Article

# Lower bounds on information rates for distributed computation via noisy channels

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## Abstract

We study a network of n nodes communicating via channels. The objective of each of the nodes is to compute a given function of the data in the network. Using Information Theoretic inequalities, we derive a lower bound to the information that must be communicated between nodes for the mean square error in their estimates to converge to zero. We use this bound to express a bound on the rate of the channel code when the mean square error is required to converge to zero exponentially with some rate. We also show how the bound can be applied on different cut-sets of a communication network to determine a lower bound to computation time until convergence of the error in the nodes' estimates to a prescribed interval around zero. Finally, we present a particular scenario for which our bound on the computation time is tight.

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... However, the complexity of network computing has restricted prior work to the analysis of elementary networks. Networks with noisy links were studied in [3,14,16,17,19,26,35,37,50] and distributed computation in networks using gossip algorithms was studied in [4-6, 9, 27, 36]. ...
... Given a (k, n) network code, every edge e ∈ E carries a vector z e of at most n alphabet symbols, 3 which is obtained by evaluating the encoding function h (e) on the set of vectors carried by the in-edges to the node and the node's message vector if it is a source. The objective of the receiver is to compute the target function f of the source messages, for any arbitrary message generator α. ...
... That is, each element of A p is a possible pair of input edge-vectors to the receiver when the target function value equals p. Let j denote the number of components of p that are either 0 or 3. Without loss of generality, suppose the first j components of p belong to {0, 3} and definew (3) ...
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The following network computing problem is considered. Source nodes in a directed acyclic network generate independent messages and a single receiver node computes a target function f of the messages. The objective is to maximize the average number of times f can be computed per network usage, i.e., the “computing capacity”. The network coding problem for a single-receiver network is a special case of the network computing problem in which all of the source messages must be reproduced at the receiver. For network coding with a single receiver, routing is known to achieve the capacity by achieving the network min-cut upper bound. We extend the definition of min-cut to the network computing problem and show that the min-cut is still an upper bound on the maximum achievable rate and is tight for computing (using coding) any target function in multi-edge tree networks. It is also tight for computing linear target functions in any network. We also study the bound's tightness for different classes of target functions. In particular, we give a lower bound on the computing capacity in terms of the Steiner tree packing number and a different bound for symmetric functions. We also show that for certain networks and target functions, the computing capacity can be less than an arbitrarily small fraction of the min-cut bound.
... However, the complexity of network computing has restricted prior work to the analysis of elementary networks. Networks with noisy links were studied in [3,14,16,17,19,26,35,37,50] and distributed computation in networks using gossip algorithms was studied in [4-6, 9, 27, 36]. ...
... Given a (k, n) network code, every edge e ∈ E carries a vector z e of at most n alphabet symbols, 3 which is obtained by evaluating the encoding function h (e) on the set of vectors carried by the in-edges to the node and the node's message vector if it is a source. The objective of the receiver is to compute the target function f of the source messages, for any arbitrary message generator α. ...
... That is, each element of A p is a possible pair of input edge-vectors to the receiver when the target function value equals p. Let j denote the number of components of p that are either 0 or 3. Without loss of generality, suppose the first j components of p belong to {0, 3} and definew (3) ...
... We note here that for this case, by an Information Theoretic lower bound derived in [2] we have that the computation time is lower bounded as ...
... Combining the result of Theorem IV.1 with that of Theorem III.2 yields Theorem II.5. Comparison with a lower bound obtained via Information Theoretic inequalities in [2] reveals that the reciprocal dependence between computation time and graph conductance in the upper bound of Theorem II.5 matches the lower bound. Hence the upper bound is tight in capturing the effect of the graph conductance Φ(P ). ...
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... Our work is related to that of Ramamoorthy [11], who studied computing the parity of a collection of binary sources in a network with two sources and arbitrary number of receivers, or vice versa; however, he considered only the existence of a solution, rather than the rate at which the solution can be computed. Problems related to function computation have been studied in a such areas as communication complexity [8], [12], average consensus [4], [6], and distributed computation [3], [10]. The reader is referred to [7] for a review of various approaches to the problem. ...
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