Approximate List-Decoding and Uniform Hardness Amplification

1
Approximate List-Decoding and Uniform Hardness
Amplification

• Russell Impagliazzo (UCSD)
• Ragesh Jaiswal (UCSD)
• Valentine Kabanets (SFU)

2
Hardness Amplification
f
F
Hard function
Harder function

• Given a hard function we can get an even harder
  function

3
Hardness
0, 1n
0, 1n
s
f
d.2n

• A function f is called d-hard for circuits of
  size s (Algorithm with running time t), if any
  circuit of size s (Algorithm with running time t)
  makes mistake in predicting the function on at
  least d fraction of the inputs

4
XOR Lemma
0, 1nk
0, 1n
f
f?k
f
f
f
0/1
k
XOR
0/1
f?k0, 1nk 0, 1 f?k(x1,, xk)
f(x1) ? ? f(xk)

• XOR Lemma If f is d-hard for size s circuits,
  then f?k is
• (1/2 - e)-hard for size s circuits (e
  e-O(dk), s spoly(d, e))

5
XOR Lemma Proof Ideal case
C?
(which computes f?k for at least (½ e) fraction
of inputs)
A
whp
C (which computes f for at least (1 - d)
fraction of inputs)
6
XOR Lemma Proof
A lesser nonuniform reduction
C?
(which computes f?k for at least (½ e) fraction
of inputs)
A
Advice (Advicepoly(1/e))
whp
C1
Cl
C (which computes f for at least (1 - d)
fraction of inputs)
One of them computes f for at least (1 - d)
fraction of inputs
l 2Advice 2poly(1/e)
7
Optimal List Size

• Question What is the reduction in the list size
  we should target?
• A good combinatorial answer using error
  correcting codes

C?
A
whp
C1
Cl
8
XOR-based Code T03

• Think of a binary message msg on M2n bits as a
  truth-table of a Boolean function f.
• The code of msg is of length Mk where
  code(x1,,xk) f(x1) ? ? f(xk)

x (x n)
msg
f(x)
x (x1, , xk)
code
f(x1) ? ? f(xk)
9
List Decoder
(1/2 ?)
m
c
w
XOR Encoding
Decoding
channel
m1,,ml
(1 - d)

• Decoder
• Local
• Approximate
• List

Information theoretically l should be O(1/?2)
10
The List Size

• The proof of Yaos XOR Lemma yields an
• approximate local list-decoding algorithm for
• the XOR-code defined above
• But the list size is 2poly(1/?) rather than
  the
• optimal poly(1/?)
• Goal Match the information theoretic bound
• on list-decoding i.e. get advice of length
• log(1/?)

11
The Main Result
12
The Main Result
C? ((½ e)-computes f?k)
A
Advice(Advice log(1/e))
whp
C ((1 - d)-computes f)

• e poly(1/k), d O(k-0.1)
• Running time of A and size of C is at most
  poly(C?, 1/e)

13
The Main Result
C? ((½ e)-computes f?k)
A
w.p. poly(e)
C ((1 - d)-computes f)

• e poly(1/k), d O(k-0.1)
• Running time of A and size of C is at most
  poly(C?, 1/e)

14
The Main Result
C?((½ e)-computes f?k)
Advice(Advice log(1/e))
A
A
whp
w.p. poly(e)
C ((1 - d)-computes f)
Cl
C1
l poly(1/e)
At least one of them (1 - ?)-computes f
Advice efficient XOR Lemma

• We get a list size of poly(1/e)
• which is optimal but
• e is large e poly(1/k)

15
Uniform Hardness Amplification
16
Uniform Hardness Amplification

• What we want

f hard wrt BPP
g harder wrt BPP

• What we get

Advice efficient XOR Lemma
f hard wrt BPP/log
g harder wrt BPP
17
Uniform Hardness Amplification

• What we can do

BDCGL92
f ? NP hard wrt BPP
f ? NP hard wrt BPP/log
Advice efficient XOR Lemma
Simple average-case reduction
g ? PNP harder wrt BPP
g ? ?? harder wrt BPP
h ? PNP hard wrt BPP
1/nc
½ - 1/nd

• g not necessarily ? NP but g ? PNP
• PNP poly-time TM which can make polynomially
  many
• parallel Oracle queries to an NP oracle

Trevisan gives a weaker reduction (from 1/nc to
(1/2 1/(log n)a) hardness) but within NP.
18
Techniques
19
Techniques

• Advice efficient Direct Product Theorem
• A Sampling Lemma
• Learning without Advice
• Self-generated advice
• Fault tolerant learning using faulty advice

20
Direct Product Theorem
0, 1nk
0, 1n
f
fk
f
f
f
0/1
k
concatenation
fk0, 1nk 0, 1k fk(x1,, xk)
f(x1) f(xk)
0, 1k

• Direct Product Theorem If f is dhard for size
  s circuits, then fk is
• (1 - e)-hard for size s circuits (e
  e-O(dk), s spoly(d, e))
• Goldreich-Levin Theorem XOR Lemma and Direct
  Product Theorem
• are saying the same thing

21
XOR Lemma from Direct Product Theorem
C? ((½ e)-computes f?k)

• Using Goldreich-Levin Theorem

A1
whp
CDP (poly(e)-computes fk)
A2
w.p. poly(e)
C ((1 - d)-computes f)

• e poly(1/k), d O(k-0.1)

22
LEARN from IW97
CDP (?-computes fk)
LEARN IW97
Advice n/?2 pairs of (x, f(x)) for independent
uniform xs
whp
C ((1 - d)-computes f)

• e e-O(dk)

23
Goal

• We want to eliminate the advice (or the (x,
  f(x)) pairs).
• In exchange we are ready to make some
  compromise on
• the success probability of the randomized
  algorithm

CDP (?-computes fk)
LEARN IW97
Advice n/?2 pairs of (x, f(x)) for independent
uniform xs
LEARN
whp
w.p. poly(?)
No advice!!!
C ((1 - d)-computes f)

• e e-O(dk)

• e poly(1/k), d O(k-0.1)

24
Self-generated advice
25
Imperfect samples

• We want to use the circuit CDP to generate n/?2
  pairs (x, f(x)) for independent uniform xs
• We will settle for n/?2 pairs (x,bx)
• The distribution on xs is statistically close
  to uniform and
• for most xs we have bx f(x).
• Then run a fault-tolerant version of LEARN on CDP
  and the generated pairs (x,bx)

26
How to generate imperfect samples
27
A Sampling Lemma
xk
x1
x2
x3
2nk

• D is a Uniform Distribution

nk
28
A Sampling Lemma
xk
x1
x2
x3

• G gt ? 2nk
• Stat-Dist(D, U) lt ((log 1/?)/k)1/2

G
nk
29
Getting Imperfect Samples

• G subset of inputs on which CDP(x) fk(x)
• G gt ? 2nk
• Pick a random k-tuple x, then pick a random
  subtuple x of size k1/2
• With probability ?, x lands in the good set G
• Conditioned on this, the Sampling Lemma says that
  x is close to being uniformly distributed
• If k1/2 gt the number of samples required by
  LEARN, then done!
• Else

30
Direct Product Amplification

• CDP CDP which poly(e)-computes fk
• where (k)1/2 gt n/e2
• ??
• CDP CDP such that for at least poly(e)
  fraction of k-tuples, x
• CDP(x) and fk(x) agree on most bits

31
Putting Everything Together
32
CDP for fk
CDP for fk
DP Amplification
Sampling
pairs (x,bx)
Fault tolerant LEARN
with probability gt poly(?)
circuit C (1-?)-computes f
Repeat poly(1/?) times to get a list containing a
good circuit for f, w.h.p.
33
Open Questions
34
Open Questions

• Advice efficient XOR Lemma for smaller ?
• For e gt exp(-ka) we get a quasi-polynomial list
  size
• Can we get an advice efficient hardness
  amplification result using a monotone combination
  function m (instead of ?)?
• Some results Buresh-Oppenheim, Kabanets,
  Santhanam use monotone list-decodable codes to
  re-prove Trevisans results for amplification
  within NP

35
Thank You