Complexity of distributions and average-case hardness Dmitry - - PowerPoint PPT Presentation

Complexity of distributions and average-case hardness

Dmitry Sokolov joint work with Dmitry Itsykson and Alexander Knop

• St. Petersburg Department of V. A. Steklov Institute of Mathematics

Problems in Theoretical Computer Science Moscow December 18-20, 2015

Sokolov D. | Complexity of distributions and average-case hardness 1/11

Definitions

Ensembles of distributions Ensemble of distributions D = {Dn}∞

n=1.

D ∈ Samp[nk] ⇔ there is a randomized O(nk)-time algorithm A such that Dn and A(1n) are equally distributed. PSamp =

k

Samp[nk]. Heuristic computations L is a language, D is an ensemble of distributions. (L, D) ∈ HeurδDTime[nk] ⇔ there is O(nk)-time algorithm A such that Pr

x←Dn[A(x) = L(x)] > 1 − δ.

HeurδP =

k

HeurδDTime[nk].

Sokolov D. | Complexity of distributions and average-case hardness 2/11

Definitions

Ensembles of distributions Ensemble of distributions D = {Dn}∞

n=1.

D ∈ Samp[nk] ⇔ there is a randomized O(nk)-time algorithm A such that Dn and A(1n) are equally distributed. PSamp =

k

Samp[nk]. Heuristic computations L is a language, D is an ensemble of distributions. (L, D) ∈ HeurδDTime[nk] ⇔ there is O(nk)-time algorithm A such that Pr

x←Dn[A(x) = L(x)] > 1 − δ.

HeurδP =

k

HeurδDTime[nk].

Sokolov D. | Complexity of distributions and average-case hardness 2/11

“Easy” problems

Theorem (Gurevich and Shelah, 1987) Let HP denote the language of Hamiltonian graphs. Then (HP, U) ∈ Heur

1 2O(√n) DTime[n].

Theorem (Babai, Erdos and Selkow, 1980) Let GI denote the language of pairs of isomorphic graphs. Then (GI, U) ∈ Heur

1 7 √n DTime[n]. Sokolov D. | Complexity of distributions and average-case hardness 3/11

“Easy” problems

Theorem (Gurevich and Shelah, 1987) Let HP denote the language of Hamiltonian graphs. Then (HP, U) ∈ Heur

1 2O(√n) DTime[n].

Theorem (Babai, Erdos and Selkow, 1980) Let GI denote the language of pairs of isomorphic graphs. Then (GI, U) ∈ Heur

1 7 √n DTime[n]. Sokolov D. | Complexity of distributions and average-case hardness 3/11

Goal and Result

Goal Result For every k there is a language L, ensemble D and small δ such that: For every 0 < a < b there is a language L, ensemble D and δ = o(1) such that:

1 D ∈ PSamp; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nk] we

have that (L, R) ∈ HeurδDTime[n].

1 D ∈ Samp[nlogb n]; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nloga n]

we have that (L, R) ∈ HeurδDTime[n].

Sokolov D. | Complexity of distributions and average-case hardness 4/11

Goal and Result

Goal Result For every k there is a language L, ensemble D and small δ such that: For every 0 < a < b there is a language L, ensemble D and δ = o(1) such that:

1 D ∈ PSamp; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nk] we

have that (L, R) ∈ HeurδDTime[n].

1 D ∈ Samp[nlogb n]; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nloga n]

we have that (L, R) ∈ HeurδDTime[n].

Sokolov D. | Complexity of distributions and average-case hardness 4/11

Goal and Result

Goal Result For every k there is a language L, ensemble D and small δ such that: For every 0 < a < b there is a language L, ensemble D and δ = o(1) such that:

1 D ∈ PSamp; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nk] we

have that (L, R) ∈ HeurδDTime[n].

1 D ∈ Samp[nlogb n]; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nloga n]

we have that (L, R) ∈ HeurδDTime[n].

Sokolov D. | Complexity of distributions and average-case hardness 4/11

Goal and Result

Goal Result For every k there is a language L, ensemble D and small δ such that: For every 0 < a < b there is a language L, ensemble D and δ = o(1) such that:

1 D ∈ PSamp; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nk] we

have that (L, R) ∈ HeurδDTime[n].

1 D ∈ Samp[nlogb n]; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nloga n]

we have that (L, R) ∈ HeurδDTime[n].

Sokolov D. | Complexity of distributions and average-case hardness 4/11

Goal and Result

Goal Result For every k there is a language L, ensemble D and small δ such that: For every 0 < a < b there is a language L, ensemble D and δ = o(1) such that:

1 D ∈ PSamp; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nk] we

have that (L, R) ∈ HeurδDTime[n].

1 D ∈ Samp[nlogb n]; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nloga n]

we have that (L, R) ∈ HeurδDTime[n].

Sokolov D. | Complexity of distributions and average-case hardness 4/11

Goal and Result

Goal Result For every k there is a language L, ensemble D and small δ such that: For every 0 < a < b there is a language L, ensemble D and δ = o(1) such that:

1 D ∈ PSamp; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nk] we

have that (L, R) ∈ HeurδDTime[n].

1 D ∈ Samp[nlogb n]; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nloga n]

we have that (L, R) ∈ HeurδDTime[n].

Sokolov D. | Complexity of distributions and average-case hardness 4/11

Goal and Result

Goal Result For every k there is a language L, ensemble D and small δ such that: For every 0 < a < b there is a language L, ensemble D and δ = o(1) such that:

1 D ∈ PSamp; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nk] we

have that (L, R) ∈ HeurδDTime[n].

1 D ∈ Samp[nlogb n]; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nloga n]

we have that (L, R) ∈ HeurδDTime[n].

Sokolov D. | Complexity of distributions and average-case hardness 4/11

Goal and Result

Goal Result For every k there is a language L, ensemble D and small δ such that: For every 0 < a < b there is a language L, ensemble D and δ = o(1) such that:

1 D ∈ PSamp; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nk] we

have that (L, R) ∈ HeurδDTime[n].

1 D ∈ Samp[nlogb n]; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nloga n]

we have that (L, R) ∈ HeurδDTime[n].

Sokolov D. | Complexity of distributions and average-case hardness 4/11

Goal and Result

Goal Result For every k there is a language L, ensemble D and small δ such that: For every 0 < a < b there is a language L, ensemble D and δ = o(1) such that:

1 D ∈ PSamp; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nk] we

have that (L, R) ∈ HeurδDTime[n].

1 D ∈ Samp[nlogb n]; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nloga n]

we have that (L, R) ∈ HeurδDTime[n].

Sokolov D. | Complexity of distributions and average-case hardness 4/11

Past and Present

Gutfreund et al., 2007 Result If NP BPP, there exists ensemble D such that: For every 0 < a < b there is a language L, ensemble D and δ = o(1) such that:

1 D ∈ Samp[quasi-poly]; 2 for every L ∈ NP-complete

and every α(n) = o(1) (L, D) / ∈ Heurα(n)BPP.

1 D ∈ Samp[nlogb n]; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nloga n]

we have that (L, R) ∈ HeurδDTime[n].

Sokolov D. | Complexity of distributions and average-case hardness 5/11

Past and Present

Gutfreund et al., 2007 Result If NP BPP, there exists ensemble D such that: For every 0 < a < b there is a language L, ensemble D and δ = o(1) such that:

1 D ∈ Samp[quasi-poly]; 2 for every L ∈ NP-complete

and every α(n) = o(1) (L, D) / ∈ Heurα(n)BPP.

1 D ∈ Samp[nlogb n]; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nloga n]

we have that (L, R) ∈ HeurδDTime[n].

Sokolov D. | Complexity of distributions and average-case hardness 5/11

Past and Present

Gutfreund et al., 2007 Result If NP BPP, there exists ensemble D such that: For every 0 < a < b there is a language L, ensemble D and δ = o(1) such that:

1 D ∈ Samp[quasi-poly]; 2 for every L ∈ NP-complete

and every α(n) = o(1) (L, D) / ∈ Heurα(n)BPP.

1 D ∈ Samp[nlogb n]; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nloga n]

we have that (L, R) ∈ HeurδDTime[n].

Sokolov D. | Complexity of distributions and average-case hardness 5/11

Past and Present

Gutfreund et al., 2007 Result If NP BPP, there exists ensemble D such that: For every 0 < a < b there is a language L, ensemble D and δ = o(1) such that:

1 D ∈ Samp[quasi-poly]; 2 for every L ∈ NP-complete

and every α(n) = o(1) (L, D) / ∈ Heurα(n)BPP.

1 D ∈ Samp[nlogb n]; 2 (L, D) /

∈ Heur1−δP;

3 for every R ∈ Samp[nloga n]

we have that (L, R) ∈ HeurδDTime[n].

Sokolov D. | Complexity of distributions and average-case hardness 5/11

Past and Present

Gutfreund et al., 2007 Result If NP BPP, there exists ensemble D such that: For every 0 < a < b there is a language L, ensemble D and δ = o(1) such that:

1 D ∈ Samp[quasi-poly]; 2 for every L ∈ NP-complete

and every α(n) = o(1) (L, D) / ∈ Heurα(n)BPP.

1 D ∈ Samp[nlogb n]; 2 (L, D) /

∈ Heur1−δR;

3 for every R ∈ Samp[nloga n]

we have that (L, R) ∈ HeurδDTime[n].

Sokolov D. | Complexity of distributions and average-case hardness 5/11

Statistical distance

Statistical distance between distributions F, G: ∆(F, G) =

• x:F(x)≥G(x)

F(x) − G(x). SD property Functions f and g satisfy SD property with parameter λ(n) (SDλ(n)(f (n), g(n))) if: there is D ∈ Samp[f (n)]; for every F ∈ Samp[g(n)], for infinitely many n: ∆(Fn, Dn) ≥ 1 − λ(n). Lemma (informal) Hierarchy theorem for (nlogb n, nloga n) ⇔ SDλ(n)(nlogb n, nloga n) property, where λ(n) → 0.

Sokolov D. | Complexity of distributions and average-case hardness 6/11

Statistical distance

Statistical distance between distributions F, G: ∆(F, G) =

• x:F(x)≥G(x)

F(x) − G(x). SD property Functions f and g satisfy SD property with parameter λ(n) (SDλ(n)(f (n), g(n))) if: there is D ∈ Samp[f (n)]; for every F ∈ Samp[g(n)], for infinitely many n: ∆(Fn, Dn) ≥ 1 − λ(n). Lemma (informal) Hierarchy theorem for (nlogb n, nloga n) ⇔ SDλ(n)(nlogb n, nloga n) property, where λ(n) → 0.

Sokolov D. | Complexity of distributions and average-case hardness 6/11

Samplable distributions hierarchy

SD property Functions f and g satisfy SD property with parameter λ(n) (SDλ(n)(f (n), g(n))) if: there is D ∈ Samp[f (n)]; for every F ∈ Samp[g(n)], for infinitely many n: ∆(Fn, Dn) ≥ 1 − λ(n). Theorem (Watson, 2013) For any a > 0, k > 0 and ǫ > 0, SD 1

k +ǫ(poly(n), na) is true. The

size of support equals k. Theorem (Itsykson, Knop, Sokolov, 2015) For every a, b, c such that 0 < a < b and c > 0, SD

1 2(log log log n)c (nlogb n, nloga n) is true. Sokolov D. | Complexity of distributions and average-case hardness 7/11

Samplable distributions hierarchy

SD property Functions f and g satisfy SD property with parameter λ(n) (SDλ(n)(f (n), g(n))) if: there is D ∈ Samp[f (n)]; for every F ∈ Samp[g(n)], for infinitely many n: ∆(Fn, Dn) ≥ 1 − λ(n). Theorem (Watson, 2013) For any a > 0, k > 0 and ǫ > 0, SD 1

k +ǫ(poly(n), na) is true. The

size of support equals k. Theorem (Itsykson, Knop, Sokolov, 2015) For every a, b, c such that 0 < a < b and c > 0, SD

1 2(log log log n)c (nlogb n, nloga n) is true. Sokolov D. | Complexity of distributions and average-case hardness 7/11

Samplable distributions hierarchy

SD property Functions f and g satisfy SD property with parameter λ(n) (SDλ(n)(f (n), g(n))) if: there is D ∈ Samp[f (n)]; for every F ∈ Samp[g(n)], for infinitely many n: ∆(Fn, Dn) ≥ 1 − λ(n). Theorem (Watson, 2013) For any a > 0, k > 0 and ǫ > 0, SD 1

k +ǫ(poly(n), na) is true. The

size of support equals k. Theorem (Itsykson, Knop, Sokolov, 2015) For every a, b, c such that 0 < a < b and c > 0, SD

1 2(log log log n)c (nlogb n, nloga n) is true. Sokolov D. | Complexity of distributions and average-case hardness 7/11

Samplable distributions hierarchy

SD property Functions f and g satisfy SD property with parameter λ(n) (SDλ(n)(f (n), g(n))) if: there is D ∈ Samp[f (n)]; for every F ∈ Samp[g(n)], for infinitely many n: ∆(Fn, Dn) ≥ 1 − λ(n). Theorem (Watson, 2013) For any a > 0, k > 0 and ǫ > 0, SD 1

k +ǫ(poly(n), na) is true. The

size of support equals k. Theorem (Itsykson, Knop, Sokolov, 2015) For every a, b, c such that 0 < a < b and c > 0, SD

1 2(log log log n)c (nlogb n, nloga n) is true. Sokolov D. | Complexity of distributions and average-case hardness 7/11

Proof of the Watson theorem for k = 2

Theorem (Watson, 2013) For any a > 0 and ǫ > 0, SD 1

2+ǫ(poly(n), na) is true.

1 Let A1, . . . , An, . . . is an enumeration of all algorithms. 2 Consider sequences ni, n∗ i such that n1 = 1, ni+1 = n∗ i + 1 and

n∗

i = 2na+1

i

.

3 Consider the following algorithm (on input 1n):

find i such that ni ≤ n ≤ n∗

i ;

if n = n∗

i return b ∈ {0, 1} such that Pr[Ai(1ni) = b] ≤ 1 2;

else run Ai(1n+1) 8 log ǫ

ǫ2

times and return the majority of answers.

Sokolov D. | Complexity of distributions and average-case hardness 8/11

Proof of the Watson theorem for k = 2

Theorem (Watson, 2013) For any a > 0 and ǫ > 0, SD 1

2+ǫ(poly(n), na) is true.

1 Let A1, . . . , An, . . . is an enumeration of all algorithms. 2 Consider sequences ni, n∗ i such that n1 = 1, ni+1 = n∗ i + 1 and

n∗

i = 2na+1

i

.

3 Consider the following algorithm (on input 1n):

find i such that ni ≤ n ≤ n∗

i ;

if n = n∗

i return b ∈ {0, 1} such that Pr[Ai(1ni) = b] ≤ 1 2;

else run Ai(1n+1) 8 log ǫ

ǫ2

times and return the majority of answers.

Sokolov D. | Complexity of distributions and average-case hardness 8/11

Proof of the Watson theorem for k = 2

Theorem (Watson, 2013) For any a > 0 and ǫ > 0, SD 1

2+ǫ(poly(n), na) is true.

1 Let A1, . . . , An, . . . is an enumeration of all algorithms. 2 Consider sequences ni, n∗ i such that n1 = 1, ni+1 = n∗ i + 1 and

n∗

i = 2na+1

i

.

3 Consider the following algorithm (on input 1n):

find i such that ni ≤ n ≤ n∗

i ;

if n = n∗

i return b ∈ {0, 1} such that Pr[Ai(1ni) = b] ≤ 1 2;

else run Ai(1n+1) 8 log ǫ

ǫ2

times and return the majority of answers.

Sokolov D. | Complexity of distributions and average-case hardness 8/11

Proof of the Watson theorem for k = 2

Theorem (Watson, 2013) For any a > 0 and ǫ > 0, SD 1

2+ǫ(poly(n), na) is true.

1 Let A1, . . . , An, . . . is an enumeration of all algorithms. 2 Consider sequences ni, n∗ i such that n1 = 1, ni+1 = n∗ i + 1 and

n∗

i = 2na+1

i

.

3 Consider the following algorithm (on input 1n):

find i such that ni ≤ n ≤ n∗

i ;

if n = n∗

i return b ∈ {0, 1} such that Pr[Ai(1ni) = b] ≤ 1 2;

else run Ai(1n+1) 8 log ǫ

ǫ2

times and return the majority of answers.

Sokolov D. | Complexity of distributions and average-case hardness 8/11

PComp

Samp Ensemble of distributions D = {Dn}∞

n=1.

D ∈ Samp[nk] ⇔ there is a randomized O(nk)-time algorithm A such that Dn and A(1n) are equally distributed. Comp Ensemble of distributions D = {Dn}∞

n=1. Dn is concentrated on

{0, 1}n. D ∈ Comp[nk] ⇔ there is a O(nk)-time algorithm A such that A(x) =

y≤x

D|x|(y). PComp =

k

Comp[nk].

Sokolov D. | Complexity of distributions and average-case hardness 9/11

PComp

Samp Ensemble of distributions D = {Dn}∞

n=1.

D ∈ Samp[nk] ⇔ there is a randomized O(nk)-time algorithm A such that Dn and A(1n) are equally distributed. Comp Ensemble of distributions D = {Dn}∞

n=1. Dn is concentrated on

{0, 1}n. D ∈ Comp[nk] ⇔ there is a O(nk)-time algorithm A such that A(x) =

y≤x

D|x|(y). PComp =

k

Comp[nk].

Sokolov D. | Complexity of distributions and average-case hardness 9/11

Results for PComp

Theorem For all a > 0 there exists D ∈ PComp such that for all F ∈ Comp[na] for infinitely many n: ∆(Dn, Fn) ≥ 1 − 2−n. Theorem For all a > 0 there exists a language L and D ∈ PComp such that: (L, F) ∈ HeurO( 1

2n )DTime[n] for all F ∈ Comp[na];

(L, D) / ∈ Heur1−

1 2n−1 R. Sokolov D. | Complexity of distributions and average-case hardness 10/11

Results for PComp

Theorem For all a > 0 there exists D ∈ PComp such that for all F ∈ Comp[na] for infinitely many n: ∆(Dn, Fn) ≥ 1 − 2−n. Theorem For all a > 0 there exists a language L and D ∈ PComp such that: (L, F) ∈ HeurO( 1

2n )DTime[n] for all F ∈ Comp[na];

(L, D) / ∈ Heur1−

1 2n−1 R. Sokolov D. | Complexity of distributions and average-case hardness 10/11

Open question

Improve Watson’s theorem for λ → 0. Prove hierarchy theorem for PSamp.

Sokolov D. | Complexity of distributions and average-case hardness 11/11