Uncertainty Quantification in AI

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

2. 2 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

3. 3 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

4. 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

5. Deep Networks cannot look beyond their horizon Motivation 5 90% cat 10% dog

6. Deep Networks cannot look beyond their horizon Motivation 6 40% cat 60% dog

7. Deep Networks cannot look beyond their horizon Motivation 7 ?

8. 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 8

9. Simple Regression Problem Interpolation 9

10. Simple Regression Problem Deep Networks don’t extrapolate Neural Arithmetic Logic Units, NIPS'18, Andrew Trask et. al.10

11. Simple Regression Problem Deep Networks don’t extrapolate 11

12. Simple Regression Problem Uncertainty about interpolation and extrapolation 12

13. Types of Uncertainty 13

15. Methods for Uncertainty Quantification 16 Relaxation of mathematical assumptions about data Gaussian Processes Deep Ensembles / Dropout Ensembles Quantile Regression Monte Carlo Dropout

17. A Gaussian Process can be thought of as a random function which is defined by its mean and covariance functions Gaussian Processes 18 Definition

18. Gaussian Processes 19 Example

19. Gaussian Processes 20 Inference

20. Gaussian Processes 21 Inference

21. Gaussian Processes 22 Inference with perfect interpolation

22. Gaussian Processes 23 Inference with noisy observations

23. Gaussian Processes 24 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

25. MC Dropout Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning, ICML 2016, Yarin Gal et. al. 26 ...

27. 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

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

30. Dropout Ensembles 31 The best of both worlds? ...

32. Using the cumulative distribution function (cdf) of a random variable Y, we define the quantile: Loss function to estimate quantile: Quantile Regression 33

33. Intuition behind Quantile Regression 34 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

34. Intuition behind Quantile Regression 35 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

35. Intuition behind Quantile Regression 36 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

36. Intuition behind Quantile Regression 37 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

37. Now the 0.75th Quantile 38 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

38. Now the 0.75th Quantile 39 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

40. 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

41. Experiments 42 Dataset

42. 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 43 Network setup and hyperparameters

43. Mean squared error (MSE) Mean negative log likelihood (MNLL) Mean Kullback-Leibler (KL) divergence Experiments 44 Measures for generalization quality

44. Experiments 45 Interpolation

45. Experiments 46 They still don’t extrapolate and they don’t quite realize

46. Experiments 47 Gaussian Process

47. Experiments Convergence 48

48. Experiments 49 Heteroscedastic noise

49. Experiments 50 Non-Gaussian noise

50. Experiments 51 Uncertainty split aleatoric epistemic

51. Summary 52 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

53. › 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 54 There is work to be done

54. › Bayesian Neural Networks (e.g. with PyMC) › Sparse Gaussian Process approximations › Gaussian Processes on top of neural networks Outlook 55 Other approaches

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