Marc LELARGE and Andrei BURSUC | Deep Learning Do It Yourself | 15.2 Uncertainty estimation - MCDropout

15.2 Towards deep learning for the real world

Andrei Bursuc

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Towards deep learning for the real world

i.e., beyond cats and dogs

Andrei Bursuc

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Motivation

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Deep Learning is great:

conceptually simple and modular
scales well with data
awesome software tools
huge community and interest
potentially real world impact

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Deep Learning is great:

conceptually simple and modular
scales well with data
awesome software tools
huge community and interest
potentially real world impact

... but has several problems

uninterpretable black-boxes
needs a lot of data
mostly empirical
what does a model not know?
can be fooled easily

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The world is a complex environment

Covering this diversity with (sufficient) data and labels is highly challenging

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Dealing with uncertainty

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Why should I care about uncertainty?

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Motivation

In May 2016, there was the first fatality from an assisted driving system, caused by the perception system confusing the white side of a trailer for bright sky.

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Motivation

An image classification system erroneously identifies two African Americans as gorillas, raising concerns of racial discrimination.

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What do we mean by uncertainty?

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Marc LELARGE and Andrei BURSUC | Deep Learning Do It Yourself | 15.2 Uncertainty estimation - MCDropout

What do we mean by uncertainty?

Return a distribution over predictions instead of a single prediction:

classification: output a label and its confidence
regression: output a mean and a variance

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Good uncertainty estimates tell us when we can trust the predictions of our model.

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Marc LELARGE and Andrei BURSUC | Deep Learning Do It Yourself | 15.2 Uncertainty estimation - MCDropout

What do we mean by Out-of-Distribution Robustness?

I.I.D $\text{ }p_{\text{test}}(\mathbf{x}, y) = p_{\text{train}}(\mathbf{x}, y)$
(I.I.D. = Indepedent and Identically Distributed)

O.O.D $\text{ }p_{\text{test}}(\mathbf{x}, y) =\not\ p_{\text{train}}(\mathbf{x}, y)$

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What do we mean by Out-of-Distribution Robustness?

I.I.D $\text{ }p_{\text{test}}(\mathbf{x}, y) = p_{\text{train}}(\mathbf{x}, y)$
(I.I.D. = Indepedent and Identically Distributed)

O.O.D $\text{ }p_{\text{test}}(\mathbf{x}, y) =\not\ p_{\text{train}}(\mathbf{x}, y)$

Examples of dataset shift:

covariate shift: distribution of features $p(\mathbf{x})$ changes and $p(y \vert \mathbf{x})$ , i.e., labels, is fixed
open-set recognition: new classes may appear at test time
label shift: distribution of labels $p(y)$ changes and $p(\mathbf{x} \vert y)$ labels, is fixed

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Marc LELARGE and Andrei BURSUC | Deep Learning Do It Yourself | 15.2 Uncertainty estimation - MCDropout

Varying corruption intensity for dataset shift

Samples from ImageNet-C

D. Hendrycks & T. Dietterich, Benchmarking Neural Network Robustness to Common Corruptions and Perturbations, ICLR 2019

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Varying corruption intensity for dataset shift

Corruption types for ImageNet-C

D. Hendrycks & T. Dietterich, Benchmarking Neural Network Robustness to Common Corruptions and Perturbations, ICLR 2019

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Neural nets do not generalize under covariate shift

Accuracy drops with increasing shift on ImageNet-C
Uncertainty quality degrades, making overconfident errors

Y. Ovadia et al., Can You Trust Your Model's Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift, NeurIPS 2019

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Neural nets assign high confidence predictions to OOD data

Example images where model assigns ${>}99.5\%$ confidence

A. Nguyen et al., Deep Neural Networks are Easily Fooled: High Confidence Predictions for Unrecognizable Images, CVPR 2015

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Neural nets assign high confidence predictions to OOD data

J.Z. Liu et al., Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness, arXiv 2020

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$\text{Calibration Error} = \vert \underbrace{\text{Confidence}}_{\text{predicted probability of correctness}} - \underbrace{\text{Accuracy}}_{\text{observed frequency of correctness}} \vert$

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Calibration

Calibration: of the times your model predicts something with $90\%$ confidence, is it right $90\%$ of the time?

Calibration of weather forecasts

Nate Silver, The singal and the noise

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Most neural networks output probability distributions, e.g., over object categories. Are these calibrated?

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Measuring calibration: Expected Calibration Error

$\text{ECE} = \sum_{b=1}^{b}\frac{n_b}{N}\vert \text{acc}(b) -\text{conf}(b) \vert$

Bin the probabilities into $B$ bins
Compute the within-bin accuracy and within-bin predicted confidence
Average the calibration error across bins (weighted by number of points in each bin)

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Calibration

Most neural networks output probability distributions, e.g., over object categories. Are these calibrated?

C. Guo et al., On Calibration of Modern Neural Networks, ICML 2017

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Why is this happening now?

The effect of network depth (far left), width (middle left), Batch Normalization (middle right), and weight decay (far right) on miscalibration, as measured by ECE (lower is better).

C. Guo et al., On Calibration of Modern Neural Networks, ICML 2017

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Why is this happening now?

The effect of network depth (far left), width (middle left), Batch Normalization (middle right), and weight decay (far right) on miscalibration, as measured by ECE (lower is better).

C. Guo et al., On Calibration of Modern Neural Networks, ICML 2017

We kind of got too good at training these beasts

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Applications

Autonomous vehicles: dataset shift: location, weather, time of day; use model uncertainty to decide when to trust model or hand-over to human

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Applications

Autonomous vehicles: dataset shift: location, weather, time of day; use model uncertainty to decide when to trust model or hand-over to human
Healthcare: model uncertainty for trusting the model or calling doctor; reject low-quality inputs

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Applications

Autonomous vehicles: dataset shift: location, weather, time of day; use model uncertainty to decide when to trust model or hand-over to human
Healthcare: model uncertainty for trusting the model or calling doctor; reject low-quality inputs
Chatbots: detect unknown sentences

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Marc LELARGE and Andrei BURSUC | Deep Learning Do It Yourself | 15.2 Uncertainty estimation - MCDropout

Applications

Autonomous vehicles: dataset shift: location, weather, time of day; use model uncertainty to decide when to trust model or hand-over to human
Healthcare: model uncertainty for trusting the model or calling doctor; reject low-quality inputs
Chatbots: detect unknown sentences
Active Learning: use model uncertainty to decide which training examples are worth labeling

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Marc LELARGE and Andrei BURSUC | Deep Learning Do It Yourself | 15.2 Uncertainty estimation - MCDropout

Applications

Autonomous vehicles: dataset shift: location, weather, time of day; use model uncertainty to decide when to trust model or hand-over to human
Healthcare: model uncertainty for trusting the model or calling doctor; reject low-quality inputs
Chatbots: detect unknown sentences
Active Learning: use model uncertainty to decide which training examples are worth labeling
Bayesian Optimization: optimize an expensive black-box function by finding which configurations to explore next

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Marc LELARGE and Andrei BURSUC | Deep Learning Do It Yourself | 15.2 Uncertainty estimation - MCDropout

Applications

Autonomous vehicles: dataset shift: location, weather, time of day; use model uncertainty to decide when to trust model or hand-over to human
Healthcare: model uncertainty for trusting the model or calling doctor; reject low-quality inputs
Chatbots: detect unknown sentences
Active Learning: use model uncertainty to decide which training examples are worth labeling
Bayesian Optimization: optimize an expensive black-box function by finding which configurations to explore next
Reinforcement Learning: use uncertainty for exploration vs. exploitation trade-off

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Sources of uncertainty

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There are two main types of uncertainties each with its own pecularities

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Case 1

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Case 1

Problems caused by sensor quality, natural randomness, that cannot be explained by our data.

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Case 1

Problems caused by sensor quality, natural randomness, that cannot be explained by our data.

Aleatoric / Data uncertainty

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Case 1

Problems caused by sensor quality, natural randomness, that cannot be explained by our data.

Aleatoric / Data uncertainty

aleator (lat.) = dice player

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Case 1

Problems caused by sensor quality, natural randomness, that cannot be explained by our data.

Aleatoric / Data uncertainty

aleator (lat.) = dice player
cannot be reduced, but can be learned
useful for:
- large data situation, where model uncertainty is low
- real-time processing, cheaper to compute than model uncertainty

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Case 1'

Similarly looking objects also fall into this category

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Similarly looking objects also fall into this category

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Aleatoric uncertainty

Distinct classes

Credit: A. Malinin

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Aleatoric uncertainty

Distinct classes

Credit: A. Malinin

Overlapping classes

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In urban scenes this type of uncertainty is frequently caused by similarly-looking classes:

pedestrian - cyclist - person on trottinette/scooter
road - sidewalk

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Aleatoric uncertainty

Credit: A. Malinin

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Aleatoric uncertainty

Credit: A. Malinin

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Aleatoric uncertainty

Low entropy

High entropy

Credit: A. Malinin

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In layman words data uncertainty is called the: known unknown

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Case 2

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Case 2

Lack of knowledge about the process that generated the data

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Case 2

Lack of knowledge about the process that generated the data

Epistemic/Knowledge uncertainty

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Case 2

Lack of knowledge about the process that generated the data

Epistemic/Knowledge uncertainty

episteme (gr.) = knowledge

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Case 2

Lack of knowledge about the process that generated the data

Epistemic/Knowledge uncertainty

episteme (gr.) = knowledge
disappears given enough data
useful for:
- detecting samples far from the training distribution
- small datasets with little annotated data

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Case 2

Epistemic error decreases when you gather more points:

Slide credit: Marcin Mozejko

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Image credit: Marcin Mozejko

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Case 2'

Let us consider a neural network model trained with several pictures of dog breeds.

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Case 2'

Let us consider a neural network model trained with several pictures of dog breeds.

We ask the model to decide on a dog breed using a photo of a cat.
What would you want the model to do?

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Case 2'

Let us consider a neural network model trained with several pictures of dog breeds.

Out-of-distribution uncertainty

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In layman words, knowledge uncertainty is called the: unknown unknown

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Epistemic uncertainty

Credit: A. Malinin

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Epistemic uncertainty

Unseen classes

Credit: A. Malinin

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Epistemic uncertainty

Unseen classes

Credit: A. Malinin

Unseen variations of seen classes

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"Our model exhibits in (d) increased aleatoric uncertainty on object boundaries and for objects far from the camera. Epistemic uncertainty accounts for our ignorance about which model generated our collected data. In (e) our model exhibits increased epistemic uncertainty for semantically and visually challenging pixels. The bottom row shows a failure case of the segmentation model when the model fails to segment the footpath due to increased epistemic uncertainty, but not aleatoric uncertainty."

A. Kendall and Y. Gal, What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?, NeurIPS 2017.

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Measuring the quality of the uncertainty can be challenging due to lack of ground truth, i.e., no “right answer” in some cases

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MC-Dropout

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Dropout

First "deep" regularization technique
Remove units at random during the forward pass on each sample
Put them all back during test

Dropout: A Simple Way to Prevent Neural Networks from Overfitting, Srivastava et al., JMLR 2014

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Dropout

Interpretation

Reduces the network dependency to individual neurons and distributes representation
More redundant representation of data

Ensemble interpretation

Equivalent to training a large ensemble of shared-parameters, binary-masked models
Each model is only trained on a single data point
A network with dropout can be interpreted as an ensemble of $2^N$ models with heavy weight sharing (Goodfellow et al., 2013)

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Dropout

>>> x = torch.full((3, 5), 1.0).requires_grad_() 
>>> x
tensor([[ 1., 1., 1., 1., 1.],
        [ 1., 1., 1., 1., 1.],
        [ 1., 1., 1., 1., 1.]])
>>> dropout = nn.Dropout(p = 0.75) 
>>> y = dropout(x)
>>> y
tensor([[ 0., 0., 4., 0., 4.],
        [ 0., 4., 4., 4., 0.],
        [ 0., 0., 4., 0., 0.]])
>>> l = y.norm(2, 1).sum()
>>> l.backward()
>>> x.grad
tensor([[ 0.0000, 0.0000, 2.8284, 0.0000, 2.8284]
        [ 0.0000, 2.3094, 2.3094, 2.3094, 0.0000]
        [ 0.0000, 0.0000, 4.0000, 0.0000, 0.0000]])

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Dropout

>>> x = torch.full((3, 5), 1.0).requires_grad_() 
>>> x
tensor([[ 1., 1., 1., 1., 1.],
        [ 1., 1., 1., 1., 1.],
        [ 1., 1., 1., 1., 1.]])
>>> dropout = nn.Dropout(p = 0.75) 
>>> y = dropout(x)
>>> y
tensor([[ 0., 0., 4., 0., 4.],
        [ 0., 4., 4., 4., 0.],
        [ 0., 0., 4., 0., 0.]])
>>> l = y.norm(2, 1).sum()
>>> l.backward()
>>> x.grad
tensor([[ 0.0000, 0.0000, 2.8284, 0.0000, 2.8284]
        [ 0.0000, 2.3094, 2.3094, 2.3094, 0.0000]
        [ 0.0000, 0.0000, 4.0000, 0.0000, 0.0000]])

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Dropout

For a given network

model = nn.Sequential(nn.Linear(10, 100), nn.ReLU(), 
                      nn.Linear(100, 50), nn.ReLU(),
                      nn.Linear(50, 2));

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Dropout

For a given network

model = nn.Sequential(nn.Linear(10, 100), nn.ReLU(), 
                      nn.Linear(100, 50), nn.ReLU(),
                      nn.Linear(50, 2));

we can simply add dropout layers

model = nn.Sequential(nn.Linear(10, 100), nn.ReLU(), 
                      nn.Dropout(),
                      nn.Linear(100, 50), nn.ReLU(), 
                      nn.Dropout(),
                      nn.Linear(50, 2));

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Dropout

A model using dropout has to be set in train or test mode

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Dropout

A model using dropout has to be set in train or test mode

The method nn.Module.train(mode) recursively sets the flag training to all sub-modules.

>>> dropout = nn.Dropout()
>>> model = nn.Sequential(nn.Linear(3, 10), dropout, nn.Linear(10, 3)) 
>>> dropout.training
True
>>> model.train(False)
Sequential (
(0): Linear (3 -> 10) (1): Dropout (p = 0.5) (2): Linear (10 -> 3)
)
>>> dropout.training 
False

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Dropout

A model using dropout has to be set in train or test mode

>>> dropout = nn.Dropout()
>>> model = nn.Sequential(nn.Linear(3, 10), dropout, nn.Linear(10, 3)) 
>>> x = torch.full((1, 3), 1.0) 
>>> model.train()
Sequential (
(0): Linear (3 -> 10) (1): Dropout (p = 0.5) (2): Linear (10 -> 3)
)
>>> model(x)
tensor([[ 0.5360, -0.5225, -0.5129]], grad_fn=<ThAddmmBackward>)
>>> model(x)
tensor([[ 0.6134, -0.6130, -0.5161]], grad_fn=<ThAddmmBackward>)

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Dropout

A model using dropout has to be set in train or test mode

>>> dropout = nn.Dropout()
>>> model = nn.Sequential(nn.Linear(3, 10), dropout, nn.Linear(10, 3)) 
>>> x = torch.full((1, 3), 1.0) 
>>> model.train()
Sequential (
(0): Linear (3 -> 10) (1): Dropout (p = 0.5) (2): Linear (10 -> 3)
)
>>> model(x)
tensor([[ 0.5360, -0.5225, -0.5129]], grad_fn=<ThAddmmBackward>)
>>> model(x)
tensor([[ 0.6134, -0.6130, -0.5161]], grad_fn=<ThAddmmBackward>)
>>>
>>> model.eval()
Sequential (
(0): Linear (3 -> 10) (1): Dropout (p = 0.5) (2): Linear (10 -> 3)
)
>>> model(x)
tensor([[ 0.5772, -0.0944, -0.1168]], grad_fn=<ThAddmmBackward>)
>>> model(x)
tensor([[ 0.5772, -0.0944, -0.1168]], grad_fn=<ThAddmmBackward>)

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How can we get uncertainties from standard networks?

Standard Neural Network

Bayesian Neural Network

Dropout as Bayesian approximation: representing model uncertainty in deep learning, Y. Gal, ICML 2016

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Standard Neural Network

Bayesian Neural Network

Image credit: Eric Ma

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From Bayesian Neural Networks to Dropout

Gal and Ghahramani build upon the ensembling view of Dropout and show that when training a network with dropout with a standard classification or regression objective, one is actually implicitly doing variational inference to match the posterior distribution of the weights.

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Uncertainty estimates from dropout

Proper epistemic uncertainty estimates at $\mathbf{x}$ can be obtained in a principled way using Monte-Carlo integration:

Draw $T$ sets of network parameters $\hat{\theta}_t$ from $q(\theta;\nu)$ .
Compute the predictions for the $T$ networks, $\{ f(\mathbf{x};\hat{\theta}_t) \}_{t=1}^T$ .
Approximate the predictive mean and variance as follows: $\begin{aligned} \mathbb{E}_{p(y|\mathbf{x},\mathbf{X},\mathbf{Y})}\left[y\right] &\approx \frac{1}{T} \sum_{t=1}^T f(\mathbf{x};\hat{\theta}_t) \\ \mathbb{V}_{p(y|\mathbf{x},\mathbf{X},\mathbf{Y})}\left[y\right] &\approx \sigma^2 + \frac{1}{T} \sum_{t=1}^T f(\mathbf{x};\hat{\theta}_t)^2 - \hat{\mathbb{E}}\left[y\right]^2 \end{aligned}$

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Yarin Gal's demo.

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Uncertainty estimates from dropout

class SimpleModel(nn.Module):
  def __init__(self, p, decay):
    super(SimpleModel, self).__init__()
    self.dropout_p = p
    self.decay = decay
    self.f = nn.Sequential(
      nn.Linear(1,20),
      nn.ReLU(),
      nn.Dropout(p=self.dropout_p),
      nn.Linear(20, 20),
      nn.ReLU(),
      nn.Dropout(p=self.dropout_p),
      nn.Linear(20,1)
    )
  def forward(self, X):
    return self.f(X)

def uncertainty_estimate(X, model, iters=200, l2=0.01):
  model.train()
  outputs = np.hstack([model(X[:, np.newaxis]).data.numpy() \
              for i in range(iters)])
  y_mean = outputs.mean(axis=1)
  y_variance = outputs.var(axis=1)
  tau = l2 * (1. - model.dropout_p) / (2. * N * model.decay)
  y_variance += (1. / tau)
  y_std = np.sqrt(y_variance)
  return y_mean, y_std

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Marc LELARGE and Andrei BURSUC | Deep Learning Do It Yourself | 15.2 Uncertainty estimation - MCDropout

Results

Y. Gal, Dropout as Bayesian approximation: representing model uncertainty in deep learning, ICML 2016

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Marc LELARGE and Andrei BURSUC | Deep Learning Do It Yourself | 15.2 Uncertainty estimation - MCDropout

Results

Y. Gal, Dropout as Bayesian approximation: representing model uncertainty in deep learning, ICML 2016

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Marc LELARGE and Andrei BURSUC | Deep Learning Do It Yourself | 15.2 Uncertainty estimation - MCDropout

Pixel-wise depth regression

A. Kendall and Y. Gal, What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?, NeurIPS 2017.

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Marc LELARGE and Andrei BURSUC | Deep Learning Do It Yourself | 15.2 Uncertainty estimation - MCDropout

Combinining heteroscedastic and epistemic uncertainty

Semantic Segmentation performance on CamVid

What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?, A. Kendall and Y. Gal, NeurIPS 2017

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Marc LELARGE and Andrei BURSUC | Deep Learning Do It Yourself | 15.2 Uncertainty estimation - MCDropout

Combinining heteroscedastic and epistemic uncertainty

Monocular Depth Regression Performance

What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?, A. Kendall and Y. Gal, NeurIPS 2017

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Marc LELARGE and Andrei BURSUC | Deep Learning Do It Yourself | 15.2 Uncertainty estimation - MCDropout

Comparing heteroscedastic and epistemic uncertainty

Aleatoric vs. Epistemic Uncertainty for Out of Dataset Examples

Aleatoric uncertainty remains constant while epistemic uncertainty increases for out of dataset examples!

What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?, A. Kendall and Y. Gal, NeurIPS 2017

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Marc LELARGE and Andrei BURSUC | Deep Learning Do It Yourself | 15.2 Uncertainty estimation - MCDropout

Applications

Multiple follow-up papers by Gal and friends:

Concrete Dropout : learn Dropout probability for each layer using Concrete/ Gumble-Softmax trick
Active Learning with MC Dropout: select samples using uncertainty
MC Dropout for RNNs: same dropout mask across time-steps
Data efficiency in RL
Stochasticity via BatchNorm perturbation

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