[Paper Review] Optimal Error Rates for Interactive Coding I: Adaptivity and Other Settings

Mohsen Ghaffari, Bernhard Haeupler+1 more|arXiv (Cornell University)|Dec 6, 2013

Coding theory and cryptography8 references4 citations

TL;DR

This paper establishes tight bounds on the maximum tolerable error rates in adaptive and non-adaptive interactive coding, showing that adaptivity increases the error tolerance from 1/4 (non-adaptive) to 2/7 for standard errors, and from 1/2 to 2/3 when shared randomness is available. It also proves a tight bound of 1/2 for list-decodable interactive communication, with matching impossibility results for higher error rates.

ABSTRACT

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 than 1/3 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 for unbounded communication and computations, whereas for our positive results communication and computations 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.

Motivation & Objective

• To determine the optimal error rates that can be tolerated in interactive communication under adversarial errors.
• To investigate whether adaptivity—where communication order depends on history—can improve error resilience beyond non-adaptive schemes.
• To analyze the impact of shared randomness and list-decoding in interactive coding, and to establish tight upper and lower bounds.
• To resolve open questions from prior work, particularly regarding the maximal tolerable error rate in adaptive settings.
• To formalize and validate a hypothesis that intrinsic error resilience is independent of computational and communication complexity measures.

Proposed method

• Proposes a formal model for adaptive interactive coding where communication order adapts based on interaction history and error estimates.
• Introduces the adaptive exchange block—a three-round protocol with error detection and retransmission using a constant-size alphabet and shared randomness.
• Uses Chernoff bounds and probabilistic analysis to bound the probability of decoding errors in adaptive exchange blocks under adversarial corruption.
• Applies techniques from prior work (e.g., [10], [5]) to simulate protocols with detected corruptions and erasures as equivalent to half-corruptions.
• Constructs adaptive coding schemes with polynomial-time computation and constant communication rate by leveraging efficiently decodable tree codes.
• Employs indistinguishability arguments and symmetry-based reasoning to derive impossibility results, later corrected to a 1/3 bound after identifying a flaw in the original 2/7 claim.

Experimental results

Research questions

• RQ1 Can adaptivity improve the maximum tolerable error rate in interactive coding beyond the 1/4 bound of non-adaptive schemes?
• RQ2 What is the tight upper bound on the error rate that can be tolerated in adaptive coding for standard errors, and can this be achieved with efficient computation?
• RQ3 How does shared randomness affect the maximum tolerable error rate in adaptive protocols, and can this be matched with a tight impossibility result?
• RQ4 What is the optimal error rate for list-decodable interactive communication, and does adaptivity provide any advantage in this setting?
• RQ5 Is the claimed 2/7 impossibility bound for the Adaptive Message Exchange problem valid, or does it rely on an incorrect symmetry assumption?

Key findings

• Adaptive coding tolerates any error rate below 2/7 for standard errors, with a matching impossibility result of 1/3 (not 2/7 as originally claimed), correcting a flaw in the original proof.
• With shared randomness, adaptivity increases the tolerable error rate from 1/2 to 2/3, and this bound is tight—no higher rate is possible.
• For list-decodable interactive communication, the maximum tolerable error rate is 1/2, and this bound holds with or without adaptivity.
• The original claim of a 2/7 impossibility result for the Adaptive Message Exchange problem was incorrect due to an unjustified symmetry assumption in the proof; the correct impossibility bound is 1/3.
• The adaptive exchange block ensures that decoding errors occur with probability at most 3δ per block, and with high probability, both parties decode correctly if at most one transmission is corrupted.
• Using adaptive exchange blocks and existing error-tolerant simulation techniques, the paper constructs a robust adaptive protocol with constant communication rate and polynomial-time computation, achieving failure probability 2^−Ω(n).