Fooling Gaussian PTFs via Local Hyperconcentration

Li-Yang Tan (Stanford)

Joint work with Ryan O’Donnell (CMU) and Rocco Servedio (Columbia)

Polynomial threshold functions (PTFs) over Rn

§ High-dimensional (𝑛 → ∞) geometric objects

§ Intensively studied within TCS and beyond TCS

polynomial

F (x) = sign(p(x))

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𝐹 𝑥 = −1p : Rn ! R

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𝐹 𝑥 = 1

p(x) =x21

a2+

x22

b2+

x23

c2� 1

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F : Rn ! {±1}

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living in R3

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Main complexity measure for this talk: degree of p

Think of d as growing with n = dimension of Rn

For concreteness, can think of d = log n

More complex

F (x) = sign(p(x)), deg(p) d

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d=1 d=2 d=5 d=6

. . . . . .

... then F accepts (Δ + 0.01) fraction of points

Discrepancy Sets for PTFs over Gaussian space

For all degree-d PTFs F,if F ’s Gaussian volume is Δ

Consider Rn endowed with N(0,1)n. Want small set of points such that:

Random set of points works great. Want explicit set.

–Rn

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Pseudorandom generators over Gaussian space

��� Ex⇠N(0,1)n

[F (x)]� Es⇠{0,1}r

[F (G(s))]��� "

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Definition: An ε-PRG for a class C is an explicit function G : {0,1}r Rn

such that: for every function F in C, !

C = { degree-d PTFs } This work: r = “seed length”

= log(size of discrepancy set)

Rn

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Our main result: PRG for PTFs over Gaussian space

§ Previous best seed lengths: 2!(#) dependence

§ Probabilistic method: 𝑂(𝑑 log 𝑛 + log(1/𝜀))

§ Gaussian space special case of Boolean space

§ Current best Boolean seed length: still 2!(#)

An ε-PRG for degree-d PTFs over R% with seed length:

𝑑𝜀

!(#$% &)⋅ log 𝑛

Comparison with prior results

The rest of this talk

§ Key technical ingredient: New structural lemma about polynomials

“An arbitrary polynomial, when hit with a mild random restriction, collapses to a simple polynomial

with many enjoyable properties”

§ Sketch of the proof of this lemma

With this lemma, still lots of technical work to get PRG!

Bulk of project and paper

Concentrated polynomials

Space of all degree-d polynomials

Concentrateddegree-d polynomials

Var[p] ⌧ E[p]2

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A polynomial p is concentrated if

§ “Morally constant”, i.e. “morally degree 0”

§ sign(p(x)) = sign(E[p]) for most x’s

Our new structural lemma for polynomials

An arbitrary degree-d polynomial, when hit with a mild random restriction,

becomes concentrated with high probability. Arbitrary degree-d

polynomial p

Concentrateddegree-d polynomials

Theorem:

In fact, we show that they become hyperconcentrated:

Hyperconcentrated polynomials

HyperVar[p] ⌧ E[p]2

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Random restrictions over Gaussian space

Definition: Let p be a polynomial. A λ-random restriction of p is the polynomial

q(x) = p(p�x+

p1� �y), y ⇠ N(0, 1)n.

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§ The smaller 𝜆 is, the “harsher” the random restriction

§ q = p’s behavior in a local neighborhood around y, where 𝜆 = radius of neighborhood

Our new structural lemma, restated

Arbitrary degree-d polynomial p

Hyperconcentrateddegree-d polynomials

“For most points y, the polynomial p behaves like a constant function in a fairly large neighborhood around y.”

𝜆 = 𝑑!"#$ %

𝜆 = 𝑑!"#$ %

Local Hyperconcentration Theorem

The proof of our Local Hyperconcentration Theorem

Degree-d polynomial Hyperconcentrated

(“morally degree 0”)

Thanks Avi!

“morally” degree !

"

“morally” degree !

#

Degree-d polynomial

. . .

§ Hermite analysis

§ Carbery-Wright anticoncentration inequality

Technical ingredients:

log 𝑑 iterations

Open Problems

Arbitrary degree-d polynomial p

Hyperconcentrateddegree-d polynomials

𝜆 = 𝑑!"#$ %§ Conjecture: 𝜆 = 𝑑*! + suffices

§ Random restriction lemma for polynomials over Boolean space?

Local Hyperconcentration Theorem

Thank you!