11 research outputs found
The Price of Uncertain Priors in Source Coding
We consider the problem of one-way communication when the recipient does not
know exactly the distribution that the messages are drawn from, but has a
"prior" distribution that is known to be close to the source distribution, a
problem first considered by Juba et al. We consider the question of how much
longer the messages need to be in order to cope with the uncertainty about the
receiver's prior and the source distribution, respectively, as compared to the
standard source coding problem. We consider two variants of this uncertain
priors problem: the original setting of Juba et al. in which the receiver is
required to correctly recover the message with probability 1, and a setting
introduced by Haramaty and Sudan, in which the receiver is permitted to fail
with some probability . In both settings, we obtain lower bounds that
are tight up to logarithmically smaller terms. In the latter setting, we
furthermore present a variant of the coding scheme of Juba et al. with an
overhead of bits, thus also establishing the
nearly tight upper bound.Comment: To appear in IEEE Transactions on Information Theor
Communication with Contextual Uncertainty
We introduce a simple model illustrating the role of context in communication
and the challenge posed by uncertainty of knowledge of context. We consider a
variant of distributional communication complexity where Alice gets some
information and Bob gets , where is drawn from a known
distribution, and Bob wishes to compute some function (with high
probability over ). In our variant, Alice does not know , but only
knows some function which is an approximation of . Thus, the function
being computed forms the context for the communication, and knowing it
imperfectly models (mild) uncertainty in this context.
A naive solution would be for Alice and Bob to first agree on some common
function that is close to both and and then use a protocol for
to compute . We show that any such agreement leads to a large overhead
in communication ruling out such a universal solution.
In contrast, we show that if has a one-way communication protocol with
complexity in the standard setting, then it has a communication protocol
with complexity in the uncertain setting, where denotes
the mutual information between and . In the particular case where the
input distribution is a product distribution, the protocol in the uncertain
setting only incurs a constant factor blow-up in communication and error.
Furthermore, we show that the dependence on the mutual information is
required. Namely, we construct a class of functions along with a non-product
distribution over for which the communication complexity is a single
bit in the standard setting but at least bits in the
uncertain setting.Comment: 20 pages + 1 title pag
The Power of Shared Randomness in Uncertain Communication
In a recent work (Ghazi et al., SODA 2016), the authors with Komargodski and Kothari initiated the study of communication with contextual uncertainty, a setup aiming to understand how efficient communication is possible when the communicating parties imperfectly share a huge context. In this setting, Alice is given a function f and an input string x, and Bob is given a function g and an input string y. The pair (x,y) comes from a known distribution mu and f and g are guaranteed to be close under this distribution. Alice and Bob wish to compute g(x,y) with high probability. The lack of agreement between Alice and Bob on the function that is being computed captures the uncertainty in the context. The previous work showed that any problem with one-way communication complexity k in the standard model (i.e., without uncertainty, in other words, under the promise that f=g) has public-coin communication at most O(k(1+I)) bits in the uncertain case, where I is the mutual information between x and y. Moreover, a lower bound of Omega(sqrt{I}) bits on the public-coin uncertain communication was also shown.
However, an important question that was left open is related to the power that public randomness brings to uncertain communication. Can Alice and Bob achieve efficient communication amid uncertainty without using public randomness? And how powerful are public-coin protocols in overcoming uncertainty? Motivated by these two questions:
- We prove the first separation between private-coin uncertain communication and public-coin uncertain communication. Namely, we exhibit a function class for which the communication in the standard model and the public-coin uncertain communication are O(1) while the private-coin uncertain communication is a growing function of n (the length of the inputs). This lower bound (proved with respect to the uniform distribution) is in sharp contrast with the case of public-coin uncertain communication which was shown by the previous work to be within a constant factor from the certain communication. This lower bound also implies the first separation between public-coin uncertain communication and deterministic uncertain communication. Interestingly, we also show that if Alice and Bob imperfectly share a sequence of random bits (a setup weaker than public randomness), then achieving a constant blow-up in communication is still possible.
- We improve the lower-bound of the previous work on public-coin uncertain communication. Namely, we exhibit a function class and a distribution (with mutual information I approx n) for which the one-way certain communication is k bits but the one-way public-coin uncertain communication is at least Omega(sqrt{k}*sqrt{I}) bits.
Our proofs introduce new problems in the standard communication complexity model and prove lower bounds for these problems. Both the problems and the lower bound techniques may be of general interest
Universal Communication, Universal Graphs, and Graph Labeling
We introduce a communication model called universal SMP, in which Alice and Bob receive a function f belonging to a family ?, and inputs x and y. Alice and Bob use shared randomness to send a message to a third party who cannot see f, x, y, or the shared randomness, and must decide f(x,y). Our main application of universal SMP is to relate communication complexity to graph labeling, where the goal is to give a short label to each vertex in a graph, so that adjacency or other functions of two vertices x and y can be determined from the labels ?(x), ?(y). We give a universal SMP protocol using O(k^2) bits of communication for deciding whether two vertices have distance at most k in distributive lattices (generalizing the k-Hamming Distance problem in communication complexity), and explain how this implies a O(k^2 log n) labeling scheme for deciding dist(x,y) ? k on distributive lattices with size n; in contrast, we show that a universal SMP protocol for determining dist(x,y) ? 2 in modular lattices (a superset of distributive lattices) has super-constant ?(n^{1/4}) communication cost. On the other hand, we demonstrate that many graph families known to have efficient adjacency labeling schemes, such as trees, low-arboricity graphs, and planar graphs, admit constant-cost communication protocols for adjacency. Trees also have an O(k) protocol for deciding dist(x,y) ? k and planar graphs have an O(1) protocol for dist(x,y) ? 2, which implies a new O(log n) labeling scheme for the same problem on planar graphs
Error Correcting Codes for Uncompressed Messages
Most types of messages we transmit (e.g., video, audio, images, text) are not fully compressed, since they do not have known efficient and information theoretically optimal compression algorithms. When transmitting such messages, standard error correcting codes fail to take advantage of the fact that messages are not fully compressed.
We show that in this setting, it is sub-optimal to use standard error correction. We consider a model where there is a set of "valid messages" which the sender may send that may not be efficiently compressible, but where it is possible for the receiver to recognize valid messages. In this model, we construct a (probabilistic) encoding procedure that achieves better tradeoffs between data rates and error-resilience (compared to just applying a standard error correcting code).
Additionally, our techniques yield improved efficiently decodable (probabilistic) codes for fully compressed messages (the standard setting where the set of valid messages is all binary strings) in the high-rate regime
Compression in a Distributed Setting
Motivated by an attempt to understand the formation and development of (human) language, we introduce a "distributed compression" problem. In our problem a sequence of pairs of players from a set of K players are chosen and tasked to communicate messages drawn from an unknown distribution Q.
Arguably languages are created and evolve to compress frequently occurring messages, and we focus on this aspect.
The only knowledge that players have about the distribution Q is from previously drawn samples, but these samples differ from player to player.
The only common knowledge between the players is restricted to a common prior distribution P and some constant number
of bits of information (such as a learning algorithm).
Letting T_epsilon denote the number of iterations it would take for a typical player
to obtain an epsilon-approximation to Q in total variation distance, we ask
whether T_epsilon iterations suffice to compress the messages down roughly to their
entropy and give a partial positive answer.
We show that a natural uniform algorithm can compress the communication down to an average cost per
message of O(H(Q) + log (D(P || Q)) in tilde{O}(T_epsilon) iterations
while allowing for O(epsilon)-error,
where D(. || .) denotes the KL-divergence between distributions.
For large divergences
this compares favorably with the static algorithm that ignores all samples and
compresses down to H(Q) + D(P || Q) bits, while not requiring T_epsilon * K iterations that it would take players to develop optimal but separate compressions for
each pair of players.
Along the way we introduce a "data-structural" view of the task of
communicating with a natural language and show that our natural algorithm can also be
implemented by an efficient data structure, whose storage is comparable to the storage requirements of Q and whose query complexity is comparable to the lengths of the message to be
compressed.
Our results give a plausible mathematical analogy to the mechanisms by which
human languages get created and evolve, and in particular highlights the
possibility of coordination towards a joint task (agreeing on a language)
while engaging in distributed learning
Communication with Imperfectly Shared Randomness
The communication complexity of many fundamental problems reduces greatly
when the communicating parties share randomness that is independent of the
inputs to the communication task. Natural communication processes (say between
humans) however often involve large amounts of shared correlations among the
communicating players, but rarely allow for perfect sharing of randomness. Can
the communication complexity benefit from shared correlations as well as it
does from shared randomness? This question was considered mainly in the context
of simultaneous communication by Bavarian et al. (ICALP 2014). In this work we
study this problem in the standard interactive setting and give some general
results. In particular, we show that every problem with communication
complexity of bits with perfectly shared randomness has a protocol using
imperfectly shared randomness with complexity bits. We also show that
this is best possible by exhibiting a promise problem with complexity bits
with perfectly shared randomness which requires bits when the
randomness is imperfectly shared. Along the way we also highlight some other
basic problems such as compression, and agreement distillation, where shared
randomness plays a central role and analyze the complexity of these problems in
the imperfectly shared randomness model.
The technical highlight of this work is the lower bound that goes into the
result showing the tightness of our general connection. This result builds on
the intuition that communication with imperfectly shared randomness needs to be
less sensitive to its random inputs than communication with perfectly shared
randomness. The formal proof invokes results about the small-set expansion of
the noisy hypercube and an invariance principle to convert this intuition to a
proof, thus giving a new application domain for these fundamental results.Comment: Updated some references and discussion w.r.t. previous wor
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