1 Setup & Literature Review. 2 SDA: analysis of decision probabilities. 3 SDA: scalability analysis of accuracy/decision time

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1 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 MURI FA978 Project Review: Behavioral Dynamics in Cooperative Control of Mixed Human/Robot Teams Center for Human and Robot Decision Dynamics, Aug 3, Sequential decision aggregation: Outline Setup & Literature Review 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 3aug / 6 Setup & Literature Review Assumptions: identical individuals, arbitrary local rule Independent information -(.( /( ( 3 Aggregation of individual decisions!"$%&"'(")(( *$%+,+"',( Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug / 6 Setup & Literature Review Assumptions: identical individuals, arbitrary local rule Independent information -(.( /( ( 3 Aggregation of individual decisions!"$%&"'(")(( *$%+,+"',( Group decision rule = SDA algorithm q out of rule: decision as soon as q nodes report concordant opinion Fastest rule fastest node decides for network (q = ) Majority rule network agrees with majority decision (q = / ) Goal : characterize decision probabilities of SDA as function of: threshold and SDM decision probabilities Goal : express accuracy & decision time as function of: decision threshold group size Group decision rule = SDA algorithm q out of rule: decision as soon as q nodes report concordant opinion Fastest rule fastest node decides for network (q = ) Majority rule network agrees with majority decision (q = / ) Goal : characterize decision probabilities of SDA as function of: threshold and SDM decision probabilities Goal : express accuracy & decision time as function of: decision threshold group size Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug 3 / 6 Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug 3 / 6

2 Setup & Literature Review Literature review Assumptions: identical individuals, arbitrary local rule Independent information -(.( /( ( 3 Aggregation of individual decisions!"$%&"'(")(( *$%+,+"',( Group decision rule = SDA algorithm q out of rule: decision as soon as q nodes report concordant opinion Fastest rule fastest node decides for network (q = ) Majority rule network agrees with majority decision (q = / ) Goal : characterize decision probabilities of SDA as function of: threshold and SDM decision probabilities Goal : express accuracy & decision time as function of: decision threshold group size Distributed/decentralized detection P. K. Varshney. Distributed Detection and Data Fusion. Signal Processing and Data Fusion. Springer Verlag, 996 V. V. Veeravalli, T. Başar, and H. V. Poor. Decentralized sequential detection with sensors performing sequential tests. Math Control, Signals & Systems, 7(4):9 3, J.. Tsitsiklis. Decentralized detection. In H. V. Poor and J. B. Thomas, editors, Advances in Statistical Signal Processing, volume, pages , J.-F. Chamberland and V. V. Veeravalli. Decentralized detection in sensor networks. IEEE Trans Signal Processing, ():47 46, 3 Social networks D. Acemoglu, M. A. Dahleh, I. Lobel, and A. Ozdaglar. Bayesian learning in social networks. Working Paper 44, ational Bureau of Economic Research, May 8 Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug 3 / 6 Literature review For decentralized detection, with conditional independence of observations: Tsitsiklis 93: Bayesian decision problem with fusion center. For large networks identical local decision rules are asymptotically optimal Varshney 96: on non-identical decision rules with q out of, threshold rules are optimal at the nodes levels finding optimal thresholds requires solving + equations Varshney 96: on optimal fusion rules for identical local decisions, q out of is optimal at the fusion center level Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug 4 / 6 Literature review For decentralized detection, with conditional independence of observations: Tsitsiklis 93: Bayesian decision problem with fusion center. For large networks identical local decision rules are asymptotically optimal Varshney 96: on non-identical decision rules with q out of, threshold rules are optimal at the nodes levels finding optimal thresholds requires solving + equations Varshney 96: on optimal fusion rules for identical local decisions, q out of is optimal at the fusion center level Contributions today arbitrary decision makers (rather than optimal local rules) sequential aggregation (rather than complete aggregation) scalability analysis of accuracy / decision time Contributions today arbitrary decision makers (rather than optimal local rules) sequential aggregation (rather than complete aggregation) scalability analysis of accuracy / decision time Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug / 6 Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug / 6

3 Today s Outline Model of sequential decision maker p i j (t) for a Gaussian distribution umber of observations (t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etup & Literature Review SDA: analysis of decision probabilities Sequential decision maker (SDM) p i j (t) := Probability say Hi given Hj at time t + + p i j = p i j (t), E[T Hi] = t ( p i (t) + p i (t) ) t= t= 3 SDA: scalability analysis of accuracy/decision time Assume knowledge of {p i j (t)}t for individual SDM, known exactly, calculated numerically, or measured empirically 4 Conclusions and future directions p (t) Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug 6 / 6 Sequential decision aggregation: Intermediate events Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug 7 / 6 Sequential decision aggregation: Intermediate events!"$ '( ')!"$%&$ FIG.. Bimodal firing states of model excitatory neuron. A: responses o single excitatory neuron to a brief stimulus are shown'( in the absence of recurr synaptic inputs and the fluctuating components of background synaptic inputs ( frozen condition). External input was set as I ext na (top), for which response was not bistable, and I ext. na (bottom), ') for which the respon was bistable. In the latter case, neuronal firing was terminated by a hyperpolariz input. Horizontal bars show the duration of the stimuli. B: model neuron w bistability repeats noise-driven transitions between the!"$%&$ baseline and elevated fir states under the influences of continuous synaptic bombardments (top). Monit ing!"$ the intracellular calcium density enables us to distinguish the epochs of elevated firing state (bottom, gray shades). C: presence of the distinct firing sta results in a bimodal consecutive firing-rate distribution. D: bimodal firing-r distribution is shown at I ext.3 na. Hz) and an elevated firing state (3 Hz) in a network of bina units (Fig. C). This mechanism works well when the occu rence of each transition is equally probable at an arbitrary tim point during a delay period. The consecutive rate distributio!"$!"$ J europhysiol VO aggregate states and divide in groups characterized by count calculate the probability of transition between the different groups characterize two states for network decisions H and H aggregate states and divide in groups characterized by count calculate the probability of transition between the different groups characterize two states for network decisions H and H Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug 8 / 6 Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug 8 / 6

4 Sequential decision aggregation: Computational approach Sequential decision aggregation: Computational approach Goal: as function of SDM decision probabilities {p i j (t)}t, compute SDA decision probabilities {p i j (t;, q)}t Goal: as function of SDM decision probabilities {p i j (t)}t, compute SDA decision probabilities {p i j (t;, q)}t General result: q out of decision probabilities p i j (t;, q) = α(t, s, s)β s= s= s + i j (t, s, s) s / + s=q ᾱ(t, s) s β i j (t, s) General result: q out of decision probabilities p i j (t;, q) = α(t, s, s)β s= s= s + i j (t, s, s) s / + s=q ᾱ(t, s) s β i j (t, s) As function of t and sizes, formulas for α, β, ᾱ, and β computational complexity linear in Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug 9 / 6 Sequential decision aggregation: Computational approach As function of t and sizes, formulas for α, β, ᾱ, and β computational complexity linear in Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug 9 / 6 Illustration of results Goal: as function of SDM decision probabilities {p i j (t)}t, compute SDA decision probabilities {p i j (t;, q)}t General result: q out of decision probabilities p i j (t;, q) = α(t, s, s)β s= s= s + i j (t, s, s) s / + s=q ᾱ(t, s) s β i j (t, s) As function of t and sizes, formulas for α, β, ᾱ, and β computational complexity linear in E[T] 3 3 q.9.9 q Expected Decision Time Probability of correct decision (H : µ = ) and (H : µ = ) SPRT with pf = pm =. Gaussian noise (µ, σ), σ = and µ {, } P[say H H ] 3 Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug 9 / 6 Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug / 6

5 Today s Outline Asymptotic results for the Fastest rule Setup & Literature Review SDA: analysis of decision probabilities 3 SDA: scalability analysis of accuracy/decision time Expected Decision Time: lim E [T H,, fastest] = earliest possible decision time Accuracy: =: tmin = min{t either p (t) or p (t) } lim p (, fastest) = {, if p (tmin) > p (tmin), if p (tmin) < p (tmin) 4 Conclusions and future directions SDA accuracy is function of (SDM probability at tmin), not of (SDA cumulative probability)! hence, SDA accuracy is not monotonic with 3 hence, SDA accuracy is unrelated to SDM accuracy for large Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug / 6 Asymptotic results for the Fastest rule Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug / 6 Asymptotic results for the Majority rule Expected Decision Time: lim E [T H,, fastest] = earliest possible decision time Accuracy: =: tmin = min{t either p (t) or p (t) } lim p (, fastest) = {, if p (tmin) > p (tmin), if p (tmin) < p (tmin) SDA accuracy is function of (SDM probability at tmin), not of (SDA cumulative probability)! hence, SDA accuracy is not monotonic with 3 hence, SDA accuracy is unrelated to SDM accuracy for large Expected Decision Time: Assume p > p and define t < := max{t p () + + p (t) < /}, t > := min{t p () + + p (t) > /} Then [ ] lim E T H,, majority = ( ) t < + t > + Accuracy: Monotonicity with group size and, as, if p < / p (, majority), if p > / /(π) (4p ), if p < /4 Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug / 6 Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug 3 / 6

Asymptotic results for the Majority rule Lessons learned about SDA Expected Decision Time: Assume p > p and define t < := max{t p () + + p (t) < /}, t > := min{t p () + + p (t) > /} Accuracy Expected

Accuracy: Monotonicity with group size and, as, if p < / p (, majority), if p > / /(π) (4p ), if p < /4 Probability Expected number of wrong decision of observations.. Comparison: Fastest vs.

Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug 4 / 6 Conclusions and future directions E[T] 4 3 to compare different thresholds, re-scale local accuracy the group accuracy is now same (eg,

6 Asymptotic results for the Majority rule Lessons learned about SDA Expected Decision Time: Assume p > p and define t < := max{t p () + + p (t) < /}, t > := min{t p () + + p (t) > /} Accuracy Expected decision time Fastest SDM accuracy at tmin earliest possible decision time tmin Majority exponentially better than SDM average of half-times t <, t > Then [ ] lim E T H,, majority = ( ) t < + t > + Accuracy: Monotonicity with group size and, as, if p < / p (, majority), if p > / /(π) (4p ), if p < /4 Probability Expected number of wrong decision of observations.. Comparison: Fastest vs. Majority 3 4 Fastest rule 6 umber of decision makers Majority rule umber of decision makers Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug 3 / 6 A fair comparison Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug 4 / 6 Conclusions and future directions E[T] 4 3 to compare different thresholds, re-scale local accuracy the group accuracy is now same (eg, low or high) compare the decision time network accuracy=.99 Fastest rule Majority rule for most cases majority rule is best for some small inaccurate networks, fastest rule is best E[T] etwork accuracy.9 Fastest rule Majority rule Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug / 6 Summary fundamental understanding of sequential aggregation applicable to broad range of agent models, eg, mixed networks applicable to family of threshold-based rules 3 tradeoffs in fastest vs majority 4 role of time in sequential aggregation Future directions models with heterogeneous agents models with interactions between agents 3 models with correlated information 4 how to use this analysis for design Dandach, Carli, Bullo (UCSB) Sequential Decision Aggregation 3aug 6 / 6

Accuracy and Decision Time for Decentralized Implementations of the Sequential Probability Ratio Test

Accuracy and Decision Time for Decentralized Implementations of the Sequential Probability Ratio Test 21 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July 2, 21 ThA1.3 Accuracy Decision Time for Decentralized Implementations of the Sequential Probability Ratio Test Sra Hala