Information Equals Amortized Communication (1106.3595v1)

Abstract: We show how to efficiently simulate the sending of a message M to a receiver who has partial information about the message, so that the expected number of bits communicated in the simulation is close to the amount of additional information that the message reveals to the receiver. This is a generalization and strengthening of the Slepian-Wolf theorem, which shows how to carry out such a simulation with low amortized communication in the case that M is a deterministic function of X. A caveat is that our simulation is interactive. As a consequence, we prove that the internal information cost (namely the information revealed to the parties) involved in computing any relation or function using a two party interactive protocol is exactly equal to the amortized communication complexity of computing independent copies of the same relation or function. We also show that the only way to prove a strong direct sum theorem for randomized communication complexity is by solving a particular variant of the pointer jumping problem that we define. Our work implies that a strong direct sum theorem for communication complexity holds if and only if efficient compression of communication protocols is possible.

• The paper demonstrates that the internal information cost of an interactive communication protocol equals the amortized communication complexity for multiple instances.
• It introduces a simulation protocol showing that expected communication closely matches the new information revealed, generalizing the Slepian-Wolf theorem for interactive settings.
• The work links the strong direct sum theorem for communication complexity to efficient protocol compression, highlighting the Correlated Pointer Jumping problem as a key challenge.

Information Equals Amortized Communication: An Analysis

The paper "Information Equals Amortized Communication" by Mark Braverman and Anup Rao provides a significant contribution to our understanding of communication complexity, particularly in the context of interactive protocols. This work explores the relationship between information theory and communication, focusing on how information can be translated into effective communication strategies in distributed systems.

The authors present an innovative method to simulate message transmission between a sender and a receiver who possesses partial information about the message. This simulation ensures that the expected number of bits transmitted is close to the additional information revealed to the receiver. The results generalize the Slepian-Wolf theorem, demonstrating that the internal information cost in computing any function or relation using a two-party interactive protocol is equivalent to the amortized communication complexity of computing multiple independent copies of the same function or relation.

Key Findings

1. Interactive Protocol Simulation: The paper introduces an interactive protocol that mimics the effect of sending a message $M$. The expected communication closely matches the new information the receiver learns from $M$, with an inefficiency that is sublinear.
2. Internal Information Cost: A critical result of the paper is the equivalence between the internal information cost and the amortized communication complexity. This means that the information revealed through a protocol is inherently tied to the communication term required when computing multiple instances of a problem.
3. Strong Direct Sum Theorem: The authors argue that a strong direct sum theorem for randomized communication complexity exists if and only if efficient compression of communication protocols can be achieved. To prove this, they identify a specific problem—Correlated Pointer Jumping—that acts as a pivotal challenge for establishing such a theorem.
4. Efficiency in Compression: The paper underscores the importance of compressing communication protocols effectively according to their internal information cost. This approach links to the broader goal of aligning communication cost with the inherent informational content of a protocol.

Practical and Theoretical Implications

The theoretical implications of the paper are profound, offering a clearer understanding of how information theory can directly inform communication efficiency in interactive settings. The results suggest that internal information cost, a key measure of information revealed, is not an abstract concept but a real factor that can determine communication requirements in practical applications.

From a practical perspective, the findings provide a blueprint for designing communication-efficient protocols, especially in distributed computing environments where data must be shared across different systems. By aligning protocol design with the information structure of the underlying tasks, systems can achieve significant reductions in communication overhead.

Future Developments

This research sets the stage for several future exploration avenues in AI and computation theory. One potential area for further investigation is the development of more sophisticated methods for protocol compression that are both robust and flexible. Additionally, exploring the limits of the Correlated Pointer Jumping problem will provide more insights into the boundaries of communication complexity.

In summary, Braverman and Rao's work bridges a critical gap in understanding how information and communication interplay in computing contexts. It establishes a vital link between theory and practice and offers new tools for advancing our capacity to design efficient data exchange protocols. As AI systems continue to become more intricate and distributed, these insights will be invaluable for ensuring they operate efficiently and effectively.