67,491 research outputs found
Optimal Error Rates for Interactive Coding I: Adaptivity and Other Settings
We consider the task of interactive communication in the presence of
adversarial errors and present tight bounds on the tolerable error-rates in a
number of different settings.
Most significantly, we explore adaptive interactive communication where the
communicating parties decide who should speak next based on the history of the
interaction. Braverman and Rao [STOC'11] show that non-adaptively one can code
for any constant error rate below 1/4 but not more. They asked whether this
bound could be improved using adaptivity. We answer this open question in the
affirmative (with a slightly different collection of resources): Our adaptive
coding scheme tolerates any error rate below 2/7 and we show that tolerating a
higher error rate is impossible. We also show that in the setting of Franklin
et al. [CRYPTO'13], where parties share randomness not known to the adversary,
adaptivity increases the tolerable error rate from 1/2 to 2/3. For
list-decodable interactive communications, where each party outputs a constant
size list of possible outcomes, the tight tolerable error rate is 1/2.
Our negative results hold even if the communication and computation are
unbounded, whereas for our positive results communication and computation are
polynomially bounded. Most prior work considered coding schemes with linear
amount of communication, while allowing unbounded computations. We argue that
studying tolerable error rates in this relaxed context helps to identify a
setting's intrinsic optimal error rate. We set forward a strong working
hypothesis which stipulates that for any setting the maximum tolerable error
rate is independent of many computational and communication complexity
measures. We believe this hypothesis to be a powerful guideline for the design
of simple, natural, and efficient coding schemes and for understanding the
(im)possibilities of coding for interactive communications
Optimal Error Rates for Interactive Coding II: Efficiency and List Decoding
We study coding schemes for error correction in interactive communications.
Such interactive coding schemes simulate any -round interactive protocol
using rounds over an adversarial channel that corrupts up to
transmissions. Important performance measures for a coding scheme are its
maximum tolerable error rate , communication complexity , and
computational complexity.
We give the first coding scheme for the standard setting which performs
optimally in all three measures: Our randomized non-adaptive coding scheme has
a near-linear computational complexity and tolerates any error rate with a linear communication complexity. This improves over
prior results which each performed well in two of these measures.
We also give results for other settings of interest, namely, the first
computationally and communication efficient schemes that tolerate adaptively, if only one party is required to
decode, and if list decoding is allowed. These are the
optimal tolerable error rates for the respective settings. These coding schemes
also have near linear computational and communication complexity.
These results are obtained via two techniques: We give a general black-box
reduction which reduces unique decoding, in various settings, to list decoding.
We also show how to boost the computational and communication efficiency of any
list decoder to become near linear.Comment: preliminary versio
Adaptive Protocols for Interactive Communication
How much adversarial noise can protocols for interactive communication
tolerate? This question was examined by Braverman and Rao (IEEE Trans. Inf.
Theory, 2014) for the case of "robust" protocols, where each party sends
messages only in fixed and predetermined rounds. We consider a new class of
non-robust protocols for Interactive Communication, which we call adaptive
protocols. Such protocols adapt structurally to the noise induced by the
channel in the sense that both the order of speaking, and the length of the
protocol may vary depending on observed noise.
We define models that capture adaptive protocols and study upper and lower
bounds on the permissible noise rate in these models. When the length of the
protocol may adaptively change according to the noise, we demonstrate a
protocol that tolerates noise rates up to . When the order of speaking may
adaptively change as well, we demonstrate a protocol that tolerates noise rates
up to . Hence, adaptivity circumvents an impossibility result of on
the fraction of tolerable noise (Braverman and Rao, 2014).Comment: Content is similar to previous version yet with an improved
presentatio
Interactive Channel Capacity Revisited
We provide the first capacity approaching coding schemes that robustly
simulate any interactive protocol over an adversarial channel that corrupts any
fraction of the transmitted symbols. Our coding schemes achieve a
communication rate of over any
adversarial channel. This can be improved to for
random, oblivious, and computationally bounded channels, or if parties have
shared randomness unknown to the channel.
Surprisingly, these rates exceed the interactive channel capacity bound
which [Kol and Raz; STOC'13] recently proved for random errors. We conjecture
and to be the optimal rates for their respective settings
and therefore to capture the interactive channel capacity for random and
adversarial errors.
In addition to being very communication efficient, our randomized coding
schemes have multiple other advantages. They are computationally efficient,
extremely natural, and significantly simpler than prior (non-capacity
approaching) schemes. In particular, our protocols do not employ any coding but
allow the original protocol to be performed as-is, interspersed only by short
exchanges of hash values. When hash values do not match, the parties backtrack.
Our approach is, as we feel, by far the simplest and most natural explanation
for why and how robust interactive communication in a noisy environment is
possible
Simulating Noisy Channel Interaction
We show that rounds of interaction over the binary symmetric channel
with feedback can be simulated with
rounds of interaction over a noiseless channel. We also introduce a more
general "energy cost" model of interaction over a noisy channel. We show energy
cost to be equivalent to external information complexity, which implies that
our simulation results are unlikely to carry over to energy complexity. Our
main technical innovation is a self-reduction from simulating a noisy channel
to simulating a slightly-less-noisy channel, which may have other applications
in the area of interactive compression
Error Correcting Codes for Distributed Control
The problem of stabilizing an unstable plant over a noisy communication link
is an increasingly important one that arises in applications of networked
control systems. Although the work of Schulman and Sahai over the past two
decades, and their development of the notions of "tree codes"\phantom{} and
"anytime capacity", provides the theoretical framework for studying such
problems, there has been scant practical progress in this area because explicit
constructions of tree codes with efficient encoding and decoding did not exist.
To stabilize an unstable plant driven by bounded noise over a noisy channel one
needs real-time encoding and real-time decoding and a reliability which
increases exponentially with decoding delay, which is what tree codes
guarantee. We prove that linear tree codes occur with high probability and, for
erasure channels, give an explicit construction with an expected decoding
complexity that is constant per time instant. We give novel sufficient
conditions on the rate and reliability required of the tree codes to stabilize
vector plants and argue that they are asymptotically tight. This work takes an
important step towards controlling plants over noisy channels, and we
demonstrate the efficacy of the method through several examples.Comment: 39 page
- …