The 1’st annual (?) workshop

The 1’st annual (?) workshop

Communication under Channel Uncertainty:Oblivious channels Michael Langberg California Institute of Technology

X Y Coding theory y m  {0,1}k Noise x = C(m)  {0,1}n decode m Error correcting codes C: {0,1}k {0,1}n

Communication channels X Y x e y=xe • Design of C depends on properties of channel. • Channel W: • W(e|x) = probability that error e is imposed by channel when x=C(m) is transmitted. • In this case y=xe is received. • BSCp: Binary Symmetric Channel. • Each bit flipped with probability p. • W(e|x)=p|e|(1-p)n-|e|

X Y Success criteria C: {0,1}k {0,1}n • Let D:{0,1}n {0,1}k be a decoder. • C is said to allow the communicationofmover W (with D) if Pre[D(C(m)e)=m] ~ 1. • Probability over W(e|C(m)). BSCp[Shannon]: exist codes with rate ~ 1-H(p) (optimal). • C is said to allow the communicationof{0,1}kover W (with D) if Prm,e[D(C(m)e)=m] ~ 1. • Probability uniform over {0,1}k and over W(e|C(m)). • Rate of C is k/n. e x=C(m) y=xe

X Y 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 {0,1}k over channel family W if there exists a decoder D s.t. for each WW: C,D allow communication of {0,1}k over W. ?

X Y 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(e|x)=0 if |e|>pn. • Wp = family of all p-channels. • Communicating overWp: design a code that enables communication no matter which p-fraction of bits are flipped. Power constrain on W

Communicating over Wp • Communicating overWp: 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 < 1-H(p) * * * * * * * * * C Min. distance Wp X Y {0,1}n

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 theassumption that the channel has a limited view of the transmitted codeword.

Oblivious channels X Y • Communicating overWp: 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.

Oblivious channels: model • A channel W is oblivious if W(e|x) is independent of x. • BSCp is an oblivious channel. • A channel W is partially-oblivious if the dependence of W(e|x) on x is limited: • Intuitively I(e,x) is small. • Partially oblivious - definition: • For each x: W(e|x)=Wx(e) is a distribution over {0,1}n. • Limit the size of the family {Wx|x}. Let W0 and W1 be two distributions over errors. Define W as follows: W(e|x) = W0(e) if the first bit of x is 0. W(e|x) = W1(e) if the first bit of x is 1. W is almost completely oblivious. X Y

Families of oblivious channels • A family of channelsW* 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. • BSCpis an oblivious channel “in” Wp. • Rate on BSCp ~ 1-H(p). • Natural question: Can this be extended to any family of oblivious channels?

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* : {Wx|x} of size at most 2n. Achievable rate ~ 1-H(p)- (if  < (1-H(p))/3). • Sketch proof for oblivious W*.

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.

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).

Proof sketch X Y x e y=x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.

Proof of Step 2 X Y x e y=x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 x=C(m): D(xe)=m. • Codeword x=C(m) is disturbed if D(xe)m. • RandomC: 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

Concentration X Y x e y=x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

Lipschitz coefficient X Y x e y=x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

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 y=x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

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).

The 1’st annual (?) workshop