Modeling uncertainty in deep learning
15 likes3,168 views
The document discusses the uncertainties in deep learning, highlighting examples of model uncertainty and the application of Bayesian neural networks for managing epistemic and aleatoric uncertainties. It covers various methodologies, including dropout as a Bayesian approximation, variational inference, and the use of mixture density networks to quantify uncertainties in predictive tasks. Additionally, it addresses approaches for safe reinforcement learning and the implications of these uncertainties in practical scenarios.
![Kendal & Gal (2017)
45
Aleatoric & epistemic uncertainties
ˆW ⇠ q(W)
x
ˆy ˆW(x) ˆ2
ˆW
(x)
[ˆy, ˆ2
] = f
ˆW
(x)
L =
1
N
NX
i=1
log N(yi; ˆy ˆW(x), ˆ2
ˆW
(x))](https://image.slidesharecdn.com/uncertainty-in-deep-learning-180106133211/85/Modeling-uncertainty-in-deep-learning-45-320.jpg)
![Kendal & Gal (2017)
46
Heteroscedastic uncertainty as loss attenuation
ˆW ⇠ q(W)
x
ˆy ˆW(x) ˆ2
ˆW
(x)
[ˆy, ˆ2
] = f
ˆW
(x)](https://image.slidesharecdn.com/uncertainty-in-deep-learning-180106133211/85/Modeling-uncertainty-in-deep-learning-46-320.jpg)
![Kendal & Gal (2017)
47
Aleatoric & epistemic uncertainties
ˆW ⇠ q(W)
x
ˆy ˆW(x) ˆ2
ˆW
(x)
[ˆy, ˆ2
] = f
ˆW
(x)
Var(y) ⇡
1
T
TX
t=1
ˆy2
t
TX
t=1
ˆyt
!2
+
1
T
TX
t=1
ˆ2
t
Epistemic unct. Aleatoric unct,](https://image.slidesharecdn.com/uncertainty-in-deep-learning-180106133211/85/Modeling-uncertainty-in-deep-learning-47-320.jpg)
Modeling uncertainty in deep learning
- 1. Uncertainties in Deep Learning in a nutshell Sungjoon Choi CPSLAB, SNU
- 2. Introduction 2 The first fatality from an assisted driving system
- 3. Introduction 3 Google Photos identified two black people as 'gorillas'
- 5. Contents 5
- 6. Contents 6 Bayesian Neural Network with variational inference and re-parametrization trick Bootstrapping- based uncertainty modeling Bayesian Neural Network modeling epistemic and aleatoric uncertainties Application to Safe RL Novelty Detection using auto-encoder Mixture Density Network modeling epistemic and aleatoric uncertainties
- 7. 7 Y. Gal, Uncertainty in Deep Learning, 2016
- 8. Gal (2016) 8 Model uncertainty 1. Given a model trained with several pictures of dog breeds, a user asks the model to decide on a dog breed using a photo of a cat.
- 9. Gal (2016) 9 Model uncertainty 2. We have three different types of images to classify, cat, dog, and cow, where only cat images are noisy.
- 10. Gal (2016) 10 Model uncertainty 3. What is the best model parameters that best explain a given dataset? what model structure should we use?
- 11. Gal (2016) 11 Model uncertainty 1. Given a model trained with several pictures of dog breeds, a user asks the model to decide on a dog breed using a photo of a cat. 2. We have three different types of images to classify, cat, dog, and cow, where only cat images are noisy. 3. What is the best model parameters that best explain a given dataset? what model structure should we use?
- 12. Gal (2016) 12 Model uncertainty 1. Given a model trained with several pictures of dog breeds, a user asks the model to decide on a dog breed using a photo of a cat. 2. We have three different types of images to classify, cat, dog, and cow, where only cat images are noisy. 3. What is the best model parameters that best explain a given dataset? what model structure should we use? Out of distribution test data Aleatoric uncertainty Epistemic uncertainty
- 13. Gal (2016) 13 Dropout as a Bayesian approximation “We show that a neural network with arbitrary depth and non-linearities, with dropout applied before every weight layer, is mathematically equivalent to an approximation to a well known Bayesian model.”
- 14. Gal (2016) 14 Dropout as a Bayesian approximation The resulting formulations are surprisingly simple.
- 15. Gal (2016) 15 Bayesian Neural Network Posterior p(w|X, Y) Prior p(w) In Bayesian inference, we aim to find a posterior distribution over the random variables of our interest given a prior distribution which is intractable in many case.
- 16. Gal (2016) 16 Bayesian Neural Network p(y⇤ |x⇤ , X, Y) = Z p(y⇤ |x⇤ , w)p(w|X, Y)dw Posterior p(w|X, Y) Inference Prior p(w) Note that even when given a posterior distribution, exact inference is very likely to be intractable as it contains integral with respect to the distribution over latent variables.
- 17. Gal (2016) 17 Bayesian Neural Network p(y⇤ |x⇤ , X, Y) = Z p(y⇤ |x⇤ , w)p(w|X, Y)dw Posterior p(w|X, Y) Inference Variational Inference KL (q✓(w)||p(w|X, Y)) = Z q✓(w) log q✓(w) p(w|X, Y) dw Prior p(w) Variational inference is used to approximate the (intractable) posterior distribution with (tractable) variational distribution with respect to the KL divergence.
- 18. Gal (2016) 18 Bayesian Neural Network p(y⇤ |x⇤ , X, Y) = Z p(y⇤ |x⇤ , w)p(w|X, Y)dw Posterior p(w|X, Y) Inference Variational Inference KL (q✓(w)||p(w|X, Y)) = Z q✓(w) log q✓(w) p(w|X, Y) dw ELBO Z q✓(w) log p(Y|X, w)dw KL(q✓(w)||p(w)) Prior p(w) Minimizing the KL divergence is equivalent to maximizing the evidence lower bound (ELBO) which also contains the integral with respect to the distribution over latent variables.
- 19. Gal (2016) 19 Bayesian Neural Network p(y⇤ |x⇤ , X, Y) = Z p(y⇤ |x⇤ , w)p(w|X, Y)dw Posterior p(w|X, Y) Inference Variational Inference KL (q✓(w)||p(w|X, Y)) = Z q✓(w) log q✓(w) p(w|X, Y) dw ELBO Z q✓(w) log p(Y|X, w)dw KL(q✓(w)||p(w)) Prior p(w) Instead of the posterior distribution, we only need a likelihood to compute the ELBO.
- 20. Gal (2016) 20 Bayesian Neural Network p(y⇤ |x⇤ , X, Y) = Z p(y⇤ |x⇤ , w)p(w|X, Y)dw Posterior p(w|X, Y) Inference Variational Inference KL (q✓(w)||p(w|X, Y)) = Z q✓(w) log q✓(w) p(w|X, Y) dw ELBO Z q✓(w) log p(Y|X, w)dw KL(q✓(w)||p(w)) ELBO (reparametrization) Z p(✏) log p(Y|X, w)d✏ KL(q✓(w)||p(w)) w = g(✓, ✏) Prior p(w)
- 21. Gal (2016) 21 Bayesian Neural Network ELBO Z q✓(w) log p(Y|X, w)dw KL(q✓(w)||p(w)) Re-parametrized ELBO Z p(✏) log p(Y|X, w)d✏ KL(q✓(w)||p(w)) w = g(✓, ✏) (Re-parametrization trick)
- 22. Gal (2016) 22 Gaussian process approximation MC approximation Apply to Gaussian processes GP Marginal likelihood
- 23. Gal (2016) 23 Bayesian Neural Network with dropout
- 24. Gal (2016) 24 Bayesian Neural Network with dropout p(y|fg(✓,ˆ✏) (x)) = N(y; ˆy✓(x), ⌧ 1 ID) Likelihood
- 25. Gal (2016) 25 Bayesian Neural Network with dropout Re-parametrized ELBO Z p(✏) log p(Y|X, w)d✏ KL(q✓(w)||p(w)) Re-parametrized likelihood Prior
- 26. Gal (2016) 26 Bayesian Neural Network with dropout
- 27. Gal (2016) 27 Predictive mean and uncertainties
- 28. 28 P. McClure, Representing Inferential Uncertainty in Deep Neural Networks Through Sampling, 2017
- 29. McClure & Kriegeskorte (2017) 29 Different variational distributions
- 30. McClure & Kriegeskorte (2017) 30 Results on MNIST Without noticeable performance degradation, the proposed methods are able to quantify the level of uncertainty.
- 31. 31 Anonymous, Bayesian Uncertainty Estimation for Batch Normalized Deep Networks, 2018
- 32. Anonymous (2018) 32 Monte Carlo Batch Normalization (MCBN)
- 33. Anonymous (2018) 33 Batch normalized deep nets as Bayesian modeling Learnable parameter Stochastic parameter
- 34. Anonymous (2018) 34 Batch normalized deep nets as Bayesian modeling
- 35. Anonymous (2018) 35 MCBN to Bayesian SegNet
- 36. 36 B. Lakshminarayanan et al., Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles, 2017
- 37. Lakshminarayanan et al. (2017) 37 Proper scoring rule “A scoring rule assigns a numerical score to a predictive distribution rewarding better calibrated predictions over worse. (…) It turns out many common neural network loss functions are proper scoring rules.”
- 38. Lakshminarayanan et al. (2017) 38 Density network x µ✓(x) ✓(x) L = 1 N NX i=1 log N(yi; µ✓(xi), 2 ✓(xi)) f✓(x)
- 39. Lakshminarayanan et al. (2017) 39 Density network L = 1 N NX i=1 log N(yi; µ✓(xi), 2 ✓(xi))
- 40. Lakshminarayanan et al. (2017) 40 Adversarial training with a fast gradient sign method “Adversarial training can be also be interpreted as a computationally efficient solution to smooth the predictive distributions by increasing the likelihood of the target around an neighborhood of the observed training examples.”
- 41. Lakshminarayanan et al. (2017) 41 Proposed method Train M different models
- 42. Lakshminarayanan et al. (2017) 42 Proposed method Empirical variance (5) Density network (1) Adversarial training Deep ensemble (5)
- 43. 43 A. Kendal and Y. Gal, What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?, 2017
- 44. Kendal & Gal (2017) 44 Aleatoric & epistemic uncertainties
- 45. Kendal & Gal (2017) 45 Aleatoric & epistemic uncertainties ˆW ⇠ q(W) x ˆy ˆW(x) ˆ2 ˆW (x) [ˆy, ˆ2 ] = f ˆW (x) L = 1 N NX i=1 log N(yi; ˆy ˆW(x), ˆ2 ˆW (x))
- 46. Kendal & Gal (2017) 46 Heteroscedastic uncertainty as loss attenuation ˆW ⇠ q(W) x ˆy ˆW(x) ˆ2 ˆW (x) [ˆy, ˆ2 ] = f ˆW (x)
- 47. Kendal & Gal (2017) 47 Aleatoric & epistemic uncertainties ˆW ⇠ q(W) x ˆy ˆW(x) ˆ2 ˆW (x) [ˆy, ˆ2 ] = f ˆW (x) Var(y) ⇡ 1 T TX t=1 ˆy2 t TX t=1 ˆyt !2 + 1 T TX t=1 ˆ2 t Epistemic unct. Aleatoric unct,
- 48. Kendal & Gal (2017) 48 Results
- 49. 49 G. Khan et al., Uncertainty-Aware Reinforcement Learning from Collision Avoidance, 2016
- 50. Khan et al. (2016) 50 Uncertainty-Aware Reinforcement Learning Uncertainty-aware collision prediction model
- 51. Khan et al. (2016) 51 Uncertainty-Aware Reinforcement Learning “Uncertainty is based on bootstrapped neural networks using dropout.” Bootstrapping? - Generate multiple datasets using sampling with replacement. - The intuition behind bootstrapping is that, by generating multiple populations and training one model per population, the models will agree in high-density areas (low uncertainty) and disagree in low-density areas (high uncertainty). Dropout? - “Dropout can be viewed as an economical approximation of an ensemble method (such as bootstrapping) in which each sampled dropout mask corresponds to a different model.”
- 52. Khan et al. (2016) 52 Uncertainty-Aware Reinforcement Learning Train B different models
- 53. 53
- 54. Richer & Roy (2017) 54 Introduction State-of-the-art deep learning methods are known to produce erratic or unsafe predictions when faced with novel inputs. Furthermore, recent ensemble, bootstrap and dropout methods for quantifying neural network uncertainty may not efficiently provide accurate uncertainty estimates when queried with inputs that are very different from their training data. We use a conventional feedforward neural network to predict collisions based on images observed by the robot, and we use an autoencoder to judge whether those images are similar enough to the training data for the resulting neural network predictions to be trusted.
- 55. Richer & Roy (2017) 55 Novelty detection Use the reconstruction error as a measure of novelty.
- 56. Richer & Roy (2017) 56 Novelty detection Use the reconstruction error as a measure of novelty.
- 57. Richer & Roy (2017) 57 Novelty detection
- 58. Richer & Roy (2017) 58 Novelty detection
- 59. Richer & Roy (2017) 59 Learning to predict collision c ˆmt it at fc(c|it, at) fp(c| ˆmt, at) fn(it) where : collision : estimated map : input image : action : neural net trained to predict collision : prior estimate of collision probability : novelty detection
- 60. Richer & Roy (2017) 60 Experiments “Using an autoencoder as a measure of uncertainty in our collision prediction network, we can transition intelligently between the high performance of the learned model and the safe, conservative performance of a simple prior, depending on whether the system has been trained on the relevant data.”
- 61. Richer & Roy (2017) 61 Experiments “In the hallway training environment, we achieved a mean speed of 3.26 m/s and a top speed over 5.03 m/s. This result significantly exceeds the maximum speeds achieved when driving in this environment under the prior estimate of collision probability before performing any learning.” “On the other hand, in the novel environment, for which our model was untrained, the novelty detector correctly identified every image as being unfamiliar. In the novel environment, we achieved a mean speed of 2.49 m/s and a maximum speed of 3.17 m/s.”
- 62. 62 S. Choi et al., Uncertainty-Aware Learning from Demonstration Using Mixture Density Networks with Sampling-Free Variance Modeling, 2017 All in this room!
- 63. Choi et al. (2017) 63 Mixture density networks x µ1(x) µ2(x) µ3(x)⇡1(x) ⇡2(x) ⇡3(x) 1(x) 2(x) 3(x) f ˆW(x) L = 1 N NX i=1 log KX j=1 ⇡j(xi)N(yi; µj(xi), 2 j (x))
- 64. 64 Choi et al. (2017) Mixture density networks
- 65. 65 Choi et al. (2017) Explained and unexplained variance “We propose a sampling-free variance modeling method using a mixture density network which can be decomposed into explained variance and unexplained variance.”
- 66. 66 Choi et al. (2017) Explained and unexplained variance “In particular, explained variance represents model uncertainty whereas unexplained variance indicates the uncertainty inherent in the process, e.g., measurement noise.”
- 67. 67 Choi et al. (2017) Analysis with Synthetic Examples The proposed uncertainty modeling method is analyzed in three different synthetic examples: 1) absence of data, 2) heavy noise, and 3) composition of functions.
- 68. 68 Choi et al. (2017) Analysis with Synthetic Examples
- 69. 69 Choi et al. (2017) Explained and unexplained variance We present uncertainty-aware learning from demonstration by using the explained variance as a switching criterion between trained policy and rule- based safe mode. Unexplained Variance Explained Variance
- 70. 70 Choi et al. (2017) Driving experiments
- 71. 71 ) ( . )