Adaptation and learning over networks under subspace constraints Part II: Performance Analysis

Nassif, Roula; Vlaski, Stefan; Sayed, Ali H.

doi:10.1109/tsp.2020.2987468

Cited by 16 publications

(22 citation statements)

References 33 publications

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“…. , w N } can be written as W = Uw and, therefore, distributed coupled optimization can be recast in the form (2). It can be verified that the coupled diffusion strategy proposed in [23] for solving this problem can be written in the form of (9) and that the (doubly-stochastic) matrix A in [23] satisfies conditions (7) and (8).…”

Section: Distributed Inference Under Subspace Constraintsmentioning

confidence: 96%

“…It can be verified that, when A satisfies (15) over a strongly connected network, the matrix A will satisfy (8), (13), (14), and (12). with overlapping parameter vectors [22]- [24] can also be recast in the form (2). This scenario is illustrated in Fig.…”

Section: Distributed Inference Under Subspace Constraintsmentioning

confidence: 99%

“…Then, building on this result, we will assess in the accompanying Part II [2] the size of this mean-square error by deriving closed-form expression for the network mean-square-deviation (MSD) defined by [4]:…”

Section: Stability Analysismentioning

confidence: 99%

“…Conditions on the network topology and signal subspace ensuring the feasibility of a distributed implementation are provided. We also show how some well-known network optimization problems, such as consensus optimization [3]- [5] and multitask smooth optimization [16], can be recast in the form (2) and addressed with the strategy proposed in this paper. The analysis in Section III of this Part I shows that, for sufficiently small µ, the proposed adaptive strategy leads to small estimation errors on the order of the small step-size.…”

Section: Introductionmentioning

confidence: 99%

“…The analysis in Section III of this Part I shows that, for sufficiently small µ, the proposed adaptive strategy leads to small estimation errors on the order of the small step-size. Building on the results of this Part I, we shall derive in Part II [2] a closed-form expression for the steady-state network mean-square-error performance. This closed form expression will reveal explicitly the influence of the data characteristics (captured by the second-order properties of the costs and second-order moments of the gradient noises) and subspace constraints (captured by U), on the network performance.…”

Section: Introductionmentioning

confidence: 99%

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Adaptation and Learning Over Networks Under Subspace Constraints—Part I: Stability Analysis

Nassif

Vlaski

Sayed

2020

IEEE Trans. Signal Process.

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This paper considers optimization problems over networks where agents have individual objectives to meet, or individual parameter vectors to estimate, subject to subspace constraints that require the objectives across the network to lie in low-dimensional subspaces. This constrained formulation includes consensus optimization as a special case, and allows for more general task relatedness models such as smoothness. While such formulations can be solved via projected gradient descent, the resulting algorithm is not distributed. Starting from the centralized solution, we propose an iterative and distributed implementation of the projection step, which runs in parallel with the stochastic gradient descent update. We establish in this Part I of the work that, for small step-sizes µ, the proposed distributed adaptive strategy leads to small estimation errors on the order of µ. We examine in the accompanying Part II [2] the steady-state performance. The results will reveal explicitly the influence of the gradient noise, data characteristics, and subspace constraints, on the network performance. The results will also show that in the small step-size regime, the iterates generated by the distributed algorithm achieve the centralized steady-state performance.where J k (·) is the cost function at agent k, N is the number of agents in the network, and w ∈ C L is the global parameter vector, which all agents need to agree upon-see Fig. 1 (middle). Each agent seeks to estimate w o through local computations and communications among neighboring agents without the need to know any of the costs besides their own. Among many useful strategies that have been proposed in the literature [3]-[10], diffusion strategies [3]-[5] are particularly attractive since they are scalable, robust, and enable continuous learning and adaptation in response to drifts in the location of the minimizer. However, there exist many network applications that require more complex models and flexible algorithms than consensus implementations since their agents may involve the need to estimate and track multiple distinct, though related, objectives. For instance, in distributed power system state estimation, the local state vectors to be estimated at neighboring control centers may overlap partially since the areas in a power system are interconnected [11], [12]. Likewise, in monitoring applications, agents need to track the movement of multiple correlated targets and to exploit the correlation profile in the data for enhanced accuracy [13], [14]. Problems of this kind, where nodes need to infer multiple, though related, parameter vectors, are referred to as multitask problems. Existing strategies to address multitask problems generally exploit prior knowledge on how the tasks across the network relate to each other [11]-[29]. For example, one way to model relationships among tasks is to formulate convex optimization problems with appropriate co-regularizers between neighboring agents [13], [16]-[19]. Graph spectral regularization can also be used in order t...

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