (R. 2014)

For a convex symmetric set K ⊆ Rn with γn(K) ≥ e−δn,

• ne can find a y ∈ K ∩ [−1, 1]n with |{i : yi = ±1}| ≥ εn in

poly-time. Algorithm: (1) take a random x∗ ∼ γn (2) compute y∗ = argmin{x∗ − y2 | y ∈ K ∩ [−1, 1]n} K x∗ y∗ [−1, 1]n

Isoperimetric inequality

◮ For set K

K

Isoperimetric inequality

◮ For set K, let K∆ := {x : d(x, K) ≤ ∆}

∆ K∆ K

Isoperimetric inequality

◮ For set K, let K∆ := {x : d(x, K) ≤ ∆}

∆ K∆ K

Lemma

γn(K) ≥ e−δn = ⇒ γn(K3

√ δn) ≥ 1 − e−δn

Isoperimetric inequality

◮ For set K, let K∆ := {x : d(x, K) ≤ ∆}

∆ K∆ K

Lemma

γn(K) ≥ e−δn = ⇒ γn(K3

√ δn) ≥ 1 − e−δn ◮ Isoperimetric inequality: worst case are half planes!

z

1 √ 2πe−z2/2

γn = e−δn γn = e−δn

Isoperimetric inequality

◮ For set K, let K∆ := {x : d(x, K) ≤ ∆}

∆ K∆ K

Lemma

γn(K) ≥ e−δn = ⇒ γn(K3

√ δn) ≥ 1 − e−δn ◮ Isoperimetric inequality: worst case are half planes!

z

1 √ 2πe−z2/2

γn = e−δn γn = e−δn ≤ 3 √ δn

Analysis

K [−1, 1]n

Analysis

◮ W.h.p. x∗ − y∗2 ≥ 1 5

√n K x∗ y∗ [−1, 1]n

Analysis

◮ W.h.p. x∗ − y∗2 ≥ 1 5

√n

◮ For any Q with γn(Q) ≥ e−o(n):

Pr[d(x∗, Q) ≥ 1

5 · √n] ≤ e−Ω(n)

K x∗ y∗ [−1, 1]n

Analysis

◮ W.h.p. x∗ − y∗2 ≥ 1 5

√n

◮ For any Q with γn(Q) ≥ e−o(n):

Pr[d(x∗, Q) ≥ 1

5 · √n] ≤ e−Ω(n) ◮ Def. I∗ := {i : |y∗ i | = 1}

Suppose |I∗| ≤ εn K x∗ y∗ [−1, 1]n

Analysis

◮ W.h.p. x∗ − y∗2 ≥ 1 5

√n

◮ For any Q with γn(Q) ≥ e−o(n):

Pr[d(x∗, Q) ≥ 1

5 · √n] ≤ e−Ω(n) ◮ Def. I∗ := {i : |y∗ i | = 1}

Suppose |I∗| ≤ εn

◮ K(I∗) := K ∩ {|xi| ≤ 1 : i ∈ I∗}

K(I∗) K x∗ y∗ [−1, 1]n

Analysis

◮ W.h.p. x∗ − y∗2 ≥ 1 5

√n

◮ For any Q with γn(Q) ≥ e−o(n):

Pr[d(x∗, Q) ≥ 1

5 · √n] ≤ e−Ω(n) ◮ Def. I∗ := {i : |y∗ i | = 1}

Suppose |I∗| ≤ εn

◮ K(I∗) := K ∩ {|xi| ≤ 1 : i ∈ I∗} ◮ x∗ − y∗2 = d(y∗, K(I∗))

K(I∗) K x∗ y∗ [−1, 1]n

Analysis

◮ W.h.p. x∗ − y∗2 ≥ 1 5

√n

◮ For any Q with γn(Q) ≥ e−o(n):

Pr[d(x∗, Q) ≥ 1

5 · √n] ≤ e−Ω(n) ◮ Def. I∗ := {i : |y∗ i | = 1}

Suppose |I∗| ≤ εn

◮ K(I∗) := K ∩ {|xi| ≤ 1 : i ∈ I∗} ◮ x∗ − y∗2 = d(y∗, K(I∗))

K(I∗) K x∗ y∗ [−1, 1]n

◮ K(I∗) still large:

γn(K(I∗)) ≥ γn(K) · (γn(strip of width 1))εn ≥ e−2δn

Analysis

◮ W.h.p. x∗ − y∗2 ≥ 1 5

√n

◮ For any Q with γn(Q) ≥ e−o(n):

Pr[d(x∗, Q) ≥ 1

5 · √n] ≤ e−Ω(n) ◮ Def. I∗ := {i : |y∗ i | = 1}

Suppose |I∗| ≤ εn

◮ K(I∗) := K ∩ {|xi| ≤ 1 : i ∈ I∗} ◮ x∗ − y∗2 = d(y∗, K(I∗))

K(I∗) K x∗ y∗ [−1, 1]n

◮ K(I∗) still large:

γn(K(I∗)) ≥ γn(K) · (γn(strip of width 1))εn ≥ e−2δn

◮ W.h.p. d(x∗, K(I∗)) ≤

√ 3δ · √n ≪ 1

5

√n

Analysis

◮ W.h.p. x∗ − y∗2 ≥ 1 5

√n

◮ For any Q with γn(Q) ≥ e−o(n):

Pr[d(x∗, Q) ≥ 1

5 · √n] ≤ e−Ω(n) ◮ Def. I∗ := {i : |y∗ i | = 1}

Suppose |I∗| ≤ εn

◮ K(I∗) := K ∩ {|xi| ≤ 1 : i ∈ I∗} ◮ x∗ − y∗2 = d(y∗, K(I∗))

K(I∗) K x∗ y∗ [−1, 1]n

◮ K(I∗) still large:

γn(K(I∗)) ≥ γn(K) · (γn(strip of width 1))εn ≥ e−2δn

◮ W.h.p. d(x∗, K(I∗)) ≤

√ 3δ · √n ≪ 1

5

√n

◮ Union bound over all |I| ≤ εn:

Pr

|I|≤εn

d(x∗, K(I)) < 1 5 √n

• ≤ e−Ω(n)

Application to approximation algorithms

◮ Classical technique: For

K = {x ∈ Rn | Ax = b} with n

2 constraints

→ basic solution x′ has ≤ n

2 fractional

variables.

b

x

Application to approximation algorithms

◮ Classical technique: For

K = {x ∈ Rn | Ax = b} with n

2 constraints

→ basic solution x′ has ≤ n

2 fractional

variables. Ax = b

b

x

Application to approximation algorithms

◮ Classical technique: For

K = {x ∈ Rn | Ax = b} with n

2 constraints

→ basic solution x′ has ≤ n

2 fractional

variables. Ax = b

b

x

b x′

Application to approximation algorithms

◮ Classical technique: For

K = {x ∈ Rn | Ax = b} with n

2 constraints

→ basic solution x′ has ≤ n

2 fractional

variables. Ax = b

b

x

b x′

◮ Lovett-Meka rounding:

◮ x′ has n( 1

2 + ε) fractional variables

◮ Additional: For |vi, x − x′| ≤ λi for unit vectors

satisfying m

i=1 e−λ2

i /16 ≤ c(ε) · n

Application to approximation algorithms

◮ Classical technique: For

K = {x ∈ Rn | Ax = b} with n

2 constraints

→ basic solution x′ has ≤ n

2 fractional

variables. Ax = b

b

x

b x′

◮ Lovett-Meka rounding:

◮ x′ has n( 1

2 + ε) fractional variables

◮ Additional: For |vi, x − x′| ≤ λi for unit vectors

satisfying m

i=1 e−λ2

i /16 ≤ c(ε) · n

Better approximation algorithms for

◮ Bin Packing [R. ’13] ◮ Broadcast scheduling

[Bansal, Charikar, Krishnaswamy, Li ’14]

Open problems

Setting: Given a set system S1, . . . , Sm ⊆ [n] where each element is in at most t sets. Beck-Fiala Conjecture: disc ≤ O( √ t)

Open problems

Setting: Given a set system S1, . . . , Sm ⊆ [n] where each element is in at most t sets. Beck-Fiala Conjecture: disc ≤ O( √ t) Known bounds:

◮ 2t (constructive [Beck-Fiala ’81]) ◮ O(

√ t log n) (constructive)

◮ O(√t · log n) (non-constructive [Banaszczyk ’97])

Open problems

Setting: Given a set system S1, . . . , Sm ⊆ [n] where each element is in at most t sets. Beck-Fiala Conjecture: disc ≤ O( √ t) Known bounds:

◮ 2t (constructive [Beck-Fiala ’81]) ◮ O(

√ t log n) (constructive)

◮ O(√t · log n) (non-constructive [Banaszczyk ’97])

Open problem:

◮ Show that disc ≤ o(t)

Open problems

Setting: Given a set system S1, . . . , Sm ⊆ [n] where each element is in at most t sets. Beck-Fiala Conjecture: disc ≤ O( √ t) Known bounds:

◮ 2t (constructive [Beck-Fiala ’81]) ◮ O(

√ t log n) (constructive)

◮ O(√t · log n) (non-constructive [Banaszczyk ’97])

Open problem:

◮ Show that disc ≤ o(t)

Thanks for your attention