Pseudorandom Generators for Combinatorial Shapes

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Parikshit Gopalan, MSR SVC

Raghu Meka, UT AustinOmer Reingold, MSR SVC

David Zuckerman, UT Austin

PRGs for Small Space?

Poly. width ROBPs. Nis-INW best.

Is RL = L?

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Saks-Zhou: Nis 90, INW94: PRGs for polynomial width ROBP’s with seed .

Can do O(log n) for these!

Small-Bias

Comb. Rectangles

Modular Sums

0/1 Halfspaces

Combinatorial shapes: unifies and generalizes

all.

What are Combinatorial Shapes?

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Fooling Linear Forms

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For

Question: Can we have this “pseudorandomly”?

Generate ,

Why Fool Linear Forms?

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Special case: small-bias spaces

Symmetric functions on subsets.Previous best: Nisan90, INW94.

Been difficult to beat Nisan-INW barrier for natural cases.

Question: Generate ,

Combinatorial Rectangles

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What about

Applications: Volume estimation, integration.

Combinatorial Shapes

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Combinatorial Shapes

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PRGs for Combinatorial Shapes

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Unifies and generalizesCombinatorial rectangles – sym. function h is

ANDSmall-bias spaces – m = 2, h is parity0-1 halfspaces – m = 2, h is shifted majority

Thm: PRG for (m,n)-Comb. shapes with

seed .

Previous Results

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Reference Function Class Seed LengthNis90, INW94 All ShapesLLSZ92 Comb. Rects, Hitting

setsEGL+92, ASWZ96, Lu02

Comb. RectanglesNN93, LRTV09, MZ09

Modular Sums

M., Zuckerman 10

Halfspaces

Our Results

Discrete Central Limit TheoremSum of ind. random variables ~ Gaussian

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Thm:

Discrete Central Limit TheoremClose in stat. distance to binomial

distribution

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Optimal error: .• Proof analytical - Stein’s method (Barbour-

Xia98).

Thm:

This Talk

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1. PRGs for Cshapes with m = 2.Illustrates main ideas for general case.

2. PRG for general Cshapes.

3. Proof of discrete central limit theorem.

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Question: Generate ,

Fooling Cshapes for m = 2 ~ Fooling 0/1 linear forms in TV.

Fooling Linear Forms in TV

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1. Fool linear forms with small test sizes.Bounded independence, hashing.

2. Fool 0-1 linear forms in cdf distance.PRG for halfspaces: M., Zuckerman

3. PRG on n/2 vars + PRG fooling in cdf PRG for linear forms, large test sets.

Thm MZ10: PRG for halfspaces with seed

3. Convolution Lem: close in cdf to close in TV.

Analysis of recursionElementary proof of discrete CLT.

Question: Generate ,

Recursion Step for 0-1 Linear Forms

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For intuition consider

X1 Xn/2+1 Xn… Xn/2 …

PRG -fool in TV PRG -fool in CDF

PRG -fool in TV

True randomness

PRG -fool in TV

Recursion Step: Convolution Lemma

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Lem:

Convolution Lemma

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Problem: Y could be even, Z odd.Define Y’:Approach:

Lem:

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Convexity of : Enough to study

Recursion for General Case

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Problem: Test set skewed to first half.

Solution: Do the partitioning randomly. Test set splits evenly to each half.Can’t use new bits for every step.

Analysis: Induction. Balance out test set. Final Touch: Use Nisan-INW across recursions.

Recursion for General Case

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X1

Xn

X2 … X

3

X1 Xi …

MZ on n/2 Vars

Xj …

MZ on n/4 Vars

…

Truly random

Geometric dec. blocks via Pairwise

Permutations

Fool 0-1 Linear forms in TV with seed

This Talk

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1. PRGs for Cshapes with m = 2.Illustrates main ideas for general case.

2. PRG for general Cshapes.

3. Proof of discrete central limit theorem.

From Shapes to Sums

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From m = 2 to General m

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Test set Large vs Small

For large: true ~ binomial

For small: k-wise

High or Low VarianceVar. high: shift-

invarianceFor small: k-wise

1. PRG fooling low variance CSums.Sandwiching poly., bounded independence.

2. PRG fooling high var. CSums in cdf.Same generator, similar analysis.

3. PRG on n/2 vars + PRG fooling in cdf PRG for high variance CSums

PRGs for CShapes

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3. Convolution Lemma. Work with shift invariance.Balance out variances (ala test set

sizes).

Low Variance Combinatorial Sums

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Need to look at the generator for halfspaces.

Some notation: Pairwise-indep. hash familyk-wise independent generatorWe use

INW on top to choose z’s.

Core Generator

x1

x2

x3 … x

nx5

x4

xk … x

1x3

xk

x5

x4

x2

1 2 t

… xn… x

5x4

x2

2 t

xnxn

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Randomness:

Low Variance Combinatorial Sums

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Why easy for m = 2? Low var. ~ small test setTest set well spread out: no bucket more than

O(1).O(1)-independence suffices.x

1x3

xk

1

… … x5

x4

x2

2 t

xn

x3

xk

x5

Low Variance Combinatorial Sums

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For general m: can have small biases.Each coordinate has non-zero but small bias.

x1

x3

xk

1

… … x5

x4

x2

2 t

xn

Low Variance Combinatorial Sums

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Total variance Variance in each bucket !Let’s exploit that.

x1

x3

xk

1

… … x5

x4

x2

2 t

xn

Low Variance Combinatorial Sums

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Use 22-wise independence in each bucket.

Union bound across buckets.Proof of lemma: sandwiching

polynomials.

Summary of PRG for CSums

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1. PRGs for low-var CSumsBounded independence, hashingSandwiching polynomials

2. PRGs for high-var CSums in cdfPRG for halfspaces

3. PRG on n/2 vars + PRG in cdf PRG for high-var CSums.

PRG for CSums

This Talk

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1. PRGs for Cshapes with m = 2.Illustrates main ideas for general case.

2. PRG for general Cshapes.

3. Proof of discrete central limit theorem.

Discrete Central Limit TheoremClose in stat. distance to binomial

distribution

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Thm:

Lem:

Convolution Lemma

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Same mean, variance

All four approx.same means,

variances

Discrete Central Limit Theorem

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Discrete Central Limit Theorem

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By CLT: small.By unimodality: shift

invariant.

Hence proved!General integer valued case

similar.

All parts have similar means and variances

Open Problems

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Optimal dependence on error rate?Non-explicit: Solve for halfspaces

More general/better notions of symmetry?Capture “order oblivious” small space.

Better PRGs for Small Space?

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Thank You