A simulation of an interactive protocol entails the use of interactive communication to produce the output of the protocol to within a fixed statistical distance ε. Recent works have proposed that the information complexity of the protocol plays a central role in characterizing the minimum number of bits that the parties must exchange for a successful simulation, namely the distributional communication complexity of simulating the protocol. Several simulation protocols have been proposed with communication complexity depending on the information complexity of the simulated protocol. However, in the absence of any general lower bounds for distributional communication complexity, the conjectured central role of information complexity is far from settled. We fill this gap and show that the distributional communication complexity of ε-simulating a protocol is bounded below by the ε-tail λε of the information complexity density, a random variable with information complexity as its expected value. For protocols with bounded number of rounds, we give a simulation protocol that yields a matching upper bound. Thus, it is not information complexity but λε that governs the distributional communication complexity.
A simulation of an interactive protocol entails the use of interactive communication to produce the output of the protocol to within a fixed statistical distance ε. Recent works have proposed that the information complexity of the protocol plays a central role in characterizing the minimum number of bits that the parties must exchange for a successful simulation, namely the distributional communication complexity of simulating the protocol. Several simulation protocols have been proposed with communication complexity depending on the information complexity of the simulated protocol. However, in the absence of any general lower bounds for distributional communication complexity, the conjectured central role of information complexity is far from settled. We fill this gap and show that the distributional communication complexity of ε-simulating a protocol is bounded below by the ε-tail λε of the information complexity density, a random variable with information complexity as its expected value. For protocols with bounded number of rounds, we give a simulation protocol that yields a matching upper bound. Thus, it is not information complexity but λε that governs the distributional communication complexity.
A simulation of an interactive protocol entails the use of interactive communication to produce the output of the protocol to within a fixed statistical distance epsilon. Recent works have proposed that the information complexity of the protocol plays a central role in characterizing the minimum number of bits that the parties must exchange for a successful simulation, namely the distributional communication complexity of simulating the protocol. Several simulation protocols have been proposed with communication complexity depending on the information complexity of the simulated protocol. However, in the absence of any general lower bounds for distributional communication complexity, the conjectured central role of information complexity is far from settled. We fill this gap and show that the distributional communication complexity of epsilon-simulating a protocol is bounded below by the epsilon-tail lambda(epsilon) of the information complexity density, a random variable with information complexity as its expected value. For protocols with bounded number of rounds, we give a simulation protocol that yields a matching upper bound. Thus, it is not information complexity but lambda(epsilon), that governs the distributional communication complexity. As applications of our bounds, in the amortized regime for product protocols, we identify the exact second order term, together with the precise dependence on epsilon. For general protocols such as a mixture of two product protocols or for the amortized case when the repetitions are not independent, we derive a general formula for the leading asymptotic term. These results sharpen and significantly extend known results in the amortized regime. In the single-shot regime, our lower bound sheds light on the dependence of communication complexity on epsilon. We illustrate this with an example that exhibits an arbitrary separation between distributional communication complexity and information complexity for all sufficiently small epsilon.
Two parties observing correlated random variables seek to run an interactive communication protocol. How many bits must they exchange to simulate the protocol, namely to produce a view with a joint distribution within a fixed statistical distance of the joint distribution of the input and the transcript of the original protocol? We present an information spectrum approach for this problem whereby the information complexity of the protocol is replaced by its information complexity density. Our single-shot bounds relate the communication complexity of simulating a protocol to tail bounds for information complexity density. As a consequence, we obtain a strong converse and characterize the second-order asymptotic term in communication complexity for independent and identically distributed observation sequences. Furthermore, we obtain a general formula for the rate of communication complexity, which applies to any sequence of observations and protocols. Connections with results from theoretical computer science and implications for the function computation problem are discussed.
Two parties observing correlated data seek to exchange their data using interactive communication. How many bits must they communicate? We derive a lower bound on the minimum number of bits that is based on relating the data exchange problem to the secret key agreement problem. Furthermore, we propose an interactive protocol for data exchange which increases the communication size in steps until the task is done and matches the performance of our lower bound. Our single-shot analysis applies to all discrete random variables and yields upper and lower bound of a similar form. In fact, the bounds are asymptotically tight and lead to a characterization of the optimal rate of communication needed for data exchange for a general sequence such as mixture of IID random variables as well as the optimal second-order asymptotic term in the length of communication needed for data exchange for the IID random variables, when the probability of error is fixed. This gives a precise characterization of the asymptotic reduction in the length of optimal communication due to interaction; in particular, two-sided Slepian-Wolf compression is strictly suboptimal.
In this work we study the effect of rounds of interaction on the common randomness generation (CRG) problem. In the CRG problem, two parties, Alice and Bob, receive samples Xi and Yi, respectively, where (Xi, Yi) are drawn jointly from a source distribution μ. The two parties wish to agree on a common random key consisting of many bits of randomness, by exchanging messages that depend on each party's respective input and the previous messages. In this work we study the amortized version of the problem, i.e., the number of bits of communication needed per random bit output by Alice and Bob, in the limit as the number of bits generated tends to infinity. The amortized version of the CRG problem has been extensively studied in the information theory literature, though very little was known about the effect of interaction on this problem. Recently Bafna et al. (SODA 2019) considered the non-amortized version of the problem (so here the goal of the interaction is to generate a fixed number of random bits): they gave a family of sources μr,n parameterized by r,n ϵ ℕ, such that with r + 2 rounds of communication one can generate n bits of common randomness with this source with O(r log n) communication, whereas with roughly r/2 rounds the communication complexity is Ω(n/ poly log n). Note in particular that their source is designed with the target number of bits in mind and hence the result does not apply to the amortized setting. In this work we strengthen the work of Bafna et al. in two ways: First we show that the results extend to the classical amortized setting. We also reduce the gap between the round complexity in the upper and lower bounds to an additive constant. Specifically we show that for every pair r, n ϵ ℕ the (amortized) communication complexity to generate Ω(n) bits of common randomness from the source μr,n using r + 2 rounds of communication is O(r log n) whereas the amortized communication required to generate the same amount of randomness from r rounds is . Our techniques exploit known connections between information complexity and CRG, and the main novelty is our ability to analyze the information complexity of protocols getting inputs from the source μr,n.
Two parties observing correlated data seek to exchange their data using interactive communication. How many bits must they communicate? We propose a new interactive protocol for data exchange, which increases the communication size in steps until the task is done. We also derive a lower bound on the minimum number of bits that is based on relating the data exchange problem to the secret key agreement problem. Our single-shot analysis applies to all discrete random variables and yields upper and lower bounds of a similar form. In fact, the bounds are asymptotically tight and lead to a characterization of the optimal rate of communication needed for data exchange for a general source sequence, such as a mixture of independent and identically distributed (IID) random variables as well as the optimal second-order asymptotic term in the length of communication needed for data exchange for IID random variables, when the probability of error is fixed. This gives a precise characterization of the asymptotic reduction in the length of optimal communication due to interaction; in particular, two-sided Slepian-Wolf compression is strictly suboptimal.
We study the round complexity of two-party protocols for generating a random n-bit string such that the output is guaranteed to have bounded bias (according to some measure) even if one of the two parties deviates from the protocol (even using unlimited computational resources). Specifically, we require that the output's statistical difference from the uniform distribution on zon is bounded by a constant less than 1.We present a protocol for the above problem that has 2 log*n+O(1) rounds, improving a 2n-round protocol that follows from the work of Goldreich, Goldwasser, and Linial (FOCS'91). Like the GGL protocol, our protocol actually provides a stronger guarantee, ensuring that the output lands in any set T⊆zon of density μ with probability at most O(√μ+δ), where δ is an arbitarily small constant.We then prove a matching lower bound, showing that any protocol guaranteeing bounded statistical difference requires at least log*n - log* log*n-O(1) rounds. As far as we know, this is the first nontrivial lower bound on the round complexity of random selection protocols (of any type) that does not impose additional constraints (e.g. on communication or "simulatability").We also state several results for the case when the output's bias is measured by the maximum multiplicative factor by which a party can increase the probability of a set T ⊆ zon.