1. Degree and Sensitivity: tails of two distributions
Parikshit Gopalan Microsoft Research
Rocco Servedio Columbia Univ.
Avi Wigderson IAS, Princeton
and
Avishay Tal IAS, Princeton*
(*see ECCC version)

2. (Real) degree of Boolean functions
f : {-1,1}n  {-1,1} in R[x1,x2,…,xn]
deg(f) = min d: ∃ Real polynomial p of degree d
such that p(x)=f(x) x  {-1,1}n
Ex1: Maj(x,y,z) = ½ (x+y+z – xyz) deg=3
Ex2: NAE(x,y,z) = ½ (xy+yz+xz –1) deg=2
pf = unique multilinear p s.t. p(x)=f(x) x{-1,1}n
pf = T [n] f’(T) ∏iT xi f’(T) = Fourier coefficients
deg(f) = deg(pf) = max |T|: f’(T) ≠ 0
Of course, this is in PPM

3. Complexity measures f : {-1,1}n  {-1,1}
D(f) – Deterministic decision tree complexity
R(f) – Probabilistic decision tree complexity
Q(f) – Quantum decision tree complexity
N(f) – Certificate complexity
deg(f) – Real degree
deg∞(f) – L∞ approximate degree
bs(f) – Block sensitivity ……
[Nisan,…] All parameters are polynomially related
sen(f) – Sensitivity ??
Of course, this is in PPM
independent of n

4. Sensitivity Gf f : {-1,1}n  {-1,1} s(x) = sensitivity of x
= vertex degree of x in Gf
sen(f) = maxx s(x)
[Nisan-Szegedy] sen(f) ≤ deg(f)2
[Sens-Conjecture] deg(f) ≤ sen(f)c
Understand Gf !!
New parameters
Of course, this is in PPM
low sensitivity = smooth
Smooth
= Simple

5. Fourier dist. & Real approx.
f : {-1,1}n  {-1,1} L2-approximation
degε (f) = min d: ∃ Real polynomial q of degree d
such that Ex {-1,1}n [|q(x)-f(x)|2] ≤ ε
T f’(T)2 =1 Fourier dist: T [n] with prob f’(T)2
εt = |T|>t f’(T)2 : Tails of the Fourier distribution
degε(f) = min d: εd ≤ ε
- Best approximator q is a truncation of pf
- deg0(f) = deg(f)
Of course, this is in PPM

6. Main result f : {-1,1}n  {-1,1} deg0(f) = deg(f)
degε(f) = min d: εd ≤ ε (best approximation in L2)
[Sens-Conj] deg0(f) ≤ sen(f)c
[Thm1] ε>0 degε (f) ≤ sen(f) log(1/ε)
[Thm2] This is “optimal”:
∃c>0,δ<1 degε (f) ≤ sen(f)c log(1/ε)δ 
 [Sens-Conj]
Of course, this is in PPM

7. Sensitive trees f : {-1,1}n  {-1,1} A sensitive tree in Gf
is a subgraph H of Gf
H is a tree
dimensions of edges(H) distinct
ts(f) = max {dim H: H sens tree}
sen(f) = max {dim H: H sens star}
[Thm3] deg(f) ≤ ts(f)2
[TS-Conj] deg(f) ≤ ts(f) 1 -1 2 2 3 3 1 2 1 3

8. Moments: Fourier vs. Sensitivity
f : {-1,1}n  {-1,1}, pf = T f’(T)XT
D: draw T [n] with probability f’(T)2
Dk=ED[|T|k] : Fourier moments
S: draw x  {-1,1}n uniformly.
Sk=Ex[s(x)k]: Sensitivity moments
[Moment-Conj] k Dk ≤ akSk (independent of f,n)
[Fact] D1 = S1 (Total influence), [Kalai] D2 = S2
[Thm4] deg(f) ≤ ts(f)  [Moment-Conj]
Average-case
variants of
deg(f) & sens(f)
Of course, this is in PPM

9. Proof of the main result
f : {-1,1}n  {-1,1}
degε(f) = min d: εd ≤ ε (best approximation in L2)
[Thm1] ε>0 degε (f) ≤100 sen(f) log(1/ε)
s k
(t=100sk)
εt = |T|>t f’(T)2 = PrD[|T|>t] = PrD[|T|k>tk] ≤
≤ E[|T|k]/tk ≤ ………… ≤ exp(-k)
≤(n/t)k Pr[deg(fρ) = k] ≤
(1) (2)
Of course, this is in PPM
Random restriction
Leaving k var alive

10. Random restrictions ρ : {x1,x2,…,xn}  {-1,1,*} at random from
Rk= {ρ = (K,y) : K=ρ-1(*), |K|=k, y  {-1,1}n-K }
(1) ED[|T|k] ≤ nk Prρ[deg(fρ) = k]
Proof: Prρ [deg(fρ) = k] = Pr[f’ρ(K) ≠0]
≥ 2-2k E[f’ρ(K)2] Granularity of Fourier
= T Prρ [KT] fρ(T)2 Heredity of Fourier
= (k/n)k ED[|T|k]
Of course, this is in PPM

11. A switching lemma (2) Prρ[deg (fρ) = k] ≤ (10sk/n)k
(2’) Prρ[ts (fρ) = k] ≤ (10sk/n)k
Bad = { ρ : ts (fρ) = k }  Rk
Bad  [2n]×[s]k×[2]k
ρ  A DFS path in the sensitive tree
|Bad|/|Rk| < (10sk/n)k
[Moment-Conj]  Hastad’s switching lemma
In paper we use
“proper walks”
Has max degree ≤ s
Of course, this is in PPM

12. Applications
Learning algorithm for low-sensitivity functions in time (1/ε)poly(s)
Under uniform dist: Uses [LMN]
Exact learning alg: Uses [GNSTW]
New proof of the switching lemma
Better bounds on Entropy-Influence conj:
[EntInf-Conj] Ent(f) ≤ c.Inf(f)
[Fact] Ent(f) ≤ c.Inf(f).log n
[EntInf-Conj] Ent(f) ≤ c.Inf(f).log sen(f)

13. Conclusions & open problems
Prove consequences of [Sens-Conj]
[GSTW] degε (f) ≤ sen(f) log(1/ε)
[GNSTW] depth(f) ≤ poly(sen(f))
Prove [Sens-Conj] !!!
If not…
deg(f) ≤ ts(f)? Relate new parameters
k ED[|T|k] ≤ ak Ex[s(x)k] ?
k Ex[s(x)k] ≤ bk ED[|T|k] ?