Sequential Decision Aggregation: Accuracy and Decision Time Sandra H. Dandach, Ruggero Carli and Francesco Bullo Center for Control, Dynamical Systems & Computation University of California at Santa Barbara http://motion.me.ucsb.edu MURI FA95500710528 Project Review: Behavioral Dynamics in Cooperative Control of Mixed Human/Robot Teams Center for Human and Robot Decision Dynamics, Aug 13, 2010 Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 13aug2010 1 / 16 Sequential decision aggregation: Outline 1 Setup & Literature Review 2 SDA: analysis of decision probabilities 3 SDA: scalability analysis of accuracy/decision time 4 Conclusions and future directions Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 13aug2010 2 / 16 Setup & Literature Review Assumptions: 1 N identical individuals, arbitrary local rule 2 Independent information 3 Aggregation of individual decisions !"##$%&"’ ") *$%+,+"’, - . / 0 Group decision rule = SDA algorithm q out of N rule: decision as soon as q nodes report concordant opinion Fastest rule fastest node decides for network (q = 1) Majority rule network agrees with majority decision (q = N/2) Goal #1: characterize decision probabilities of SDA as function of: threshold and SDM decision probabilities Goal #2: express accuracy & decision time as function of: decision threshold × group size Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 13aug2010 3 / 16 Setup & Literature Review Assumptions: 1 N identical individuals, arbitrary local rule 2 Independent information 3 Aggregation of individual decisions !"##$%&"’ ") *$%+,+"’, - . / 0 Group decision rule = SDA algorithm q out of N rule: decision as soon as q nodes report concordant opinion Fastest rule fastest node decides for network (q = 1) Majority rule network agrees with majority decision (q = N/2) Goal #1: characterize decision probabilities of SDA as function of: threshold and SDM decision probabilities Goal #2: express accuracy & decision time as function of: decision threshold × group size Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 13aug2010 3 / 16
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Sequential Decision Aggregation:Accuracy and Decision Time
Sandra H. Dandach, Ruggero Carli and Francesco Bullo
Center for Control,Dynamical Systems & Computation
University of California at Santa Barbara
http://motion.me.ucsb.edu
MURI FA95500710528 Project Review: Behavioral Dynamics inCooperative Control of Mixed Human/Robot Teams
Center for Human and Robot Decision Dynamics, Aug 13, 2010
1 P. K. Varshney. Distributed Detection and Data Fusion. Signal Processing and DataFusion. Springer Verlag, 1996
2 V. V. Veeravalli, T. Basar, and H. V. Poor. Decentralized sequential detection withsensors performing sequential tests. Math Control, Signals & Systems, 7(4):292–305, 1994
3 J. N. Tsitsiklis. Decentralized detection. In H. V. Poor and J. B. Thomas, editors,Advances in Statistical Signal Processing, volume 2, pages 297–344, 1993
4 J.-F. Chamberland and V. V. Veeravalli. Decentralized detection in sensor networks. IEEETrans Signal Processing, 51(2):407–416, 2003
Social networks
1 D. Acemoglu, M. A. Dahleh, I. Lobel, and A. Ozdaglar. Bayesian learning in socialnetworks. Working Paper 14040, National Bureau of Economic Research, May 2008
For decentralized detection, with conditional independence of observations:
Tsitsiklis ’93: Bayesian decision problem with fusion center. For largenetworks identical local decision rules are asymptotically optimal
Varshney ’96: on non-identical decision rules with q out of N,1 threshold rules are optimal at the nodes levels2 finding optimal thresholds requires solving N + 2N equations
Varshney ’96: on optimal fusion rules for identical local decisions, “qout of N” is optimal at the fusion center level
Contributions today
arbitrary decision makers (rather than optimal local rules)
sequential aggregation (rather than “complete” aggregation)
For decentralized detection, with conditional independence of observations:
Tsitsiklis ’93: Bayesian decision problem with fusion center. For largenetworks identical local decision rules are asymptotically optimal
Varshney ’96: on non-identical decision rules with q out of N,1 threshold rules are optimal at the nodes levels2 finding optimal thresholds requires solving N + 2N equations
Varshney ’96: on optimal fusion rules for identical local decisions, “qout of N” is optimal at the fusion center level
Contributions today
arbitrary decision makers (rather than optimal local rules)
sequential aggregation (rather than “complete” aggregation)
Hz) and an elevated firing state (30 Hz) in a network of binaryunits (Fig. 2C). This mechanism works well when the occur-rence of each transition is equally probable at an arbitrary timepoint during a delay period. The consecutive rate distributions
of the different graded-activity types exhibited quite differentprofiles (Fig. 2, E and F). Most characteristically, the distribu-tion obtained from the stepwise rate changes in single neuronsexhibits a trough near the peak of the distribution obtainedfrom the truly graded rate changes. Thus the rate distributionenables us to examine which type of graded activity givenspike trains are more likely to represent.
Graded activity in recurrent neural networks
We then constructed a recurrent network consisting of 500excitatory neurons and 100 inhibitory neurons (see METHODS).In the network model, excitatory neurons receive excitatoryand inhibitory recurrent synaptic inputs, excitatory and inhib-itory background synaptic inputs, and an external input toinduce graded activity. Inhibitory neurons receive synapticinput from excitatory neurons as well as excitatory and inhib-itory background synaptic inputs. Each excitatory neuronprojects to 10% of randomly chosen other excitatory neuronsand to all inhibitory neurons, whereas each inhibitory neuronprojects to all excitatory neurons, but not to other inhibitoryneurons. We note that the temporal integration performancewas relatively independent of the connectivity of synapses. In
FIG. 2. Comparison between different temporal integration mechanisms. A:graded activity may be modeled as a trial- or an ensemble-average of graduallyincreasing firing rates of individual neurons. B: climbing activity (bottom) wasconstructed from nonstationary Poisson spike trains with a gradually increas-ing mean firing rate (top). C: graded activity in our model consists oftemporally organized bimodal transitions between the baseline and elevatedfiring states. In the individual neurons, the transitions should occur at arbitrarytemporal positions with equal probabilities. Both trial average and ensembleaverage give equally good representations of graded activity in the presentmodel. D: climbing activity (bottom) was constructed from artificial bimodalPoisson spike trains showing stepwise increases in the mean firing rate (top).E: consecutive firing-rate distribution (see METHODS) exhibits a single peak inthe climbing activity shown in B. F: by contrast, the firing-rate distribution isbimodal in the climbing activity shown in D.
FIG. 1. Bimodal firing states of model excitatory neuron. A: responses of asingle excitatory neuron to a brief stimulus are shown in the absence of recurrentsynaptic inputs and the fluctuating components of background synaptic inputs (the“frozen” condition). External input was set as Iext ! 0 nA (top), for which theresponse was not bistable, and Iext ! 0.025 nA (bottom), for which the responsewas bistable. In the latter case, neuronal firing was terminated by a hyperpolarizinginput. Horizontal bars show the duration of the stimuli. B: model neuron withbistability repeats noise-driven transitions between the baseline and elevated firingstates under the influences of continuous synaptic bombardments (top). Monitor-ing the intracellular calcium density enables us to distinguish the epochs of theelevated firing state (bottom, gray shades). C: presence of the 2 distinct firing statesresults in a bimodal consecutive firing-rate distribution. D: bimodal firing-ratedistribution is shown at Iext ! 0.035 nA.
3862 H. OKAMOTO, Y. ISOMURA, M. TAKADA, AND T. FUKAI