Degree and Sensitivity: tails of two distributions

Degree and Sensitivity: tails of two distributions ParikshitGopalanMicrosoft Research Rocco ServedioColumbia Univ. Avi Wigderson IAS, Princeton and Avishay Tal IAS, Princeton* (*see ECCC version)

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 multilinearps.t.p(x)=f(x) x{-1,1}n pf= T [n]f’(T) ∏iTxif’(T)= Fourier coefficients deg(f) = deg(pf) = max |T|: f’(T)≠ 0 (Real) degree of Boolean functions

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 ?? Complexity measures independent of n

1 1 Gf f : {-1,1}n {-1,1} -1 -1 Sensitivity -1 1 • s(x)= sensitivity of x • = vertex degree of xin Gf • sen(f)= maxxs(x) • [Nisan-Szegedy] sen(f) ≤deg(f)2 • [Sens-Conjecture] deg(f) ≤sen(f)c -1 1 UnderstandGf!! New parameters low sensitivity = smooth Smooth =Simple

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

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>0degε(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] Main result

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

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

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

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] ≤ nkPrρ[deg(fρ) = k] • Proof: Prρ[deg(fρ) = k] = Pr[f’ρ(K)≠0] • ≥ 2-2k E[f’ρ(K)2] Granularity of Fourier • = TPrρ[KT] fρ(T)2Heredity of Fourier • = (k/n)k ED[|T|k] Random restrictions

(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” A switching lemma Has max degree ≤ s

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)

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 kED[|T|k] ≤ akEx[s(x)k] ? kEx[s(x)k] ≤ bkED[|T|k] ?

Degree and Sensitivity: tails of two distributions