We provide tight upper and lower bounds on the noise resilience of interactive communication over noisy channels with feedback. In this setting, we show that the maximal fraction of noise that any robust protocol can resist is 1 3. Additionally, we provide a simple and efficient robust protocol that succeeds as long as the fraction of noise is at most 1 3 - . Surprisingly, both bounds hold regardless of whether the parties send bits or symbols from an arbitrarily large alphabet. We also consider interactive communication over erasure channels. We provide a protocol that matches the optimal tolerable erasure rate of 1 2 - of previous protocols (, CRYPTO '13) but operates in a much simpler and more efficient way. Our protocol works with an alphabet of size 4, in contrast to prior protocols in which the alphabet size grows as epsilon goes to zero. Building on the above algorithm with a fixed alphabet size, we are able to devise a protocol for binary erasure channels that tolerates erasure rates of up to 1 3 - .
For erasure channels, a tight bound of @math on the erasure rate of robust protocols was given in @cite_11 . For the case of adaptive protocols, @cite_6 provided a coding scheme with a that resists a relative erasure rate of up to @math in a setting that allows parties to remain silent in an adaptive way. The case where the parties share a memoryless erasure channel with a noiseless feedback was considered by Schulman @cite_7 who showed that for any function @math , the communication complexity of solving @math in that setting equals the distributional complexity of @math (over noiseless channels), up to a factor of the channel's capacity.
@cite_7 @cite_6 @cite_11
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We provide tight upper and lower bounds on the noise resilience of interactive communication over noisy channels with feedback. In this setting, we show that the maximal fraction of noise that any nonadaptive protocol can withstand is 1 3. In addition, we provide a simple and efficient nonadaptive coding scheme that succeeds as long as the fraction of noise is at most @math . Surprisingly, both bounds hold regardless of whether the parties send bits or symbols from an arbitrarily large alphabet. We also consider interactive communication over erasure channels. We provide a coding scheme that withstands the optimal tolerable erasure rate of @math [ , IEEE Trans. Info. Theory , 2015], but operates in a much simpler and more efficient way than the previous schemes. Our coding scheme works with an alphabet of size 4, in contrast to prior schemes in which the alphabet size grows as @math . Building on the above algorithm with a fixed alphabet size, we are able to devise a protocol for binary erasure channels that tolerates erasure rates of up to @math . 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 @math . When the order of speaking may adaptively change as well, we demonstrate a protocol that tolerates noise rates up to @math . Hence, adaptivity circumvents an impossibility result of @math on the fraction of tolerable noise (Braverman and Rao, 2014). We provide the first capacity approaching coding schemes that robustly simulate any interactive protocol over an adversarial channel that corrupts any @math fraction of the transmitted symbols. Our coding schemes achieve a communication rate of @math over any adversarial channel. This can be improved to @math for random, oblivious, and computationally bounded channels, or if parties have shared randomness unknown to the channel. Surprisingly, these rates exceed the @math interactive channel capacity bound which [Kol and Raz; STOC'13] recently proved for random errors. We conjecture @math and @math 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. 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 [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.