Xiaochuan Zhao

Diffusion strategies for adaptation and learning over networks: an examination of distributed strategies and network behavior

Nature provides splendid examples of real-time learning and adaptation behavior that emerges from highly localized interactions among agents of limited capabilities. For example, schools of fish are remarkably apt at configuring their topologies almost instantly in the face of danger [1]: when a predator arrives, the entire school opens up to let the predator through and then coalesces again into a moving body to continue its schooling behavior. Likewise, in bee swarms, only a small fraction of the agents (about 5%) are informed, and these informed agents are able to guide the entire swarm of bees to their new hive [2]. It is an extraordinary property of biological networks that sophisticated behavior is able to emerge from simple interactions among lower-level agents [3].

Performance Limits for Distributed Estimation Over LMS Adaptive Networks

In this work, we analyze the mean-square performance of different strategies for distributed estimation over least-mean-squares (LMS) adaptive networks. The results highlight some useful properties for distributed adaptation in comparison to fusion-based centralized solutions. The analysis establishes that, by optimizing over the combination weights, diffusion strategies can deliver lower excess-mean-square-error than centralized solutions employing traditional block or incremental LMS strategies. We first study in some detail the situation involving combinations of two adaptive agents and then extend the results to generic N -node ad-hoc networks. In the latter case, we establish that, for sufficiently small step-sizes, diffusion strategies can outperform centralized block or incremental LMS strategies by optimizing over left-stochastic combination weighting matrices. The results suggest more efficient ways for organizing and processing data at fusion centers, and present useful adaptive strategies that are able to enhance performance when implemented in a distributed manner.

Diffusion Adaptation Over Networks Under Imperfect Information Exchange and Non-Stationary Data

Adaptive networks rely on in-network and collaborative processing among distributed agents to deliver enhanced performance in estimation and inference tasks. Information is exchanged among the nodes, usually over noisy links. The combination weights that are used by the nodes to fuse information from their neighbors play a critical role in influencing the adaptation and tracking abilities of the network. This paper first investigates the mean-square performance of general adaptive diffusion algorithms in the presence of various sources of imperfect information exchanges, quantization errors, and model non-stationarities. Among other results, the analysis reveals that link noise over the regression data modifies the dynamics of the network evolution in a distinct way, and leads to biased estimates in steady-state. The analysis also reveals how the network mean-square performance is dependent on the combination weights. We use these observations to show how the combination weights can be optimized and adapted. Simulation results illustrate the theoretical findings and match well with theory.

Clustering via diffusion adaptation over networks

Distributed processing over networks relies on in-network processing and cooperation among neighboring agents. Cooperation is beneficial when all agents share the same objective or belong to the same group. However, if agents belong to different clusters or are interested in different objectives, then cooperation can be damaging. In this work, we devise an adaptive combination rule that allows agents to learn which neighbors belong to the same cluster and which other neighbors should be ignored. In doing so, the resulting algorithm enables the agents to identify their grouping and to attain improved learning and estimation performance over networks.

Distributed Clustering and Learning Over Networks

Distributed processing over networks relies on in-network processing and cooperation among neighboring agents. Cooperation is beneficial when agents share a common objective. However, in many applications, agents may belong to different clusters that pursue different objectives. Then, indiscriminate cooperation will lead to undesired results. In this paper, we propose an adaptive clustering and learning scheme that allows agents to learn which neighbors they should cooperate with and which other neighbors they should ignore. In doing so, the resulting algorithm enables the agents to identify their clusters and to attain improved learning and estimation accuracy over networks. We carry out a detailed mean-square analysis and assess the error probabilities of Types I and II, i.e., false alarm and misdetection, for the clustering mechanism. Among other results, we establish that these probabilities decay exponentially with the step-sizes so that the probability of correct clustering can be made arbitrarily close to one.

Asynchronous Adaptation and Learning Over Networks—Part I: Modeling and Stability Analysis

In this work and the supporting Parts II and III of this paper, also in the current issue, we provide a rather detailed analysis of the stability and performance of asynchronous strategies for solving distributed optimization and adaptation problems over networks. We examine asynchronous networks that are subject to fairly general sources of uncertainties, such as changing topologies, random link failures, random data arrival times, and agents turning on and off randomly. Under this model, agents in the network may stop updating their solutions or may stop sending or receiving information in a random manner and without coordination with other agents. We establish in Part I conditions on the first and second-order moments of the relevant parameter distributions to ensure mean-square stable behavior. We derive in Part II expressions that reveal how the various parameters of the asynchronous behavior influence network performance. We compare in Part III the performance of asynchronous networks to the performance of both centralized solutions and synchronous networks. One notable conclusion is that the mean-square-error performance of asynchronous networks shows a degradation only in the order of O(ν), where ν is a small step-size parameter, while the convergence rate remains largely unaltered. The results provide a solid justification for the remarkable resilience of cooperative networks in the face of random failures at multiple levels: agents, links, data arrivals, and topology.

Single-link diffusion strategies over adaptive networks

We propose an adaptive diffusion strategy with limited communication overhead by cutting off all links but one for each node in the network. We keep the “best” neighbor that has the smallest estimated variance-product measure and ignore the other neighbors. The combination coefficients for the interacting nodes are calculated via a maximal-ratio-combining rule to minimize the steady-state meansquare-deviation. Simulation results illustrate that, with less communication overhead and less computations, the proposed algorithm performs well and outperforms other related methods with similar overheads.

Bacterial motility via diffusion adaptation

Bacteria forage by moving towards nutrient sources in a process known as chemotaxis. The bacteria follow gradient variations by tumbling or moving in straight lines. Both modes of locomotion are affected by Brownian motion. Bacteria are also capable of interactions through chemical signaling. As the bacteria swim towards nutrients, they emit chemicals that can be sensed by their neighboring bacteria and used to adjust the direction of motion. In this paper, we propose schemes for cooperation and diffusion of information [1]–[7] and study their effect on bacteria motility. Because bacteria are limited in their abilities, we restrict the sharing of information to binary choices (such as whether to run or tumble). Simulation results suggest that cooperation among bacteria is critical for effective foraging to improve their decisions of movement.

Asynchronous Adaptation and Learning Over Networks—Part III: Comparison Analysis

In Part II of this paper, also in this issue, we carried out a detailed mean-square-error analysis of the performance of asynchronous adaptation and learning over networks under a fairly general model for asynchronous events including random topologies, random link failures, random data arrival times, and agents turning on and off randomly. In this Part III, we compare the performance of synchronous and asynchronous networks. We also compare the performance of decentralized adaptation against centralized stochastic-gradient (batch) solutions. Two interesting conclusions stand out. First, the results establish that the performance of adaptive networks is largely immune to the effect of asynchronous events: the mean and mean-square convergence rates and the asymptotic bias values are not degraded relative to synchronous or centralized implementations. Only the steady-state mean-square-deviation suffers a degradation in the order of ν, which represents the small step-size parameters used for adaptation. Second, the results show that the adaptive distributed network matches the performance of the centralized solution. These conclusions highlight another critical benefit of cooperation by networked agents: cooperation does not only enhance performance in comparison to stand-alone single-agent processing, but it also endows the network with remarkable resilience to various forms of random failure events and is able to deliver performance that is as powerful as batch solutions.

Learning over social networks via diffusion adaptation

We propose a diffusion strategy to enable social learning over networks. Individual agents observe signals influenced by the state of the environment. The individual measurements are not sufficient to enable the agents to detect the true state of the environment on their own. Agents are then encouraged to cooperate through a diffusive process of self-learning and social-learning. We show that the diffusion algorithm converges almost surely to the true state. Simulation results also illustrate the superior convergence rate of the diffusion strategy over consensus-based strategies since diffusion schemes allow information to diffuse more thoroughly through the network.