The 1

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The 1st annual (?) workshop
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Communication under Channel UncertaintyOblivio
us channels
Michael Langberg
California Institute of Technology
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Coding theory
y
m ? 0,1k
Noise
x C(m) ? 0,1n
decode
m
Error correcting codes
C 0,1k
0,1n
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Communication channels
X
Y
x
e
yx?e

• Design of C depends on properties of channel.
• Channel W
• W(ex) probability that error e is imposed by
  channel when xC(m) is transmitted.
• In this case yx?e is received.
• BSCp Binary Symmetric Channel.
• Each bit flipped with probability p.
• W(ex)pe(1-p)n-e

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Success criteria
C 0,1k
0,1n

• Let D 0,1n 0,1k be a decoder.
• C is said to allow the communication of m over W
  (with D) if PreD(C(m)?e)m 1.
• Probability over W(eC(m)).

BSCp Shannon exist codes with rate 1-H(p)
(optimal).

• C is said to allow the communication of 0,1k
  over W (with D) if Prm,eD(C(m)?e)m 1.
• Probability uniform over 0,1k and over
  W(eC(m)).
• Rate of C is k/n.

e
xC(m)
yx?e
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Channel uncertainty

• What if properties of the channel are not known?
• Channel can be any channel in family W W.
• Objective design a code that will allow
  communication not matter which W is chosen in W.
• C is said to allow the communication of 0,1k
  over channel family W if there exists a decoder D
  s.t. for each W?W C,D allow communication of
  0,1k over W.

?
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The family Wp
Adversarial model in which the channel W is
chosen maliciously by an adversarial jammer
within limits of Wp.

• A channel W is a p-channel if it can only change
  a p-fraction of the bits transmitted W(ex)0 if
  egtpn.
• Wp family of all p-channels.
• Communicating over Wp design a code that enables
  communication no matter which p-fraction of bits
  are flipped.

Power constrain on W
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Communicating over Wp

• Communicating over Wp design a code C that
  enables communication no matter which p-fraction
  of bits are flipped.
• Equivalently minimum distance of C is 2pn.

• What is the maximum achievable
• rate over Wp?
• Major open problem.
• Known 1-H(2p) R lt 1-H(p)

C
Min. distance
Wp
X
Y
0,1n
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This talk
X
Y

• Communication over Wp not fully understood.
• Wp does not allow communication w/ rate 1-H(p).
• BSCp allows communication at rate 1-H(p).
• In essence BSCp?Wp (power constraint).
• Close gap by considering restriction of Wp.
• Oblivious channels
• Communication over Wp with the assumption that
  the channel has a limited view of the transmitted
  codeword.

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Recall

• Communicating over Wp design a code C that
  enables communication no matter which p-fraction
  of bits are flipped (equivalently minimum
  distance of C is 2pn).
• Known 1-H(2p) R lt 1-H(p).
• Communicating over Wp Corresponds to an energy
  constraint on the channels behavior.
• Would like to study rate achievable under
  additional limitations.

X
Y
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Oblivious channels
X
Y

• Communicating over Wp only p-fraction of bits
  can be flipped.
• Think of channel as adversarial jammer.
• Jammer acts maliciously according to codeword
  sent.
• Additional constraint Would like to limit the
  amount of information the adversary has on
  codeword x sent.
• For example
• Channel with a window view.
• In general correlation between codeword x and
  error e imposed by W is limited.

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Oblivious channels model

• A channel W is oblivious if W(ex) is independent
  of x.
• BSCp is an oblivious channel.
• A channel W is partially-oblivious if the
  dependence of W(ex) on x is limited
• Intuitively I(e,x) is small.
• Partially oblivious - definition
• For each x W(ex)Wx(e) is a distribution over
  0,1n.
• Limit the size of the family Wxx.

Let W0 and W1 be two distributions over
errors. Define W as follows W(ex) W0(e) if
the first bit of x is 0. W(ex) W1(e) if the
first bit of x is 1. W is almost completely
oblivious.
X
Y
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Families of oblivious channels

• A family of channels W?? Wp is (partially)
  oblivious if each W?W is (partially) oblivious.
• Study the rate achievable when comm. over W.
• Jammer W is limited in power and knowledge.
• BSCp is an oblivious channel in Wp.
• Rate on BSCp 1-H(p).
• Natural question Can this be extended to any
  family of oblivious channels?

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Our results
X
Y

• Study both oblivious and partially oblivious
  families.
• For oblivious families W?one can achieve rate
  1-H(p).
• For families W of partially oblivious channels
  in which
• ?W?W Wxx of size at most 2?n.
• Achievable rate 1-H(p)-? (if ? lt
  (1-H(p))/3).
• Sketch proof for oblivious W.

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Previous work

• Oblivious channels in W have been addressed by
  CsiszarNarayan as a special case of
  Arbirtrarily Varying Channels with state
  constraints.
• CsiszarNarayan show that rate 1-H(p) for
  oblivious channels in W using the method of
  types.
• Partially oblivious channels not defined
  previously.
• For partially oblivious channels CsiszarNarayan
  implicitly show 1-H(p)-30? (compare with
  1-H(p)-?).
• Our proof technique are substantially different.

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Linear codes
Linear codes work well vs. BSCp.

• Observation linear codes will not allow rate
  1-H(p) on W unless they allow comm. over Wp
  (dist. 2pn)
• Exists a codeword c of weight less than 2pn.
• Take W to be the channel that always imposes
  error e s.t. e? pn and dist(e,c) pn.
• Can show W does not allow transition of ½ the
  messages.
• No codes of distance 2pn and rate 1-H(p) ? linear
  codes do not suffice.
• Natural candidate random codes (each codeword
  chosen at random).

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Proof technique Random codes

• Let C be a code (of rate 1-H(p)) in which each
  codeword is picked at random.
• Show with high probability C allows comm. over
  any oblivious channel in W (any channel W which
  always imposes the same distribution over
  errors).
• Implies Exists a code C that allows comm. over
  W with rate 1-H(p).

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Proof sketch
X
Y
x
e
yx?e

• Show with high probability C allows comm. over
  any oblivious channel in W.
• Step 1 show that C allows comm. over W iff C
  allows comm. over channels W that always impose a
  single error e (e pn).
• Step 2 Let We be the channel that always imposes
  error e.
• Show that w.h.p. C allows comm. over We.
• Step 3 As there are only 2H(p)n channels We
  use union bound.

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Proof of Step 2
X
Y
x
e
yx?e

• Step 2 Let We be the channel that always imposes
  error e.
• Show that w.h.p. C allows comm. over We.
• Let D be the Nearest Neighbor decoder.
• By definition C allows comm. over We iff
• for most codewords xC(m) D(x?e)m.
• Codeword xC(m) is disturbed if D(x?e)?m.
• Random C expected number of disturbed
  codewords is small (i.e. in
  expectation C allows communication).
• Need to prove that number of disturbed codewords
  is small w.h.p.

e
C
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Concentration
X
Y
x
e
yx?e

• Expected number of disturbed codewords is small.
• Need to prove that number of disturbed
• codewords is small w.h.p.
• Standard tool - Concentration inequalities
• Azuma, Talagrand, Chernoff.
• Work well when random variable has small
• Lipschitz coefficient.
• Study Lipschitz coefficient of our process.

e
C
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Lipschitz coefficient
X
Y
x
e
yx?e

• Lipschitz coefficient in our setting
• Let C and C be two codes that differ
• in a single codeword.
• Lipschitz coefficient difference between
• number of disturbed codewords in C and C
• w.r.t. We.
• Can show that L. coefficient is very large.
• Cannot apply standard concentration techniques.
• What next?

e
C
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Lipschitz coefficient
Vu Random process in which Lipschitz
coefficient has small expectation and
variance will have exponential
concentration probability of deviation from
expectation is exponential in deviation. KimVu
concentration of low degree multivariate
polynomials (extends Chernoff).
X
Y
x
e
yx?e

• Lipschitz coefficient in our setting is large.
• However one may show that on average
• Lipschitz coefficient is small.
• This is done by studying the list decoding
• properties of random C.
• Once we establish that average Lipschitz
• coef. is small one may use recent concentration
  
• result of Vu to obtain proof.
• Establishing average Lipschitz coef. is
  technically involved.

e
C
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Conclusions and future research

• Theme Communication over Wp not fully
  understood. Gain understanding of certain
  relaxations of Wp.
• Seen
• Oblivious channels W? Wp.
• Allows rate 1-H(p).
• Other relaxations
• Online adversaries.
• Adversaries restricted to changing certain
  locations (unknown to X and Y).