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EE-559 Deep learning 8. Under the hood Fran cois Fleuret https://fleuret.org/dlc/ [version of: June 5, 2018] COLE POLYTECHNIQUE FDRALE DE LAUSANNE Understanding a networks behavior Fran cois Fleuret EE-559 Deep

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EE-559 – Deep learning

  • 8. Under the hood

Fran¸ cois Fleuret https://fleuret.org/dlc/

[version of: June 5, 2018]

ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE

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Understanding a network’s behavior

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Understanding what is happening in a deep architectures after training is complex and the tools we have at our disposal are limited. In the case of convolutional feed-forward networks, we can look at

  • the network’s parameters, filters as images,
  • internal activations as images,
  • distributions of activations on a population of samples,
  • derivatives of the response(s) wrt the input,
  • maximum-response synthetic samples,
  • adversarial samples.

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Given a one-hidden layer fully connected network R2 → R2

nb_hidden = 20 model = nn. Sequential ( nn.Linear (2, nb_hidden), nn.ReLU (), nn.Linear(nb_hidden , 2) )

we can represent each of its internal units as a line corresponding to { x : w · x + b = 0 }.

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Given a one-hidden layer fully connected network R2 → R2

nb_hidden = 20 model = nn. Sequential ( nn.Linear (2, nb_hidden), nn.ReLU (), nn.Linear(nb_hidden , 2) )

we can represent each of its internal units as a line corresponding to { x : w · x + b = 0 }. During training, these separations get organized so that their combination delimitates properly the different value domains in the signal space.

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Iteration 1

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Iteration 4

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Iteration 7

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Iteration 10

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Iteration 16

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Iteration 34

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Iteration 77

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Iteration 100

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Iteration 703

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Iteration 1407

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Iteration 2789

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Iteration 4999

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Iteration 1

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Iteration 4

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Iteration 7

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Iteration 10

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Iteration 16

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Iteration 34

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Iteration 100

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Iteration 272

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Iteration 556

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Iteration 887

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Iteration 2222

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Iteration 4999

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Convnet filters

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A similar analysis is complicated to conduct with real-life networks given the high dimension of the signal. The simplest approach for convnets consists of looking at the filters as images. While it is quite reasonable in the first layer, since the filters are indeed consistent with the image input, it is far less so in the subsequent layers.

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LeNet’s first convolutional layer (1 → 32), all filters

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LeNet’s second convolutional layer (32 → 64), first 32 filters out of 64

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AlexNet’s first convolutional layer (3 → 64), first 20 filters out of 64

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AlexNet’s first convolutional layer (3 → 64), first 20 filters out of 64

  • r as RGB images

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AlexNet’s second convolutional layer (64 → 192). First 15 channels (out of 64)

  • f the first 20 filters (out of 192).

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Convnet internal layer activations

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An alternative approach is to look at the activations themselves. Since the convolutional layers maintain the 2d structure of the signal, the activations can be visualized as images, where the local coding at any location

  • f an activation map is associated to the original content at that same location.

Given the large number of channels, we have to pick a few at random.

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An alternative approach is to look at the activations themselves. Since the convolutional layers maintain the 2d structure of the signal, the activations can be visualized as images, where the local coding at any location

  • f an activation map is associated to the original content at that same location.

Given the large number of channels, we have to pick a few at random. Since the representation is distributed across multiple channels, individual channel have usually no clear semantic.

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A MNIST character with LeNet (leCun et al., 1998).

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An RGB image with AlexNet (Krizhevsky et al., 2012).

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An RGB image with AlexNet (Krizhevsky et al., 2012).

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An RGB image with AlexNet (Krizhevsky et al., 2012).

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An RGB image with AlexNet (Krizhevsky et al., 2012).

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An RGB image with AlexNet (Krizhevsky et al., 2012).

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ILSVRC12 with ResNet152 (He et al., 2015).

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ILSVRC12 with ResNet152 (He et al., 2015).

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ILSVRC12 with ResNet152 (He et al., 2015).

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Yosinski et al. (2015) developed analysis tools to visit a network and look at the internal activations for a given input signal. This allowed them in particular to find units with a clear semantic in an AlexNet-like network trained on ImageNet.

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Figure 2. A view of the 13×13 activations of the 151st channel on the conv5 layer of a deep neural network trained on ImageNet, a dataset that does not contain a face class, but does contain many images with faces. The channel responds to human and animal faces and is robust to changes in scale, pose, lighting, and context, which can be discerned by a user by actively changing the scene in front of a webcam or by loading static images (e.g. of the lions) and seeing the corresponding response of the unit. Photo of lions via Flickr user arnolouise, licensed under CC BY-NC-SA 2.0.

(Yosinski et al., 2015)

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Prediction of 2d dynamics with a 18 layer residual network. Gn Sn Rn (Fleuret, 2016)

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Sn Gn Rn Ψ(Sn, Gn) (Fleuret, 2016)

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1/1024 2/1024 3/1024 511/1024 512/1024 513/1024 514/1024

. . . (Fleuret, 2016)

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(Fleuret, 2016)

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(Fleuret, 2016)

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Layers as embeddings

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In the classification case, the network can be seen as a series of processings aiming as disentangling classes to make them easily separable for the final decision. In this perspective, it makes sense to look at how the samples are distributed spatially after each layer.

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The main issue to do so is the dimensionality of the signal. If we look at the total number of dimensions in each layer:

  • A MNIST sample in a LeNet goes from 784 to up to 18k dimensions,
  • A ILSVRC12 sample in Resnet152 goes from 150k to up to 800k

dimensions. This require a mean to project a [very] high dimension point cloud into a 2d or 3d “human-brain accessible” representation

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We have already seen PCA and k-means as two standard methods for dimension reduction, but they poorly convey the structure of a smooth low-dimension and non-flat manifold.

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We have already seen PCA and k-means as two standard methods for dimension reduction, but they poorly convey the structure of a smooth low-dimension and non-flat manifold. It exists a plethora of methods that aim at reflecting in low-dimension the structure of data points in high dimension. A popular one is t-SNE developed by van der Maaten and Hinton (2008).

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Given data-points in high dimension D =

  • xn ∈ RD, n = 1, . . . , N
  • the objective of data-visualization is to find a set of corresponding

low-dimension points E =

  • yn ∈ RC , n = 1, . . . , N
  • such that the positions of the ys “reflect” that of the xs.

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The t-Distributed Stochastic Neighbor Embedding (t-SNE) proposed by van der Maaten and Hinton (2008) optimizes with SGD the yis so that the distances to close neighbors of each point are preserved.

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The t-Distributed Stochastic Neighbor Embedding (t-SNE) proposed by van der Maaten and Hinton (2008) optimizes with SGD the yis so that the distances to close neighbors of each point are preserved. It actually matches for DKL two distance-dependent distributions: Gaussian in the original space, and Student t-distribution in the low-dimension one.

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The scikit-learn toolbox http://scikit-learn.org/ is built around SciPy, and provides many machine learning algorithms, in particular embeddings, among which an implementation of t-SNE. The only catch to use it in PyTorch is the conversions to and from numpy arrays.

from sklearn.manifold import TSNE # x is the array of the

  • riginal

high -dimension points x_np = x.numpy () y_np = TSNE( n_components = 2, perplexity = 50). fit_transform (x_np) y = torch.from_numpy (y_np)

n components specifies the embedding dimension and perplexity states [crudely] how many points are considered neighbors of each point.

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t-SNE unrolling of the swiss roll (with one noise dimension)

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t-SNE unrolling of the swiss roll (with one noise dimension)

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Input

t-SNE for LeNet on MNIST

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Layer #1

t-SNE for LeNet on MNIST

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Layer #4

t-SNE for LeNet on MNIST

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Layer #7

t-SNE for LeNet on MNIST

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Layer #9

t-SNE for LeNet on MNIST

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Input

t-SNE for an home-baked resnet (no pooling, 66 layers) CIFAR10

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Layer #5

t-SNE for an home-baked resnet (no pooling, 66 layers) CIFAR10

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Layer #10

t-SNE for an home-baked resnet (no pooling, 66 layers) CIFAR10

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Layer #15

t-SNE for an home-baked resnet (no pooling, 66 layers) CIFAR10

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Layer #20

t-SNE for an home-baked resnet (no pooling, 66 layers) CIFAR10

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Layer #25

t-SNE for an home-baked resnet (no pooling, 66 layers) CIFAR10

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Layer #30

t-SNE for an home-baked resnet (no pooling, 66 layers) CIFAR10

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Layer #31

t-SNE for an home-baked resnet (no pooling, 66 layers) CIFAR10

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Layer #32

t-SNE for an home-baked resnet (no pooling, 66 layers) CIFAR10

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Layer #33

t-SNE for an home-baked resnet (no pooling, 66 layers) CIFAR10

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Layer #34

t-SNE for an home-baked resnet (no pooling, 66 layers) CIFAR10

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Layer #35

t-SNE for an home-baked resnet (no pooling, 66 layers) CIFAR10

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Layer #36

t-SNE for an home-baked resnet (no pooling, 66 layers) CIFAR10

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Layer #37

t-SNE for an home-baked resnet (no pooling, 66 layers) CIFAR10

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Occlusion sensitivity

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Another approach to understanding the functioning of a network is to look at the behavior of the network “around” an image. For instance, we can get a simple estimate of the importance of a part of the input image by computing the difference between:

  • 1. the value of the maximally responding output unit on the image, and
  • 2. the value of the same unit with that part occluded.

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Another approach to understanding the functioning of a network is to look at the behavior of the network “around” an image. For instance, we can get a simple estimate of the importance of a part of the input image by computing the difference between:

  • 1. the value of the maximally responding output unit on the image, and
  • 2. the value of the same unit with that part occluded.

This is computationally intensive since it requires as many forward passes as there are locations of the occlusion mask, ideally the number of pixels.

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Original images Occlusion mask 32 × 32

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Original images Occlusion sensitivity, mask 32 × 32, stride of 2, AlexNet

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Original images Occlusion sensitivity, mask 32 × 32, stride of 2, VGG16

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Original images Occlusion sensitivity, mask 32 × 32, stride of 2, VGG19

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Saliency maps

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An alternative is to compute the gradient of the maximally responding output unit with respect to the input (Erhan et al., 2009; Simonyan et al., 2013), e.g. ∇|xf (x; w) where f is the activation of the output unit with maximum response, and |x stresses that the gradient is computed with respect to the input x and not as usual with respect to the parameters w.

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This can be implemented by specifying that we need the gradient with respect to the input. We use here the correct unit, not the maximum response one. Using torch.autograd.grad to compute the gradient wrt the input image instead of torch.autograd.backward has the advantage of not changing the model’s parameter gradients.

input = Variable(img , requires_grad = True)

  • utput = model(input)

loss = nllloss(output , target) grad_input , = torch.autograd.grad(loss , input)

Note that since torch.autograd.grad computes the gradient of a function with possibly multiple inputs, the returned result is a tuple.

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The resulting maps are quite noisy. For instance with AlexNet:

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This is due to the local irregularity of the network’s response as a function of the input.

Figure 2. The partial derivative of Sc with respect to the RGB val- ues of a single pixel as a fraction of the maximum entry in the gradient vector, maxi

∂Sc ∂xi (t), (middle plot) as one slowly moves

away from a baseline image x (left plot) to a fixed location x + ǫ (right plot). ǫ is one random sample from N(0, 0.012). The fi- nal image (x + ǫ) is indistinguishable to a human from the origin image x.

(Smilkov et al., 2017)

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Smilkov et al. (2017) proposed to smooth the gradient with respect to the input image by averaging over slightly perturbed versions of the latter. ˜ ∇|xfy(x; w) = 1 N

N

  • n=1

∇|xfy(x + ǫn; w) where ǫ1, . . . , ǫN are i.i.d of distribution N(0, σ2I), and σ is a fraction of the gap ∆ between the maximum and the minimum of the pixel values.

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A simple version of this “SmoothGrad” approach can be implemented as follows

nb_smooth = 100 std = smooth_std * (img.max () - img.min ()) acc_grad = img.new(img.size ()).zero_ () for q in range(nb_smooth): # This should be done with mini -batches ... noisy_input = img + img.new(img.size ()).normal_ (0, std) noisy_input = Variable(noisy_input , requires_grad = True)

  • utput = model( noisy_input )

loss = nllloss(output , target) grad_input , = torch.autograd.grad(loss , noisy_input ) acc_grad += grad_input .data acc_grad = acc_grad.abs ().sum (1) # sum across channels

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Original images Gradient, AlexNet SmoothGrad, AlexNet, σ = ∆

4

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Original images Gradient, VGG16 SmoothGrad, VGG16, σ = ∆

4

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Original images Gradient, VGG19 SmoothGrad, VGG19, σ = ∆

4

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Deconvolution and guided back-propagation

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Zeiler and Fergus (2014) proposed to invert the processing flow of a convolutional network by constructing a corresponding deconvolutional network to compute the “activating pattern” of a sample. As they point out, the resulting processing is identical to a standard backward pass, except when going through the ReLU layers.

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Remember that if s is one of the input to a ReLU layer, and x the corresponding output, we have for the forward pass x = max(0, s), and for the backward ∂l ∂s = 1{s>0} ∂l ∂x .

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Zeiler and Fergus’s deconvolution can be seen as a backward pass where we propagate back through ReLU layers the quantity max

  • 0, ∂l

∂x

  • = 1{ ∂l

∂x >0}

∂l ∂x , instead of the usual ∂l ∂s = 1{s>0} ∂l ∂x .

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Zeiler and Fergus’s deconvolution can be seen as a backward pass where we propagate back through ReLU layers the quantity max

  • 0, ∂l

∂x

  • = 1{ ∂l

∂x >0}

∂l ∂x , instead of the usual ∂l ∂s = 1{s>0} ∂l ∂x . This quantity is positive for units whose output has a positive contribution to the response, kills the others, and is not modulated by the pre-layer activation s.

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Springenberg et al. (2014) improved upon the deconvolution with the guided back-propagation, which aims at the best of both worlds: Discarding structures which would not contribute positively to the final response, and discarding structures which are not already present. It back-propagates through the ReLU layers the quantity 1{s>0}1{ ∂l

∂x >0}

∂l ∂x which keeps only units which have a positive contribution and activation.

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So these three visualization methods differ only in the quantities propagated through ReLU layers during the back-pass:

  • back-propagation (Erhan et al., 2009; Simonyan et al., 2013):

1{s>0} ∂l ∂x ,

  • deconvolution (Zeiler and Fergus, 2014):

1{ ∂l

∂x >0}

∂l ∂x ,

  • guided back-propagation (Springenberg et al., 2014):

1{s>0}1{ ∂l

∂x >0}

∂l ∂x .

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These procedures can be implemented simply in PyTorch by changing the nn.ReLU ’s backward pass. The class nn.Module provides methods to register “hook” functions that are called during the forward or the backward pass, and can implement a different computation for the latter.

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SLIDE 111

For instance

>>> x = Variable(Tensor ([ 1.23 ,

  • 4.56 ]))

>>> m = nn.ReLU () >>> m(x) Variable containing: 1.2300 0.0000 [torch. FloatTensor

  • f size 2]

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For instance

>>> x = Variable(Tensor ([ 1.23 ,

  • 4.56 ]))

>>> m = nn.ReLU () >>> m(x) Variable containing: 1.2300 0.0000 [torch. FloatTensor

  • f size 2]

>>> def my_hook(module , input , output): ... print(str(m) + ’ got ’ + str(input [0]. size ())) ... >>> handle = m. register_forward_hook (my_hook) >>> m(x) ReLU () got torch.Size ([2]) Variable containing: 1.2300 0.0000 [torch. FloatTensor

  • f size 2]

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For instance

>>> x = Variable(Tensor ([ 1.23 ,

  • 4.56 ]))

>>> m = nn.ReLU () >>> m(x) Variable containing: 1.2300 0.0000 [torch. FloatTensor

  • f size 2]

>>> def my_hook(module , input , output): ... print(str(m) + ’ got ’ + str(input [0]. size ())) ... >>> handle = m. register_forward_hook (my_hook) >>> m(x) ReLU () got torch.Size ([2]) Variable containing: 1.2300 0.0000 [torch. FloatTensor

  • f size 2]

>>> handle.remove () >>> m(x) Variable containing: 1.2300 0.0000 [torch. FloatTensor

  • f size 2]

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Using hooks, we can implement the deconvolution as follows:

def relu_backward_deconv_hook (module , grad_input , grad_output ): return F.relu( grad_output [0]) , def equip_model_deconv (model): for m in model.modules (): if isinstance(m, nn.ReLU):

  • m. register_backward_hook ( relu_backward_deconv_hook )

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SLIDE 115

Using hooks, we can implement the deconvolution as follows:

def relu_backward_deconv_hook (module , grad_input , grad_output ): return F.relu( grad_output [0]) , def equip_model_deconv (model): for m in model.modules (): if isinstance(m, nn.ReLU):

  • m. register_backward_hook ( relu_backward_deconv_hook )

def grad_view(model , image_name ): to_tensor = transforms .ToTensor () img = to_tensor(PIL.Image.open(image_name )) img = 0.5 + 0.5 * (img - img.mean ()) / img.std () if torch.cuda. is_available (): img = img.cuda () input = Variable(img.view(1, img.size (0) , img.size (1) , img.size (2)), \ requires_grad = True)

  • utput = model(input)

result , = torch.autograd.grad(output.max (), input) result = result.data / result.data.max () + 0.5 return result

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SLIDE 116

Using hooks, we can implement the deconvolution as follows:

def relu_backward_deconv_hook (module , grad_input , grad_output ): return F.relu( grad_output [0]) , def equip_model_deconv (model): for m in model.modules (): if isinstance(m, nn.ReLU):

  • m. register_backward_hook ( relu_backward_deconv_hook )

def grad_view(model , image_name ): to_tensor = transforms .ToTensor () img = to_tensor(PIL.Image.open(image_name )) img = 0.5 + 0.5 * (img - img.mean ()) / img.std () if torch.cuda. is_available (): img = img.cuda () input = Variable(img.view(1, img.size (0) , img.size (1) , img.size (2)), \ requires_grad = True)

  • utput = model(input)

result , = torch.autograd.grad(output.max (), input) result = result.data / result.data.max () + 0.5 return result model = models.vgg16(pretrained = True) model.eval () model = model.features equip_model_deconv (model) result = grad_view(model , ’blacklab.jpg ’) utils.save_image (result , ’blacklab -vgg16 -deconv.png ’)

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SLIDE 117

The code is the same for the guided back-propagation, except the hooks themselves:

def relu_forward_gbackprop_hook (module , input , output):

  • module. input_kept = input [0]

def relu_backward_gbackprop_hook (module , grad_input , grad_output ): return F.relu( grad_output [0]) * F.relu(module. input_kept).sign (), def equip_model_gbackprop (model): for m in model.modules (): if isinstance(m, nn.ReLU):

  • m. register_forward_hook ( relu_forward_gbackprop_hook )
  • m. register_backward_hook ( relu_backward_gbackprop_hook )

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SLIDE 118

Original images AlexNet, max feature response, gradient

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SLIDE 119

Original images AlexNet, max feature response, deconvolution

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SLIDE 120

Original images AlexNet, max feature response, guided back-propagation

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SLIDE 121

Original images VGG16, max feature response, gradient

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SLIDE 122

Original images VGG16, max feature response, deconvolution

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SLIDE 123

Original images VGG16, max feature response, guided back-propagation

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SLIDE 124

Original images VGG19, max feature response, gradient

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SLIDE 125

Original images VGG19, max feature response, deconvolution

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SLIDE 126

Original images VGG19, max feature response, guided back-propagation

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SLIDE 127

Experiments with an AlexNet-like network. Original images + deconvolution (or filters) for the top-9 activations for channels picked randomly. (Zeiler and Fergus, 2014)

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SLIDE 128

(Zeiler and Fergus, 2014)

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SLIDE 129

Maximum response samples

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SLIDE 130

Another approach to get an intuition of the information actually encoded in the weights of a convnet consists of optimizing from scratch a sample to maximize the activation f of a chosen unit, or the sum over an activation map.

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SLIDE 131

Doing so generates images with high frequencies, which tend to activate units a

  • lot. For instance these images maximize the responses of the units “bathtub”

and “lipstick” respectively (yes, this is strange, we will come back to it).

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SLIDE 132

Since f is trained in a discriminative manner, there is no reason that a sample maximizing its response would be “realistic”.

Class 0 Class 1

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SLIDE 133

Since f is trained in a discriminative manner, there is no reason that a sample maximizing its response would be “realistic”.

Class 0 Class 1 f

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SLIDE 134

Since f is trained in a discriminative manner, there is no reason that a sample maximizing its response would be “realistic”.

Class 0 Class 1 f

ˆ x

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SLIDE 135

Since f is trained in a discriminative manner, there is no reason that a sample maximizing its response would be “realistic”.

Class 0 Class 1 p −h

We can mitigate this by adding a penalty h corresponding to a “realistic” prior and compute in the end argmax

x

f (x; w) − h(x)

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SLIDE 136

Since f is trained in a discriminative manner, there is no reason that a sample maximizing its response would be “realistic”.

Class 0 Class 1 f − h

We can mitigate this by adding a penalty h corresponding to a “realistic” prior and compute in the end argmax

x

f (x; w) − h(x)

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SLIDE 137

Since f is trained in a discriminative manner, there is no reason that a sample maximizing its response would be “realistic”.

Class 0 Class 1 f − h

ˆ x We can mitigate this by adding a penalty h corresponding to a “realistic” prior and compute in the end argmax

x

f (x; w) − h(x)

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SLIDE 138

Since f is trained in a discriminative manner, there is no reason that a sample maximizing its response would be “realistic”.

Class 0 Class 1 f − h

ˆ x We can mitigate this by adding a penalty h corresponding to a “realistic” prior and compute in the end argmax

x

f (x; w) − h(x) by iterating a standard gradient update: xk+1 = xk − η∇|x(h(xk) − f (xk; w)).

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SLIDE 139

A reasonable h penalizes too much energy in the high frequencies by integrating edge amplitude at multiple scales.

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SLIDE 140

This can be formalized as a penalty function h of the form h(x) =

  • s≥0

δs(x) − g ⊛ δs(x)2 where g is a Gaussian kernel, and δ is a factor-2 downscale operator.

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SLIDE 141

We first implement h(x) =

  • s≥0

δs(x) − g ⊛ δs(x)2 as a module:

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SLIDE 142

We first implement h(x) =

  • s≥0

δs(x) − g ⊛ δs(x)2 as a module:

class MultiScaleEdgeEnergy (nn.Module): def __init__(self): super(MultiScaleEdgeEnergy , self).__init__ () k = Tensor ([[1 , 4, 6, 4, 1]]) k_5x5_pseudo_gaussian = k.t().mm(k).view(1, 1, 5, 5) k_5x5_pseudo_gaussian /= k_5x5_pseudo_gaussian .sum ()

  • self. k_5x5_pseudo_gaussian = Parameter( k_5x5_pseudo_gaussian )

k_2x2_uniform = Tensor ([[0.25 , 0.25] , [0.25 , 0.25]]).view(1, 1, 2, 2)

  • self. k_2x2_uniform = Parameter( k_2x2_uniform )

def forward(self , x): if x.size (1) > 1: # dealing with multiple channels by unfolding them as as # many

  • ne

channel images result = self(x.view(x.size (0) * x.size (1) , 1, x.size (2) , x.size (3))) result = result.view(x.size (0) ,

  • 1).sum (1)

else: result = 0.0 while x.size (2) > 5 and x.size (3) > 5: blurry = F.conv2d(x, self. k_5x5_pseudo_gaussian , padding = 2) result += (x - blurry).view(x.size (0) ,

  • 1).pow (2).sum (1)

x = F.conv2d(x, self.k_2x2_uniform , stride = 2, padding = 1) return result

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SLIDE 143

Then, the optimization of the image per se is straightforward:

model = models.vgg16(pretrained = True) model.eval () edge_energy = MultiScaleEdgeEnergy () input = Tensor (1, 3, 224, 224).normal_ (0, 0.01) if torch.cuda. is_available (): model.cuda () edge_energy .cuda () input = input.cuda () input = Variable(input , requires_grad = True)

  • ptimizer = optim.Adam ([ input], lr = 1e -1)

for k in range (250):

  • utput = model(input)

loss = edge_energy (input) - output [0, 700] # paper towel

  • ptimizer.zero_grad ()

loss.backward ()

  • ptimizer.step ()

result = input.data result = 0.5 + 0.1 * (result - result.mean ()) / result.std () torchvision .utils. save_image (result , ’result.png ’)

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SLIDE 144

Then, the optimization of the image per se is straightforward:

model = models.vgg16(pretrained = True) model.eval () edge_energy = MultiScaleEdgeEnergy () input = Tensor (1, 3, 224, 224).normal_ (0, 0.01) if torch.cuda. is_available (): model.cuda () edge_energy .cuda () input = input.cuda () input = Variable(input , requires_grad = True)

  • ptimizer = optim.Adam ([ input], lr = 1e -1)

for k in range (250):

  • utput = model(input)

loss = edge_energy (input) - output [0, 700] # paper towel

  • ptimizer.zero_grad ()

loss.backward ()

  • ptimizer.step ()

result = input.data result = 0.5 + 0.1 * (result - result.mean ()) / result.std () torchvision .utils. save_image (result , ’result.png ’)

(take a second to think about the beauty of autograd)

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SLIDE 145

VGG16, maximizing a channel of the 4th convolution layer

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SLIDE 146

VGG16, maximizing a channel of the 7th convolution layer

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SLIDE 147

VGG16, maximizing a unit of the 10th convolution layer

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SLIDE 148

VGG16, maximizing a unit of the 13th (and last) convolution layer

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SLIDE 149

VGG16, maximizing a unit of the output layer

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SLIDE 150

VGG16, maximizing a unit of the output layer “King crab”

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SLIDE 151

VGG16, maximizing a unit of the output layer “King crab” “Samoyed” (that’s a fluffy dog)

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SLIDE 152

VGG16, maximizing a unit of the output layer

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SLIDE 153

VGG16, maximizing a unit of the output layer “Hourglass”

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SLIDE 154

VGG16, maximizing a unit of the output layer “Hourglass” “Paper towel”

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SLIDE 155

VGG16, maximizing a unit of the output layer

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SLIDE 156

VGG16, maximizing a unit of the output layer “Ping-pong ball”

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SLIDE 157

VGG16, maximizing a unit of the output layer “Ping-pong ball” “Steel arch bridge”

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SLIDE 158

VGG16, maximizing a unit of the output layer

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SLIDE 159

VGG16, maximizing a unit of the output layer “Sunglass”

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SLIDE 160

VGG16, maximizing a unit of the output layer “Sunglass” “Geyser”

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SLIDE 161

These results show that the parameters of a network trained for classification carry enough information to generate identifiable large-scale structures. Although the training is discriminative, the resulting model has strong generative capabilities. It also gives an intuition of the accuracy and shortcomings of the resulting global compositional model.

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SLIDE 162

Adversarial examples

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SLIDE 163

In spite of their good predictive capabilities, deep neural networks are quite sensitive to adversarial inputs, that is to inputs crafted to make them behave incorrectly (Szegedy et al., 2014).

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SLIDE 164

In spite of their good predictive capabilities, deep neural networks are quite sensitive to adversarial inputs, that is to inputs crafted to make them behave incorrectly (Szegedy et al., 2014). The simplest strategy to exhibit such behavior is to optimize the input to maximize the loss.

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SLIDE 165

Let x be an image, y its proper label, f (x; w) the network’s prediction, and L the cross-entropy loss. We can construct an adversarial example by maximizing the loss. To do so, we iterate a “gradient ascent” step: xk+1 = xk + η∇|xL(f (xk; w), y). After a few iterations, this procedure will reach a sample ˇ x whose class is not y.

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SLIDE 166

Let x be an image, y its proper label, f (x; w) the network’s prediction, and L the cross-entropy loss. We can construct an adversarial example by maximizing the loss. To do so, we iterate a “gradient ascent” step: xk+1 = xk + η∇|xL(f (xk; w), y). After a few iterations, this procedure will reach a sample ˇ x whose class is not y. The counter-intuitive result is that the resulting miss-classified images are indistinguishable from the original ones to a human eye.

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SLIDE 167

input = Variable(input , requires_grad = True) model = torchvision .models.alexnet(pretrained = True) cross_entropy = nn. CrossEntropyLoss ()

  • ptimizer = optim.SGD ([ input], lr = 1e -1)

if torch.cuda. is_available (): model.cuda () cross_entropy .cuda () target = model(input).data.max (1) [1]. view (-1) if torch.cuda. is_available (): target = target.cuda () target = Variable(target) for k in range (15):

  • utput = model(input)

loss = - cross_entropy (output , target)

  • ptimizer.zero_grad ()

loss.backward ()

  • ptimizer.step ()

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SLIDE 168

Original Adversarial Differences (magnified)

x−ˇ x x

1.02% 0.27%

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SLIDE 169

Predicted classes

  • Nb. iterations

Image #1 Image #2 Weimaraner desktop computer 1 Weimaraner desktop computer 2 Labrador retriever desktop computer 3 Labrador retriever desktop computer 4 Labrador retriever desktop computer 5 brush kangaroo desktop computer 6 brush kangaroo desktop computer 7 sundial desktop computer 8 sundial desktop computer 9 sundial desktop computer 10 sundial desktop computer 11 sundial desktop computer 12 sundial desktop computer 13 sundial desktop computer 14 sundial desk

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SLIDE 170

Another counter-intuitive result is that if we sample 1, 000 images on the sphere centered on x of radius 2x − ˇ x, we do not observe any change of label. x ˇ x

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SLIDE 171

Adversarial images can be pushed one step further by optimizing images from scratch with genetic optimization to maximize the network’s response (Nguyen et al., 2015)

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SLIDE 172

Dilated convolution

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SLIDE 173

Convolution operations admit one more standard parameter that we have not discussed yet: The dilation, which modulates the expansion of the filter support (Yu and Koltun, 2015). It is 1 for standard convolutions, but can be greater, in which case the resulting

  • peration can be envisioned as a convolution with a regularly sparsified filter.

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SLIDE 174

Convolution operations admit one more standard parameter that we have not discussed yet: The dilation, which modulates the expansion of the filter support (Yu and Koltun, 2015). It is 1 for standard convolutions, but can be greater, in which case the resulting

  • peration can be envisioned as a convolution with a regularly sparsified filter.

This notion comes from signal processing, where it is referred to as algorithme ` a trous, hence the term sometime used of “convolution ` a trous”.

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SLIDE 175

Input Output

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SLIDE 176

Input Output

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SLIDE 177

Input Output

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SLIDE 178

Input Output

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SLIDE 179

Input Output

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SLIDE 180

Input Output

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SLIDE 181

Input Output

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SLIDE 182

Input Output

Dilation = 1

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SLIDE 183

Input Output

Dilation = 2

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SLIDE 184

Input Output

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SLIDE 185

Input Output

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Input Output

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Input Output

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Input Output

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SLIDE 189

Input Output

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SLIDE 190

Input Output

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SLIDE 191

Input Output

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SLIDE 192

A convolution with a 1d kernel of size k and dilation d can be interpreted as a convolution with a filter of size 1 + (k − 1)d with only k non-zero coefficients. For with k = 3 and d = 4, the difference between the input map size and the

  • utput map size is 1 + (3 − 1)4 − 1 = 8.

>>> from torch import nn , Tensor >>> from torch.autograd import Variable >>> x = Variable(Tensor (1, 1, 20, 30).normal_ ()) >>> l = nn.Conv2d (1, 1, kernel_size = 3, dilation = 4) >>> l(x).size () torch.Size ([1, 1, 12, 22])

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SLIDE 193

Having a dilation greater than one increases the units’ receptive field size without increasing the number of parameters. Convolutions with stride or dilation strictly greater than one reduce the activation map size, for instance to make a final classification decision, without employing pooling operators.

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SLIDE 194

Having a dilation greater than one increases the units’ receptive field size without increasing the number of parameters. Convolutions with stride or dilation strictly greater than one reduce the activation map size, for instance to make a final classification decision, without employing pooling operators. Such networks have the advantage of simplicity:

  • non-linear operations are only in the activation function,
  • joint operations (combining multiple activations to produce one) are only

in the convolutional layers.

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SLIDE 195

The end

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SLIDE 196

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