How to send a real number using a single bit (and some shared randomness)
This addresses a fundamental communication problem in information theory, with potential applications in data compression and distributed computing, but it is incremental as it builds on known estimation approaches.
The paper tackles the problem of communicating a real number estimate using only one bit, with and without shared randomness, aiming to minimize worst-case expected squared error or variance. It proves optimality of existing methods without shared randomness and shows that even a single shared random bit reduces cost, proposing near-optimal solutions.
We consider the fundamental problem of communicating an estimate of a real number $x\in[0,1]$ using a single bit. A sender that knows $x$ chooses a value $X\in\set{0,1}$ to transmit. In turn, a receiver estimates $x$ based on the value of $X$. We consider both the biased and unbiased estimation problems and aim to minimize the cost. For the biased case, the cost is the worst-case (over the choice of $x$) expected squared error, which coincides with the variance if the algorithm is required to be unbiased. We first overview common biased and unbiased estimation approaches and prove their optimality when no shared randomness is allowed. We then show how a small amount of shared randomness, which can be as low as a single bit, reduces the cost in both cases. Specifically, we derive lower bounds on the cost attainable by any algorithm with unrestricted use of shared randomness and propose near-optimal solutions that use a small number of shared random bits. Finally, we discuss open problems and future directions.