Collaborative Localization as a Paradigm for Incremental Knowledge Fusion

CogInfoCom 2014 • 5th IEEE International Conference on Cognitive Infocommunications • November 5-7, 2014, Vietri sul Mare, Italy

Collaborative Localization as a Paradigm for Incremental Knowledge Fusion George Kampis

Paul Lukowicz

Embedded Intelligence DFKI (German Res. Center for Artificial Intelligence) Kaiserslautern, Germany Email: George.Kampis@dfki.de

Embedded Intelligence DFKI (German Res. Center for Artificial Intelligence) Kaiserslautern, Germany Email: Paul.Lukowicz@dfki.de

ITMO University, St. Petersburg, Russia

Abstract—Collaborative localization is the computation of improved spatial coordinates in mobile agents based on their physical meetings in a pedestrian dead reckoning (PDR) system.

Upon meeting the agents can exchange information about their subjective position and update it based on a simple algorithm. We show in a simulation model that the localization error diverges unless this algorithm is introduced in which case it remains bounded. We consider collaborative localization as an example of broader incremental knowledge fusion and discuss its various implications such as the importance of well-informed agents.

individual properties, objectives and actions. Decision-making, coordination and computation are distributed and usually dispersed, and interaction between the units lead to emergent or unexpected, unplanned phenomena. The agents are typically heterogeneous (such as humans, computers, robots, various artefacts and biological entities), having different, a priori uncoordinated (thus possibly inconsistent) objectives and goals.

I. I NTRODUCTION Pedestrian dead reckoning (PDR) systems are widely used for localization (based e.g. on the opportunistic use on inertial sensors built into different devices such as smart phones) but are bound to various errors. Among these, the most important is the accumulation of errors: each step adds an uncertainty in the localization so the further the user moves the less accurate the localization information becomes.

In the Smart Society and Smart City context a new idea has been tested, based on the meetings of persons using devices.

When two users come close to each other, their devices can use this for updating their own subjective position information based on the fact that (despite their error-prone subjective positions) they occupy nearly identical spatial positons. Using a simple averaging algorithm the estimated (subjective) position can be improved. In mathematical language, the probability density distributions of each system with respect to its own location using the fact of the meeting allows for the construction of a new, joint distribution that has a lower variance than the individual estimates alone. The principle has been tested on the field and is illustrated in (Fig. 1).

In this paper we introduce and discuss an agent based simulation, reproducing the earlier empirical result and providing further insights, in particular on the importance of informed agents, system size, correlations, the role of meeting densities and other parameters.

(a) The principle of collaborative localization

II. F ROM DESIGNED TO SELF - ORGANIZING COMPLEX ADAPTIVE SYSTEMS (CAS)

(b) Empirical results from [1]

CAS, or collective adaptive systems, are systems that comprise many agents, all of them coming along with their own

Fig. 1: Collaborative localization in a population of PDR systems

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G. Kampis and P. Lukowicz • Collaborative Localization as a Paradigm for Incremental Knowledge Fusion

Collaborative localization endeavours to approach CAS systems by realizing their basic potential to utilize populational, decentralized, heterogeneous and self-organizing processes.

In other words, complementing more traditional approaches based on a centralized control and planning (where the information flow is carefully designed and implemented), here we consider ad hoc and essentially unknown interactions between agents of an unspecified nature, enabling their coordination to interactively and iteratively give rise to system-level (i.e. population-level), emergent functionalities.

In the case of collaborative localization, the functionality is simple and local (i.e. knowledge of a good estimate of current position by an agent despite the lack of a central localization service). In the case of collaborative knowledge fusion, discussed in Section VII the goal is more ambitious and involves the self-organized emergence of common knowledge in the population of agents.

III. C OLLABORATIVE LOCALIZATION Collaborative localization is the problem of determination of personal geometric position based on agent-to-agent interaction using mobile devices carried by the agents such as smart phones, in such a way where little or no global (i.e. top-down) information (such as GSP positioning) is available at the level of the individual agents. The idea is that agents maintain a subjective position based on step (gait) estimates from the built-in inertial sensors of the carried devices, and upon meeting, these subjective positions can be improved (Fig. 2).

Fig. 2: Error reduction in collaborative localization.

In the Figure, Yi (t) denotes the error term (more precisely, the expected error value) of the position estimate for agent i, further, d is the distance threshold within which meeting is detected (e.g. the range of Bluetooth or similar devices for local contact), and Ycol (t) the error term for the new collective estimate. The new estimated position is calculated as xnew = (x1 + x2 )/2 and similarly ynew = (y1 + y2 )/2 where xi and yi are the subjective coordinates for agent i. Note that usually d << Yi (t) and therefore we can assume Ycol (t) to be identical for both agents. Meetings are not always as presented

on the Figure; if both agents are located on the ”same side” of the actual meeting range, the new position estimates in fact increase the error term for one of them. An analytic treatment of the problem [2] however shows that in general, the error term Y (t) decreases in a meeting as 1 Ycol (t) = √ Y (t) 2

(1)

provided that the Yi (t) are independent (in which case we can assume Yi (t) = Y (t) to be identical).

There is a knowledge representation of the problem as well.

Let Ki be the knowledge state of agent i, then we have Knew = f (K1 , K2 ) or, in general, Knew = f (K)

(2)

where f acts a local knowledge operator. For the collaborative localization example, K is belief about position and f is the averaging function (the usefulness of f is guaranteed by eq.

1). But clearly other beliefs K and functions f are possible (the usefulness of which is to be established separately).

Collaborative localization is thus a baseline case or paradigm for collaborative knowledge fusion (and both provide a ground case for CAS) because here the agents essentially exchange individual knowledge to achieve a more advanced common (and hence individual) knowledge status collectively, that is not achievable individually, and to achieve this without prior planning or coordination of agent-to-agent information flow.

IV. A N AGENT BASED MODEL The model uses a population of agents (of size N ) that navigate in 2-space. Each agent possesses a position (known to the observer) and a ”subjective position” (known to the agent).

Agents perform random motion allowed by the topology (torus, square, or a walled structure of streets or offices - the interactive model includes tools for hand-drawing them, or for importing a maze map). When an agent moves, its position is updated accordingly. Subjective position is also updated, using two kinds of errors in every step: a systematic bias (positive or negative) of step size estimate (as a model of inertia based step calculation) and a random step-uncertainty (both defined as a variable parameter). When performing random motion, agents turn by a parameter maxturn (to which the model is essentially non-sensitive but yields different moving patterns).

Agents’ meeting is understood as an event when agents are within a meeting radius d. (Note that at d = 0 agents can never meet, positions being represented as real numbers that have a negligible chance to match up exactly). Upon meeting, agents update their estimate as described above. In two different variants of the model (a difference to which the process again turns out to be essentially non-sensitive), the subjective position updating can take place for the active agent only or for both agents (the active agent is the one having the current CPU slice, which is usually a well defined entity in a model). Agents may also own a nonzero initial position error. Importantly, a part of the agent population is

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CogInfoCom 2014 • 5th IEEE International Conference on Cognitive Infocommunications • November 5-7, 2014, Vietri sul Mare, Italy

allowed to ”know” its exact position. Agents can start from the same initial point or randomly dispersed in space. An allimportant final parameter is meeting density m, which cannot be set directly, it is an emergent result of N , d and other parameters including the exact topology (think of isolated offices that segment the area versus one common room). Based on physical analogies, it can be expected that meeting density m will give the most sensitive comparisons.

elling (ABM) and simulation environment NetLogo (provided

by Northwestern University [3]) that secures tools for fast interactive development and interactive experimentation. We performed parameter sweeps using NetLogo’s built-in ”BehaviorSpace” module (on 24 AMD Opteron 6348 cores at 2,8Ghz and 64GB of RAM), and analysed the results using the R statistical programming language and environment.

TABLE I: Parameter intervals and values tested Variable meeting radius step uncertainty num-agents

Initial value 0.04 1 100

Step size 0.01 1 100

End value 0.5 10 1,000

V. R ESULTS Meeting density grows linearly with the number of agents (not shown.) A summary plot showing the effect of both is seen in Figure 4, values taken at t = 5, 000.

Fig. 3: The model GUI for interactive exploration.

The model is written in NetLogo and available from http://ccl.northwestern.edu/netlogo/models/ In interactive experiments using the GUI, a few simple facts can be readily observed. A nonzero initial position error can never disappear unless a proportion of agents is well informed (”have a GPS”). This can be also expected as in lack of external information, subjective positions can never improve beyond the best available initial estimate.

Increasing the meeting radius raises meeting density (and hence the chance to collaboratively improve the position estimate) but lowers the precision of every meeting, hence compromising the improving of the estimate, thus leading to a trade-off. Factors such as turning, topology, the initial positions of agents, and other side issues have have a negligible or zero effect (and will be omitted from the analysis below).

Understandably, higher population numbers tend, in general, to lead to higher meeting density and thus (other things being equal) to a better localization.

Most importantly, however, what we readily see in the interactive runs is that there is an important bifurcation (as seen in the field experiments) between the meeting/non-meeting regimes. Without the meetings (and the associated knowledge exchange and the resulting improvement of the subjective position estimate) the error term diverges, whereas by introducing the meetings (i.e. the information exchange and the computing of new estimates) we can keep this error bounded, with the exact expected error value depending on many parameters. All this has been made precise by parameter sweeps, detailed next.

The model can be run interactively form the GUI or ”headless” for the parameter sweeps and later analysis. The model was written in the freely available agent based mod-

Fig. 4: The effect of basic parameters: localization error decreases with meeting density and the number of agents.

The main result is best shown in time plots. Figure 5 shows 1,000 different time plots between t0 = 0 and t = 5, 000, at N = 500, using values taken from the parameter intervals indicated above. Different colors (shown on greyscale) indicate different error values. The plot proceeds from the ”nonmeeting” situation (m = 0.04) to that of frequent meetings (m = 0.5). The meeting radius axis shows 70 values as a progression from 0.04 to 0.39. Although a few outliers exist, it is well noticeable that nonzero meeting radius values yield ”flat” lines in the time plots, indicating that average localization error does not grow over time after reaching a (relatively low) maximum. On the other hand in those situations where meetings hardly if at all occur (m = 0.04) we observe a continual divergence of the average error term. The

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G. Kampis and P. Lukowicz • Collaborative Localization as a Paradigm for Incremental Knowledge Fusion

original plot is an OpenGL object and can be mouse rotated and zoomed.

A quick analysis into the (Pearson’s) correlation of the error terms Y1 (t) vs. Y2 (t) shows (Fig. 6) that the correlation indeed quickly decreases (from low medium values) to virtually zero level in realistic population numbers N . Thus, we can conclude that in real flocks of users, equipped with smart devices, such as in open spaces or public gatherings, repeated meetings and the resulting correlations do not ruin collaborative localization performance.

VI. T HE EFFECT OF WELL - INFORMED AGENTS

Fig. 5: Bifurcation between the meeting/non-meeting regimes and the noisy gradual effect of growing meeting radius.

Several other parameter sweeps have been performed but are omitted from the present report. In conclusion, the collaborative localization model is fully understood - in a wide range of parameters, it provides meaningful (i.e. constant error) localization.

However, notice that finite systems behave differently from eq. 1 and in these systems the independence Ansatz no longer holds. As a result, localization may become difficult or impossible; as a limiting case, if errors are highly correlated, no improvement can be made upon an average meeting and thus the average error may diverge. Our simulations results do not show this phenomenon for the systems studied.

The collaborative localization algorithm tested above used the ”trick” that agents that meet are at the same (or nearby) position. The broader essence of collaborative localization is, however, information exchange.

Hence it is natural to ask about the effectiveness of information spreading in the system. Specifically, fixing the number of well-informed agents (i.e. agents ”knowing” their exact position, having a zero amount of localization error) at values different from zero, we may ask how this ”supreme quality” information is spreading and used in the system. Intuition says that introducing the well-informed should decrease the average localization error in a population of agents, but how? Is there a critical transition, or is change gradual?

We have performed separate parameter sweeps for this problem at step uncertainty = 2 and meeting radius = 0.1, using the following values: Variable percent well informed number of agents

Initial value 0 100

Step size 5 100

End value 50 500

We study average localization error Y (t) at t = 5000 using 10 runs for each tested value combination. Results are shown on Figure 7.

0.10 0.05

Fig. 7: The effect of well-informed agents (from 0 to 50%) and different systems sizes.

0.00

correlation of error terms

0.15

Error correlation vs. system size (with Lowess interpolation)

200

400

600

800

1000

number of agents

Fig. 6: Decreasing error correlation as a function of the growing number of agents. Line is Lowess interpolation.

It is seen that the transitions are smooth. The effect of an increasing percentage of well-informed agents p is thus similar to that of increasing the system size N . However, note that, at the same time, the Y (t)−N section is steeper than the Y (t)−p

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CogInfoCom 2014 • 5th IEEE International Conference on Cognitive Infocommunications • November 5-7, 2014, Vietri sul Mare, Italy

section. Thus the introduction of well-informed agents has a lesser effect on the improvement than does the increase of N . (The reason is that the former does not improve meeting density but ”only” the quality of meetings, whereas the latter increases meeting frequency and hence makes information exchange more efficient).

Note that in the particular case studied, a meeting between a well-informed and a ”naive” agent does not change the perfect knowledge of the former. (In other words, once informed, always informed.)

To conclude this section, there is a trade-off between p and N . Put it differently, a few well-informed agents can (albeit with a lower efficiency) replace a higher number of other agents: having a higher proportion of well-informed in an otherwise smaller population can still bring good results.

This recognition could be utilized in the pure knowledge exchange scenarios where, in general, knowledge will not be automatically improved by sharing a geographic location.

VII. K NOWLEDGE FUSION As suggested by eq. 2, our aim is to generalize the collaborative localisation scheme to various self-organising, incremental schemes of knowledge fusion. One step towards this general end was looking into the effect of well-informed agents. If knowledge fusion is assumed to work like a ”population poll” then pure averaging effects are not detrimental. If, however, fusion is expected to produce ”true knowledge” then such well-informed agents can be crucial. Yet to be studied are the general conditions under which different varieties of incremental fusion are possible in lack of external knowledge improvement.

Similar work was started in [4], [5] in a somewhat different framework, using a P2P system for collaborative data mining.

The idea was to have a large number (perhaps millions or more) computers (nodes) where any node can send a message to any other. A fully distributed data model (horizontal data distribution) was considered. Every node has very few records, e.g. only one. By making contacts with other peers, data distribution can change. The model does not allow for moving data, however, only local exchange and local processing are possible (privacy preservation). Various self-organizing local algorithms based on ”gossip”, i.e. undirected, step-wise P2P information exchange were developed in the framework.

One particular application is to the classification problem in machine learning, Given a set of (xi , yi ) examples, where yi is the class of xi (e.g. yi is -1 or 1), we seek for a model f (), such that for all i, f (xi) = y. Here, f () is very often a parameterized function fw (), and so the classification problem becomes an error minimization problem in w. The question of ”gossip based learning” is thus how to minimise the average error in a system bases on P2P, i.e. agent-to-agent interactions.

This makes the approach similar to that considered here.

Another application concerns matrix factorization. Given R, and m × n matrix, and U , an m × k matrix, seek I (k × n) such that R ≈ U I. The idea is that using horizontal data distribution, R is nowhere represented but U (fixed) and I

(variable). If k << m, n (e.g. k = 1) then little information is stored at each node. (Even further, individual nodes may only store some elements Ui and Ij such that Rij ≈ Ui × Ij ). The ”gossip matrix” (or vector) I (or its elements Ij ) are updated by a local learning algorithm upon individual exchange with other agents. Ref. [5] discusses gossip-based local learning algorithms and conditions for their convergence.

Parallels to the work presented here are obvious and are to be explored in future publications.

VIII. C ONCLUSION AND DISCUSSION We have presented and analyzed an agent based model for collaborative localization. The latter is understood as an example of social computation where users together give rise to an intelligent or ”smart” knowledge situation without external control or planning. In collaborative localization, estimated (error-prone) spatial positions are exchanges and averaged upon meeting, leading to a recursive improvement producing a bounded error localization under a broad range of conditions. We a presented quantitative analysis showing the effect of various parameters, in particular of system size and the number of well-informed agents.

Collaborative localization was here considered a case of incremental knowledge fusion. We discussed various repercussions of its idea and its relationship to ”gossip” based selforganisation in P2P systems, to be explored later. Finally, as cognitive infocommunications deal with the use of cognitive capabilities together with ICT for communication [6], our model offers a paradigm for it where joint cognitive capacities (for knowledge formation) exceed individual ones as a consequence of ICT mediated interaction.

ACKNOWLEDGMENT We acknowledge partial funding from the European Community’s Seventh Framework Program (FP7/2007-2013) grant agreement #600854 Smart Society (http://www.smart-societyproject.eu/), by the CoCoRec (Collaborative Context Recognition in Dynamic, Multimodal Smart Environments) project supported by the German Federal Ministry of Education and Research, and by the Russian Scientific Foundation, proposal #14-21-00137.

R EFERENCES [1] K. Kloch, P. Lukowicz, C. Fischer (2011): Collaborative PDR Localisation with Mobile Phones, in: Proceedings of the 2011 15th Annual International Symposium on Wearable Computers, ISWC 11, IEEE Computer Society, Washington, DC, USA, pp. 37-40.

[2] G. Kampis, J. Kantelhardt, K. Kloch and P. Lukowicz (2013): Analytical and Simulation Models for Collaborative Localization, submitted.

[3] U. Wilensky, Netlogo, Center for Connected Learning and ComputerBased Modeling, Northwestern University. Evanston, IL. URL http://ccl.northwestern.edu/netlogo/ [4] R. Ormandi, I. Hegedus, and M. Jelasity (2011): Asynchronous peer-topeer data mining with stochastic gradient descent. In Emmanuel Jeannot, Raymond Namyst, and Jean Roman, editors, Euro-Par 2011, volume 6852 of Lecture Notes in Computer Science, pages 528?540. Springer-Verlag.

[5] M. Jelasity, A. Montresor, and O. Babaoglu (2005): Gossip-based aggregation in large dynamic networks. ACM Transactions on Computer Systems, 23(3):219?252, (doi:10.1145/1082469.1082470) [6] P. Baranyi and A. Csapo (2012): Definition and Synergies of Cognitive Infocommunications, Acta Polytechnica Hungarica, vol. 9, pp. 67?83.

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