We propose an adaptive diffusion mechanism to optimize a global cost function in a distributed manner over a network of nodes. The cost function is assumed to consist of a collection of individual components. Diffusion adaptation allows the nodes to cooperate and diffuse information in real-time; it also helps alleviate the effects of stochastic gradient noise and measurement noise through a continuous learning process. We analyze the mean-square-error performance of the algorithm in some detail, including its transient and steady-state behavior. We also apply the diffusion algorithm to two problems: distributed estimation with sparse parameters and distributed localization. Compared to well-studied incremental methods, diffusion methods do not require the use of a cyclic path over the nodes and are robust to node and link failure. Diffusion methods also endow networks with adaptation abilities that enable the individual nodes to continue learning even when the cost function changes with time. Examples involving such dynamic cost functions with moving targets are common in the context of biological networks.
constant step-sizes are necessary for continuous adaptation, learning, and tracking, which in turn enable the resulting algorithms to perform well even under data that exhibit statistical variations, measurement noise, and gradient noise. This paper is organized as follows. In Sec. II, we introduce the global cost function and approximate it by a distributed optimization problem through the use of a second-order Taylor series expansion. In Sec. III, we show that optimizing the localized alternative cost at each node k leads naturally to diffusion adaptation strategies. In Sec. IV, we analyze the mean-square performance of the diffusion algorithms under statistical perturbations when stochastic gradients are used. In Sec. V, we apply the diffusion algorithms to two application problems: sparse distributed estimation and distributed localization. Finally, in Sec. VI, we conclude the paper.
Notation. Throughout the paper, all vectors are column vectors except for the regressors {u k,i }, which are taken to be row vectors for simplicity of notation. We use boldface letters to denote random quantities (such as u k,i ) and regular font letters to denote their realizations or deterministic variables (such as u k,i ).
We write E to denote the expectation operator. We use diag{x 1 , . . . , x N } to denote a diagonal matrix consisting of diagonal entries x 1 , . . . , x N , and use col{x 1 , . . . , x N } to denote a column vector formed by stacking x 1 , . . . , x N on top of each other. For symmetric matrices X and Y , the notation X ≤ Y denotes Y -X ≥ 0, namely, that the matrix difference Y -X is positive semi-definite.
The objective is to determine, in a collaborative and distributed manner, the M ×1 column vector w o that minimizes a global cost of the form:
where J l (w), l = 1, 2, . . . , N , are individual real-valued functions, defined over w ∈ R M and assumed to be differentiable and strictly convex. Then, J glob (w) in ( 1) is also strictly convex so that the minimizer w o is unique [44]. In this article we study the important case where the component functions {J l (w)} are minimized at the same w o . This case is common in practice; situations abound where nodes in a network need to work cooperatively to attain a common objective (such as tracking a target, locating the source of chemical leak, estimating a physical model, or identifying a statistical distribution). This scenario is also frequent in the context of biological networks. For example, during the foraging behavior of an animal group, each agent in the group is interested in determining the same vector w o that corresponds to the location of the food source or the location of the predator [3]. This scenario is equally common November 27, 2024 DRAFT in online distributed machine learning problems, where data samples are often generated from the same underlying distribution and they are processed in a distributed manner by different nodes (e.g., [4], [5]).
The case where the {J l (w)} have different individual minimizers is studied in [45]; this situation is more challenging to study. Nevertheless, it is shown in [45] that the same diffusion strategies ( 18)- (19) of this paper are still applicable and nodes would converge instead to a Pareto-optimal solution.
Our strategy to optimize the global cost J glob (w) in a distributed manner is based on three steps.
First, using a second-order Taylor series expansion, we argue that J glob (w) can be approximated by an alternative localized cost that is amenable to distributed optimization -see (11). Second, each individual node optimizes this alternative cost via a steepest-descent procedure that relies solely on interactions within the neighborhood of the node. Finally, the local estimates for w o are spatially combined by each node and the procedure repeats itself in real-time.
To motivate the approach, we start by introducing a set of nonnegative coefficients {c l,k } that satisfy:
where N k denotes the neighborhood of node k (including node k itself); the neighbors of node k consist of all nodes with which node k can share information. Each c l,k represents a weight value that node k assigns to information arriving from its neighbor l. Condition (2) states that the sum of all weights leaving each node l should be one. Using the coefficients {c l,k }, we can express J glob (w) from (1) as
where
In other words, for each node k, we are introducing a new local cost function, J loc k (w), which corresponds to a weighted combination of the costs of its neighbors. Since the {c l,k } are all nonnegative and each J l (w) is convex, then J loc k (w) is also a convex function (actually, the J loc k (w) will be guaranteed to be strongly convex in our treatment in view of Assumption 1 further ahead). Now, each J loc l (w) in the second term of (3) can be approximated via a second-order Taylor series expansion as: where
) is the (scaled) Hessian matrix relative to w and evaluated at w = w o , and th
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