Smart Embedded Systems Group University of Osnabrück
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Uncertainty and Trust Estimation inIncrementally Learning Function Approximation
Andreas Buschermöhle, Jan Schoenke and Werner Brockmann
Institute of Computer ScienceSmart Embedded Systems Group
Catania, July 9th 2012
Smart Embedded Systems Group University of Osnabrück
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Overview
Motivation
Uncertainties in Learning Function Approximation
Uncertainty Estimation
Results
Conclusion and Summary
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Motivation
• Upcoming tasks are of increased complexity
Operation in natural and changing environments
Not every situation can be planned in advance
Unforeseeable interaction behavior
• Disturbances influence the system dynamically
Sensor faults, Actuator faults
Unobserved influences
…
Changing behavior Incremental learning
Disturbances Uncertainties
Safe operation always crucial
?
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Uncertainty in Learning Function Approximation
• Five sources influence incrementally learning function approximation
i. Uncertain training input data
ii. Uncertain training output data
iii. Varying target function / unobserved variables
iv. Expressiveness of the approximator
v. Sparsity of training data
• At the output this results in two effects
1. No data is available to evaluatethe approximation (ignorance)
2. Multiple output values are possible,i.e. the target output varies (conflict)
i
ii
iii
iv
v
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Related Work
• Gaussian processes [Rasm2006], [Dall2009]
Local data density reflected in output variance
Separate input variance possible
• Evidence theory [Deno1997], [Peti2004]
Local data density reflected in output bounds
Bound for training outputs
• RBF networks [Leon1992]
Local data density by counting for each RBF
Approximation uncertainty by cross-validation
v
v
ii
v
iv
Every approach is dataset-based
Not all sources of uncertainty are covered
i
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Approach
• Different measures to observe the influence
Mapped to trust signals
no trustworthiness
1 full trustworthiness
Fuzzy operators for fusion of trust signals
• Application to a zero-order Takagi-Sugeno fuzzy system
Local linear interpolation
• Learning by normalized gradient descent
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Approach
• Local Measures:
Frequency of learning stimuli
Activity sum of learning stimuli
Mean node adaptation
Mean absolute node adaptation
Direct incremental rating of node adaptation
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Evaluation
• Two measures to investigate the performance
Trust-weighted MSE Mean trust level
• Two exemplary target functions
• Simulated disturbances for cases i-iv
i. Normally distributed noise on training input
ii. Normally distributed noise on training output
iii. Additive disturbance of with unobserved input
iv. Lowered number of fuzzy rules for approximation
v. Implicitly covered in progress of learning
i
ii
iii
iv
v
∑𝑗=1
𝑁𝑡 𝜗 (𝑥 𝑗 )𝑁𝑡
( 𝑓 (𝑥 𝑗 )−~𝑓 (𝑥 𝑗)) ² ∑
𝑗=1
𝑁𝑡 𝜗 (𝑥 𝑗 )𝑁𝑡
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Resultsno disturbances
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Resultstraining input noisei
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Resultstraining output noiseii
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Resultsunobserved variableiii
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Resultsexpressiveness of the approximatoriv
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Resultsall sources of uncertaintyvi
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Conclusions
• It is possible to estimate all influences of uncertainty on-line
• Different measures account for different influences
Activity sum of learning stimuli () ignorance
Mean absolute adaptation () long term conflict
Direct incremental rating of adapt. () short term conflict
• Combination of these through trust management
redundantnon
redundant
Good combined results for all influences of uncertainty
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Combined Uncertainty Measureall sources of uncertaintyvi
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Summary
Integral view on all sources of uncertainty inincrementally learning function approximation
Measuring ignorance and conflict combined through trust management
Combined trust signal represents• all uncertainties at the output• locally• incrementally• gradually
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References
[Dall2009] Dallaire, P., Besse, C., Chaib-Draa, B.: Learning Gaussian Process Models from Uncertain Data. In: Proc. Int. Conf. on Neural Information Processing, pp. 433-440. Springer, Bangkok (2009)
[Deno1997] Denoeux, T.: Function Approximation in the Framework of Evidence Theory: A Connectionist Approach. In: Proc. Int. Conf. on Neural Networks, pp. 199-203. IEEE Press, Houston (1997)
[Leon1992] Leonard, J.A., Kramer, M.A., Ungar, L.H.: Using Radial Basis Functions to Approximate a Function and its Error Bounds. IEEE Transactions on Neural Networks, vol. 3, no. 4, 624-627. IEEE Press (1992)
[Peti2004] Petit-Renaud, S., Denoeux, T.: Nonparametric Regression Analysis of Uncertain and Imprecise Data Using Belief Functions. In: Int. J. of Approximate Reasoning 35, pp. 1-28. Elsevier (2004)
[Rasm2006] Rasmussen, C.E.,Williams, C.K.I.: Gaussian Processes for Machine Learning. MIT Press, Cambridge (2006)
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