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Are you sure about that!?
Uncertainty Quantification in AI
Florian Wilhelm Berlin, October 10th 2019

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Dr. Florian Wilhelm
Principal Data Scientist @ inovex
@FlorianWilhelm
FlorianWilhelm
florianwilhelm.info
Mathematical Modelling
Data Science to Production
Recommender Systems
Uncertainty Quantification & Causality
Python Data Stack
Maintainer PyScaffold

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Simon Bachstein
Data Scientist @ inovex
2018/07 – 2019/01 Master Thesis at inovex:
Uncertainty Quantification in Deep Learning
• Blogpost:
http://inovex.de/blog/uncertainty-quantification-deep-learning
• Master Thesis:
https://sbachstein.de/master_thesis.pdf
@simonbachstein
sbachstein
sbachstein.de

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1. Motivation
2. Methods
a. Gaussian Processes
b. Monte-Carlo Dropout
c. Deep Ensembles
d. Dropout Ensembles
e. Quantile Regression
3. Experiments
4. Conclusion & Outlook
Agenda
4

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Deep Networks cannot look beyond their horizon
Motivation
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90% cat
10% dog

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Deep Networks cannot look beyond their horizon
Motivation
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40% cat
60% dog

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Deep Networks cannot look beyond their horizon
Motivation
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?

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Boult, T. E., Cruz, S., Dhamija, A., Gunther, M., Henrydoss, J., & Scheirer, W. (2019). Learning and the Unknown: Surveying
Steps Toward Open World Recognition. Aaai, 1–8. Retrieved from www.aaai.org
Learning and the Unknown
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Simple Regression Problem
Interpolation
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Simple Regression Problem
Deep Networks don’t extrapolate
Neural Arithmetic Logic Units, NIPS'18, Andrew Trask et. al.10

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Simple Regression Problem
Deep Networks don’t extrapolate
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Simple Regression Problem
Uncertainty about interpolation and extrapolation
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Types of Uncertainty
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1. Motivation
2. Methods
a. Gaussian Processes
b. Monte-Carlo Dropout
c. Deep Ensembles
d. Dropout Ensembles
e. Quantile Regression
3. Experiments
4. Conclusion & Outlook
Agenda
14

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Methods for Uncertainty Quantification
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Relaxation of mathematical assumptions about data
Gaussian
Processes
Deep Ensembles / Dropout
Ensembles
Quantile
Regression
Monte Carlo
Dropout

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1. Motivation
2. Methods
a. Gaussian Processes
b. Monte-Carlo Dropout
c. Deep Ensembles
d. Dropout Ensembles
e. Quantile Regression
3. Experiments
4. Conclusion & Outlook
Agenda
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A Gaussian Process can be thought of as a random function
which is defined by its mean and covariance functions
Gaussian Processes
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Definition

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Gaussian Processes
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Example

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Gaussian Processes
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Inference

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Gaussian Processes
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Inference

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Gaussian Processes
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Inference with perfect interpolation

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Gaussian Processes
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Inference with noisy observations

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Gaussian Processes
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Inference
Inference using given data points can be done analytically. For
example, when assuming the (prior) mean function to be zero
everywhere, we get:
Good introduction:
Bayesian Non-parametric Models for Data Science using PyMC by Christopher Fonnesbeck
• https://www.youtube.com/watch?v=-sIOMs4MSuA
• https://de.slideshare.net/mlreview/bayesian-nonparametric-models-for-data-science-using-pymc
computationally intense

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1. Motivation
2. Methods
a. Gaussian Processes
b. Monte-Carlo Dropout
c. Deep Ensembles
d. Dropout Ensembles
e. Quantile Regression
3. Experiments
4. Conclusion & Outlook
Agenda
25

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MC Dropout
Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning, ICML 2016, Yarin Gal et. al.
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...

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1. Motivation
2. Methods
a. Gaussian Processes
b. Monte-Carlo Dropout
c. Deep Ensembles
d. Dropout Ensembles
e. Quantile Regression
3. Experiments
4. Conclusion & Outlook
Agenda
27

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Deep Ensembles
Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles, NIPS 2017, Balaji Lakshminarayanan et. al28
Custom loss function:
Capture uncertainty directly at training time

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Deep Ensembles
Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles, NIPS 2017, Balaji Lakshminarayanan et. al29
Combine an ensemble of networks

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1. Motivation
2. Methods
a. Gaussian Processes
b. Monte-Carlo Dropout
c. Deep Ensembles
d. Dropout Ensembles
e. Quantile Regression
3. Experiments
4. Conclusion & Outlook
Agenda
30

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Dropout Ensembles
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The best of both worlds?
...

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1. Motivation
2. Methods
a. Gaussian Processes
b. Monte-Carlo Dropout
c. Deep Ensembles
d. Dropout Ensembles
e. Quantile Regression
3. Experiments
4. Conclusion & Outlook
Agenda
32

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Using the cumulative distribution function (cdf) of a random variable Y, we
define the quantile:
Loss function to estimate quantile:
Quantile Regression
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Intuition behind Quantile Regression
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0.0 0.1 0.2 0.90.3 0.4 0.5 0.6 0.7 0.8 1.0
Assume median is here (𝜏 = 0.5)
𝑦 > )𝑞+(𝑥)
𝑦 ≤ )𝑞+(𝑥)
0.1
0.2
0.5
0.8
Error: 0.0 + 1.6 = 1.6

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Intuition behind Quantile Regression
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0.0 0.1 0.2 0.90.3 0.4 0.5 0.6 0.7 0.8 1.0
Assume median is here (𝜏 = 0.5)
𝑦 > )𝑞+(𝑥)
𝑦 ≤ )𝑞+(𝑥)
0.1
0.4
0.7
Error: 0.0 + 1.2 = 1.2
0.0

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Intuition behind Quantile Regression
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0.0 0.1 0.2 0.90.3 0.4 0.5 0.6 0.7 0.8 1.0
Assume median is here (𝜏 = 0.5)
𝑦 > )𝑞+(𝑥)
𝑦 ≤ )𝑞+(𝑥)
0.1
0.3
0.6
Error: 0.1 + 0.9 = 1.0
0.0

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Intuition behind Quantile Regression
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0.0 0.1 0.2 0.90.3 0.4 0.5 0.6 0.7 0.8 1.0
Assume median is here (𝜏 = 0.5)
𝑦 > )𝑞+(𝑥)
𝑦 ≤ )𝑞+(𝑥)
0.2
0.2
0.5
Error: 0.3 + 0.7 = 1.0
0.1
No change due to the linearity of the error!
+0.1
+0.1
-0.1
-0.1

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Now the 0.75th Quantile
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0.0 0.1 0.2 0.90.3 0.4 0.5 0.6 0.7 0.8 1.0
Assume 𝜏 = 0.75 is here
𝑦 > )𝑞+(𝑥)
𝑦 ≤ )𝑞+(𝑥)
0.2
0.2
0.5
0.1
Error: (1 − 0.75) ⋅ 0.3 + 0.75 ⋅ 0.7 = 0.6
Right-side error weights 3 times as much as
the left-side error

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Now the 0.75th Quantile
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0.0 0.1 0.2 0.90.3 0.4 0.5 0.6 0.7 0.8 1.0
𝜏 = 0.75
𝑦 > )𝑞+(𝑥)
𝑦 ≤ )𝑞+(𝑥)
0.5
0.1
0.2
Error: (1 − 0.75) ⋅ 1.0 + 0.75 ⋅ 0.2 = 0.4
0.4
Change in the right-side error also weights
3 times as much as the left-side error

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1. Motivation
2. Methods
a. Gaussian Processes
b. Monte-Carlo Dropout
c. Deep Ensembles
d. Dropout Ensembles
e. Quantile Regression
3. Experiments
4. Conclusion & Outlook
Agenda
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According to the function
Samples are generated as follows:
Experiments
Uncertainty in Deep Learning (Phd thesis), Yarin Gal, http://mlg.eng.cam.ac.uk/yarin/blog_2248.html41
Dataset

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Experiments
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Dataset

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Neural networks
› 2 hidden layers with 20 ReLU neurons each
› 5 networks for Deep Ensembles
› 100 iterations for Dropout predictions
› Adam optimizer with batch size of 128
› LR, weight decay, dropout probability are optimized
Gaussian Processes
› squared exponential covariance and zero mean function prior
› covariance function parameters and aleatory noise are optimized
Experiments
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Network setup and hyperparameters

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Mean squared error (MSE)
Mean negative log likelihood (MNLL)
Mean Kullback-Leibler (KL) divergence
Experiments
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Measures for generalization quality

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Experiments
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Interpolation

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Experiments
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They still don’t extrapolate and they don’t quite realize

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Experiments
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Gaussian Process

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Experiments
Convergence
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Experiments
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Heteroscedastic noise

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Experiments
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Non-Gaussian noise

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Experiments
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Uncertainty split
aleatoric epistemic

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Summary
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GP MCD DeepE DropoutE QR
Homoscedastic
noise
++ o + o o
Heteroscedastic
noise
-- - ++ + +
Non-Gaussian
noise
+ o + + -
Convergence ++ - + - +
Speed (--) + - / (+) + ++
Uncertainty split yes no yes yes no

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1. Motivation
2. Methods
a. Gaussian Processes
b. Monte-Carlo Dropout
c. Deep Ensembles
d. Dropout Ensembles
e. Quantile Regression
3. Experiments
4. Conclusion & Outlook
Agenda
53

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› Neural network approaches discussed here are very aware of
aleatory uncertainty, however, not capable of correctly estimating
epistemic uncertainty
› Gaussian Processes give clear signals about ignorance but do not
scale
A combined solution needs to be developed because
uncertainty estimation is needed in critical applications
Conclusion
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There is work to be done

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› Bayesian Neural Networks (e.g. with PyMC)
› Sparse Gaussian Process approximations
› Gaussian Processes on top of neural networks
Outlook
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Other approaches

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Thank You!
Florian Wilhelm
Principal Data Scientist
inovex GmbH
Schanzenstraße 6-20
Kupferhütte 1.13
51063 Köln
florian.wilhelm@inovex.de