Visualizing and Understanding Stochastic Depth Networks

Transcription

1 Visualizing and Understanding Stochastic Depth Networks Russell Kaplan, Raphael Palefsky-Smith, Liu Jiang Stanford University 450 Serra Mall, Stanford, CA {rjkaplan, rpalefsk, Abstract In this paper, we understand, analyze, and visualize Stochastic Depth Networks, an architecture introduced in March of Stochastic Depth Networks have enjoyed interest as a result of their significant reduction in training time while beating the then state of the art in accuracy. However, while Stochastic Depth Networks have delivered exceptional results, no academic paper has sought to understand the source of their performance or their limitations. In providing an analysis of Stochastic Depth Networks representations, error types, and strengths and weaknesses, we conduct seven experiments: t-sne on layer activations, weight activations, maximally activated images, guided backpropagation, dead neuron counting, robustness to input noise, and linear classifier probes. By specifically comparing and contrasting Stochastic Depth Networks with Fixed Depth Networks (standard residual networks), we discover that Stochastic Depth Networks have a faster training time, a lower test error, similar clustering of data, and more strongly differentiated weight activations. 1. Introduction Stochastic Depth Networks have demonstrated an impressive ability to train extremely deep neural networks. Inspired by Dropout, Stochastic Depth Networks are essentially ResNets with one small tweak: they randomly drop some of the layers at training time and replace them with the identity function [6]. Stochastic Depth Networks have been shown to reduce training time and lower generalization error, and they can train extremely deep networks. The dropping of layers also helps with gradient flow and serves as a regularizer by effectively training a random ensemble of networks that are then averaged at test time. Previous experiments support the regularization hypothesis, but many questions remain about why Stochastic Nets perform so well. We explore the inner workings of Stochastic Depth Networks through a series of seven experiments. Figure 1. Layer Dropout: the third and fifth blocks are replaced with an identity function. (Huang et al.) 2. Related Work The crux of our work involves analyzing deep networks with stochastic depth, the architecture of which is introduced by Huang et al. [6]. To address vanishing gradients and diminished forward flow, both of which are problems associated with 1

2 training deep convolutional networks with hundreds of layers, Huang et al. propose a training procedure called stochastic depth that enables the contradictory setup to train short networks and use deep networks at test time [6]. Huang et al. begin with deep networks but then randomly insert a subset of layers and bypass them with the identity function for each minibatch. The identity connections when dropping a layer are preserved such that the inputs from a previous layer are fed into the next layer in the stack. This approach is complementary to the recent success of residual networks and reduces training time while improving the test error. There are many unexplored facets of Stochastic Depth Networks. Huang et al. only experiment with architectures that use residual connections to make benchmarking against prior work easy and isolate the benefit obtained from stochastic depth [5][6]. This is useful for demonstrating improved performance, but running experiments that implement stochastic depth on networks without residual connections would be more informative. As a direct follow-up to Huang et al., our work analyzes why their procedure works so well. While Huang et al. analyze their Stochastic Depth Network architecture with standard performance techniques, they do not verify that their hypotheses for the drivers behind its high performance are actually true. For example, despite the fact that their regularizer hypothesis seems legitimate, the closest step they take towards verifying their hypothesis is showing that there is less over-fitting. While Stochastic Depth Nets have not been analyzed in great depth, other types of networks, particularly recurrent neural networks (RNNs) and convolutional neural networks (CNNs), have been explored and visualized [7][9][15]. For example, in a spirit similar to ours, Karpathy et al. use character-level language models as an interpretable testbed to provide an analysis of RNNs representations, predictions, and error types [7]. Their experiments reveal the existence of interpretable cells that keep track of long-range dependences. Their comparative analysis with finite horizon n-gram models shows that the source of LSTM improvements is long-range structural dependences. Because CNNs have demonstrated impressive classification performance on the ImageNet benchmark, there have been a few pieces of related work on visualizing and understanding CNNs and diagnosing additional possible improvements to their performance. Zeiler et al. [15] introduce a visualization technique that gives insight into the function of CNNs intermediate feature layers and the operation of the classifier. Similarly, Simonyan et al. [9] consider two visualization techniques: one that generates an image, which maximizes the class score and thus visualizes the notion of the class, and a second that computes a class saliency map, specific to a given image and class. Previous work has also proposed either new frameworks for simplifying the training of deep neural networks [6] or new methods for regularizing networks such as RNNs [8]. For example, He et al. reformulated the layers as learning residual functions referencing the layer inputs, as opposed to learning unreferenced functions [4]. Based on evaluation of residual nets of up to 152 layers on the ImageNet dataset, He et al. provide evidence that their residual networks are easier to optimize and can gain accuracy as depth increases [4]. Another paper by He et al. analyzes the propogation formulations between residual building blocks and proposes a new residual unit that makes training easier and improves generalizations [5]. A series of experiments support the importance of identity mappings, which can be used as the skip connections and afteraddition activation [5]. Interestingly enough, Huang et al.s architecture is essentially an exact copy of Hes RNN architecture, just with some stochastic layer dropping [5][6]. In that sense, our work primarily builds off of He et al.s and Huang et al.s papers. As another example, Krueger et al. propose a new method for regularizing RNNs known as zoneout [8]. Zoneout is a perunit version of stochastic depth [5]. At each timestep, zoneout stochastically forces some hidden units in order to maintain their previous values. Like dropout, zoneout improves generalization by using random noise. However, by preserving rather than dropping hidden units, information on gradient and state are more easily propogated through time. Krueger et al.s empirical investigation of various RNN regularizers shows that zoneout has significant performance improvements across tasks [8]. It is important to note that Krueger et al.s work extends the stochastic depth method to RNNs and networks with hidden state [8]. Some more recent work abstracts away from specific neural network types and attempts to avoid the overarching issue of training a new model for every individual problem. For example, Zamir et al. train a model to learn fundamental vision tasks [15]. They employ a method to learn a generic 3D representation that generalizes to unseen 3D tasks with humanlevel performance on the supervised task and without any fine-tuning needs [15]. The learned representation shows traits of abstraction abilities [15]. Zamir et al. developed independent semantic and 3D representations but integrating them is a future direction of research that we similarly hope to undertake. 3. Model Training Rather than using a pre-trained model, we opted to train our own model. We used the official code from the Huang et al. paper [4]. We train two separate 110-layer ResNets: one with stochasticity of 0.5 on a linear decay schedule, and one with 2

3 no stochasticity. The difference between the two networks lies in the layer death rate. The first network (Fixed Depth) is a conventional ResNet: every layer is trained, and there is no layer death. The second network, the Stochastic Depth Network [4], has a death rate of 0.5 that decays linearly across ResBlock layers (i.e., the later layers have a higher probability of dropping for any particular minibatch. See Huang et al. for details.) The networks are trained on CIFAR-10 with standard data augmentation. There are 45,000 training images, a 5,000-image validation set, and a 10,000-image test set. The mini batch size is 128 with 18 residual modules, and we train on an Amazon GX2-large EC2 instance with an NVIDIA K80 GPU. Training occurs for 500 epochs via a Stochastic Gradient Descent approach with a learning rate of 0.1, that decays to 0.01 and then after 250 and 375 epochs respectively, using weight decay of 1e-4 and Nesterov momentum. As expected, the Stochastic Depth epochs were quicker because we stochastically skip the forward and backward computations for some of the ResBlocks. The Stochastic Depth Networks had an average epoch time of 210 seconds while the Fixed Depth Networks had an average epoch time of 258 seconds. 4. Methods and Technical Solution 4.1. t-sne on Layer Activations Figure 2. Test Error vs Number of Epochs Trained For our first experiment, we used t-sne to visualize 64-dimensional CNN codes of 4096 different image inputs [8]. T- SNE is a visualization technique that embeds high-dimensional vectors into a low-dimensional space, while trying to preserve the relative distances between different points in the high-dimensional space in the low-dimensional projection. In our case, we embed the 64-dimensional codes into a 2-dimensional space and plot each input image according to the projection of its corresponding codes. We use a perplexity of 30 to produce t-sne plots that show embeddings of image codes from the Stochastic and Fixed Depth Networks respectively. We employed t-sne to visualize the activations of the following layers: layer 36 (after the first third of the network), layer 72 (after the second third of the network), and layer 108 (after the final spatial average pooling layer that is just before the final fully-connected layer). We take the outputs of all the neurons in each of those three layers and use them as feature vectors. In the case of layer 108, the output is a 64-length vector. In the case of layers 36 and 72, which are much earlier in the net, the output is large (greater than 16,000). Because t-sne cannot run efficiently on such a large dataset, we used Spatial Average Pooling, a 2D average-pooling operation over an input image, to drop the outputs of layer 72 and layer 36 down to 64. 3

4 4.2. Weight Activations We plot the activations of weights at different layers for a given input image. At a high level, visualizing the activations allows us to identify which neurons at different layers in the network get excited about or respond to which specific inputs Maximally Activated Images We ran the entire test set through the network, examined a single neuron, and recorded its activations for each image. We then sorted the images by this activation and selected the top five. We completed this process for two different neurons across six different layers (layers 1, 21, 41, 61, 81, and 101) on both networks Guided Backpropogation To visualize what input is maximally exciting to specific neurons, we employ guided backpropogation, a method described by Springenberg et al. [8]. Rather than masking out values corresponding to negative entries of the bottom data (backpropogation) or top gradient (deconvolutional network approach), guided backpropogation masks out the values for which at least one of these values is negative [8]. In contrast to traditional backpropogation, guided backpropogation adds an additional guidance signal, thereby preventing the backward flow of negative gradients [8]. Unlike the deconvolutional network approach, guided backpropogation works well without switches and allows for visualization of intermediate as well as last layers of networks. Firstly, we found which input images result in the highest activations for several specific neurons early and late in the network. Next, we change the backwards pass of our network so that the gradient of the layer whose neuron we wish to visualize is set to all 0s, except for the specific neuron we visualize, which has all 1s. We then modify the gradients of the previous ReLU layers Dead Neurons We run the entire test set through the network. At each layer, we note neurons that have zero activation. Because this is done after the ReLU, zero actually means less than or equal to 0. After running the entire test set through, we made note of which neurons were zero-activated for every single image. We then tallied up the number of neurons per layer, and compare between Stochastic and Fixed. Note that we chose to do a cumulative plot due to the fact that the raw graph fluctuates between 5 and 0 at almost every layer and is unreadable. By cumulative, we mean the total number of dead neurons up to and including this layer Robustness to Input Noise Huang et. al. hypothesize that stochastic depth acts as a regularizer. They cite the higher training loss but lower test error of the Stochastic Depth Network after convergence as evidence of a regularizing effect. The authors also draw a comparison to Dropout, which has regularizing benefits that are well-studied by those like Wager et al. [11][13]. We test this hypothesis by adding different types and levels of noise to image inputs and comparing the networks performance. For this experiment, we add noise to images and examine how much the noise affects error [3]. If the Stochastic Depth Network has a better test error than the Fixed Depth Network when the same amount of noise is added, the regularization hypothesis would be supported. The test error is calculated across the entire test set. To achieve an understanding of the overall effect of noise, we ran every image through the network, giving each image a different but equally strong bit of noise, and then record the overall accuracy on the entire test set. Each of the noise functions had a noise parameter that we varied (i.e. the x-axis), and as the parameter increased, the image became noisier. At this point, the images have already been mean-subtracted and have a standard deviation of 1. Thus, most of the pixel values are in the range of 0 to 1, so adding noise of 0.5 is pretty significant. For the Gaussian normally distributed noise, we add a random value with zero mean and standard deviation from our x-axis to each pixel. For the uniformly distributed noise, we added a random value to each pixel but we draw a random value in the range from 0 to the x-axis. For the Gaussian blur, we blurred the image with a Gaussian kernel using the same convolution semantics, thereby keeping the image at its initial size. The noise parameter for the Gaussian blur is σ, which controls the size of the filter. As an example, a filter with σ = 5 will combine more pixels than a filter with σ = Linear Classifier Probes Because neural network models have a reputation for being black boxes, we employ methods to better visualize and understand what is being done at each layer of a Stochastic Depth Network. One way to do this is with a technique called 4

5 linear classifier probes [1], which essentially measures how linearly separable the activations of a particular layer are into final class labels. These probes can only use the hidden units of a specific intermediate layer as discriminating features, and these probes do not affect the training phase of our models as we add them after training. Intermediate layers are particularly interesting as the first layers of a convolution network for image recognition contain relatively general filters in that they would likely continue to perform well even under a different image dataset. Furthermore, the last layers are often specific to a dataset and have to be retrained under a different dataset. Thus, intermediate layers are highly relevant in terms of pinpointing when this transition occurs and if this transformation is progressive or sudden. 5. Experimental Results 5.1. t-sne on Layer Activations As seen in Figure 3, both the Stochastic Depth Network as well as the Fixed Depth Network learned to cluster the data. Notice the clean separability of the final layer, and the different color distributions of the plots for layer 72. The cluster patterns in the t-sne of layer 72, two thirds deep into the network, show that the fixed network activations are close-by when low-level features like background color are close-by in the input space. (On a zoomed in version of this plot, one can clearly observe birds and planes intermixed in the same regions when their backgrounds are both a blue sky, for example. Similarly for deer and horses with grassy backgrounds.) This contrasts with the layer 72 t-sne for the Stochastic Depth Network, where the background colors are relatively jumbled but there are more examples of images in the same class congregating closer together when they have different background patterns and colors. One explanation of the differences in layer 72, supported by later experiments, is that the Stochastic Depth Network cares less about mastering low-level image feature extraction; it devotes more representational capacity to learning and separating higher-level features. We can begin to observe this in this t-sne plot: by layer 72, the Stochastic Depth Network no longer cares to cluster by background color, but rather it has begun to cluster by higher-level semantic significance. (Note that this does not mean the data are more linearly separable into classes with layer 72 activations in the stochastic variant; as our later linear probe experiments show, the opposite is actually the case. But we can see that the high-level features are given more representational weight at this layer, even if those representations aren t yet class-separating.) Figure 3. t-sne plots for the activations at layers 36, 72, and 113 (the final spatial average pooling layer) of 4096 test set images. 5

6 5.2. Weight Activations We observe that across all inputs we tried visualizing, late-layer weight activations are more strongly differentiated between neurons in the Stochastic Depth Network than in the Fixed Depth Network. The implication is that the same late-stage layer in the Fixed Depth Network, the activations are more diffuse across filters (i.e. no one filter is activated as strongly, and more activate weakly) versus a corresponding layer in the Stochastic Depth Network. For a clarifying illustration of this result, see Figure 4. The distribution and strength of weight activations might indicate that the Stochastic Depth Network better confidently discriminates between different classes of the input image. Another observation is that immediately after each time we double the number of filters, otherwise known as neurons or tiles, nearly half of them is are often all black. Our dead neuron distribution experiment confirms this hypothesis: there is a spike in neuron death each time we double the number of filters. Figure 4. Weight activations at various depths of the two different networks, for the same input image. Note that the actual input image was in color. 6

7 5.3. Maximally Activated Images By the end of the process, the neurons learned higher order features. Our results validate the hierarchical assumption of ConvNets. As shown in Figure 5, at Layer 1, we see very basic responses, which makes sense because it is early in the network. Specifically the Fixed Neuron 2 Layer 1 likes red objects regardless of what they are, and Stochastic Neuron 6 Layer 1 likes green objects regardless of background. Fixed Neuron 2 Layer 101 is an emu neuron, Fixed Neuron 6 Layer 101 is a car neuron, and both Stochastic Neurons Layer 101 are horse neurons. Figure 5. This chart displays the top-5 maximally activating images for various ReLU neurons within the Fixed-Depth and Stochastic-Depth networks. At each layer, we plot the images that maximally activate Neurons 2 and 6 (randomly selected). 7

8 5.4. Guided Backpropogation In both the Stochastic and Fixed Depth Networks, neurons perform as expected. The early layer neurons react strongly to color and texture, whereas the late layer neurons react to more semantically meaningful units (e.g. the wheels and headlights on cars, the heads of birds, sticks that the birds often sit on, and so forth). These results were consistent throughout the various neurons we visualized at different layers of each network. Overall, there was no strong difference between the Stochastic and Fixed Depth Networks that we observed through this method. Figure 6. Guided backpropagation visualizations of the excitations of neurons in the first and last ReLU layers of both the Fixed and Stochastic depth network. Within each block, each row represents a different neuron in the layer. The 6 tiles to the right are the top 6 maximally activating images for that neuron, and the tiles to the left are the guided backpropagation visualizations of the neuron corresponding to each of those 6 image inputs. The all-black tiles for the last row in the first ReLU of the fixed network show a dead neuron: it has an activation of 0 (and thus no gradient signal) for all images in the dataset Dead Neurons The plot in Figure 7 shows that stochasticity does not help with the dead neurons problem; in fact the problem is actually more pronounced in the early layers. Nonetheless, the Stochastic Depth Network has relatively fewer dead neurons in later layers. One intuition for this second point is that the later layers drop with higher probability due to the linear decay schedule, in which the probability of survival decays linearly as we go deeper. Because the later layers in the Stochastic Depth Network are dropped frequently, having more neurons is more important because it is less likely that they are present. 8

9 Figure 7. This plot shows the accumulation of dead neurons in each network: i.e., how many neurons up through the layer marked on the x-axis do not activate for any input image? We note that the stochastic depth network accumulates more dead neurons earlier, but the fixed depth network gains more later. They end up with a roughly equal total number of dead neurons Robustness to Input Noise As shown in Figure 8, the Stochastic Depth Network is less robust to image noise than the Fixed Depth Network for both Gaussian normally distributed noise and uniformly distributed noise. The Stochastic Depth Network performs slightly better for Gaussian blur perturbations, although it is questionable how meaningful these results are for σ > 3, given how much of the image is destroyed for larger σ. For examples, please view the images below the graphs of Figure 8. The regularization hypothesis may therefore not be universally true. This is especially apparent for low-level perturbations like image noise. The Stochastic Depth Network has nearly twice the number of dead neurons as the Fixed Depth Network in the earliest layers, as those layers are responsible for the pixel-level pattern matching that the image noise is most likely to interfere with. This, in conjunction with the dead neurons experiment described earlier, suggests that the early layers of a Stochastic Depth Network are actually less robust than those in a Fixed Depth Network. The fact that test-time performance is still generally better for Stochastic Depth suggests that perhaps having the most robust early layers is not that important. The main sources of remaining error on datasets like CIFAR may potentially lie not in problems with early layer feature activations but in layer ones. This supports the general observation made by Deng et al. in [2] that CNNs can often vastly outperform humans on fine-grained pattern recognition tasks in images (e.g. distinguishing between many close breeds of dogs) but be inferior in classification when high-level features of the image are very skewed (e.g. extreme occlusions). 9

10 Figure 8. These plots display the effect of noise on test error for both Fixed Depth and Stochasti -Depth Networks. The x-axis is the amount of noise applied to images in the test set, and the y-axis is the corresponding error on the noise-corrupted test sets. Each plot is a different type of noise Unformly Distributed, Normally Distributed, and Gaussian Blur and the images below provide an example of each noise applied at various amounts to an image Linear Classifier Probes In Figure 9, we plot the results of our linear probe experiments. Interestingly, the fixed networks intermediate layer activations are generally more linearly separable into the class labels than those of the Stochastic Depth Network. The only exception to this is at the earliest layer we probed, layer 18, and the last non-fully-connected layer of the network (the output of the 8 8 average spatial pooling layer). Clearly, the activations at the last layer will be more linearly separable for the Stochastic Depth Network, as this is the network that ultimately had lower test error. However, it is interesting that essentially all of its intermediary layers produce activations that are less separable into classes. Recall that it is not the real job of intermediate layers to produce linearly separable class activations. That is only the job of the last layer of the network; the remaining layers are simply supposed to produce the most useful possible feature activations for further processing by the next layer. Here we see that in the process of doing a better job, Stochastic Depth Networks produce less separating intermediary activations. Why does that happen, and what does it suggest? One interpretation can be made by recalling what stochastic depth actually does: by randomly dropping the activations of some layers, and only letting activations flow through the skip connection when that happens, stochasticity essentially asks more from each of the intermediate layers: be useful to the next layer, but also be useful to the layer after when the next layer is not present. We suspect this results in a kind of representational hedging : because the task demanded of each intermediate layer changes from epoch to epoch, depending on which layers are dropped, they do worse on any one individual layer s task request, like linear separability. They can be thought of as blurry representations that need to work well in multiple different contexts. Figure 9. Test errors of linear probes trained independently at different layers. Probes are trained with no pooling (which means early layer probes have many tens of thousands of parameters) and a learning rate of until convergence. 10

11 6. Conclusion We conducted seven experiments: t-sne on layer activations, weight activations, maximally activated images, guided backpropogation, dead neurons, robustness to input noise, and linear classifier probes. One of our overarching conclusions, which is supported by the overall test error and our dead neuron, t-sne, and linear probes experiments, is that Stochastic Depth Nets are less tuned for low-level feature extraction but more tuned for higher level feature differentiation. This is supported by their higher susceptibility to error after low-level noise is introduced, and the intermediate t-sne plots that show higher level features being paid attention to earlier in the network, whereas the background color is the primary clustering factor for the fixed depth network. The different in robustness to noise also adds nuance to the analysis of Huang et al. s suggestion that stochasticity acts as a regularizer. Increased regularization would normally be expected to provide greater invariance to input noise. Our interpretation is that while stochasticity still likely has a regularizing effect (as test error is lower but training loss is higher after convergence), the effect regularizes across higher-level features in the image, as opposed to low-level perturbations. Overall, it seems that the representations learned by these networks are still rather similar. The performance is different, but not drastically; maximally activating neurons and guided backpropagation visualizations do not reveal major contrasts. But there is nonetheless the hint that the distribution of representational power is slightly different for each network. Stochastic depth networks are a fascinating architectural idea and we look forward to continued research on their utility. 7. Future Work We see many promising avenues for future work and plan to conduct the following additional experiments, among others: 1. Performing analyses on datasets beyond CIFAR-10, including MNIST and (a subset of) ImageNet. This way, we can collect quantitative results independent of the specific dataset, thereby ensuring that our findings do not depend on the properties of CIFAR-10 in particular. 2. Evaluating more architectures, including fully-connected nets and nets without any residual connections. 3. Determining how well the representations learned with Stochastic Depth Networks can be used for transfer learning with new tasks. 4. Finally, as more techniques for neural network visualization and understanding are developed, we would like to apply these generalized techniques to Stochastic Depth Nets in particular, perhaps uncovering relationships that our analyses missed. References [1] G. Alain and Y. Bengio. Understanding Intermediate Layers Using Linear Classifier Probes [2] J. Deng, W. Dong, R. Socher, L.J. Li, K. Li and L. Fei-Fei, ImageNet: A Large-Scale Hierarchical Image Database. In CVPR, [3] S. Dodge and L. Karam. Understanding How Image Quality Affects Deep Neural Networks [4] K. He, X. Zhang, S. Ren, and J. Sun. Deep Residual Learning for Image Recognition. In CVPR, [5] K. He, X. Zhang, S. Ren, and J. Sun. Identity Maps in Deep Residual Networks. In ECCV, [6] G. Huang, Y. Sun, Z. Liu, D. Sedra, and K. Weinberger. Deep Networks with Stochastic Depth. In NIPS, [7] Karpathy, J. Johsnon, and F. Li. Visualizing and Understanding Recurrent Networks. In ICLR, [8] D. Krueger, T. Maharaj, J. Kramr, M. Pezeshki, N. Ballas, N. Ke, A. Goyal, Y. Bengio, H. Larochelle, A. Courville, and C. Pal. Zoneout: Regularizing RNNs by Randomly Preserving Hidden Activations [9] K. Simonyan, A. Vedaldi, and A. Zisserman. Deep Inside Convolutional Networks: Visualizing Image Classification Models and Saliency Maps. In ICLR, [10] J.T. Springenberg, A. Dosovitskiy, T. Brox, M. Riedmiller. Striving for Simplicty: The All Convolutional Net. In ICLR, [11] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, R. Salakhutdinov: Dropout: A simple way to prevent neural networks from overfitting. The Journal of Machine Learning Research 15(1): , [12] L.J.P. van der Maaten and G.E. Hinton. Visualizing High-Dimensional Data Using t-sne. Journal of Machine Learning Research 9(Nov): , [13] S. Wager, S. Wang, P. Liang. Dropout Training as Adaptive Regularization. In NIPS,

12 [14] A. Zamir, P. Agrawal, T. Wekel, J. Malik, and S. Savarese. Generic 3D Representations via Pose Estimation and Mapping. In ECCV, [15] M. Zeiler and R. Fergus. Visualizing and Understanding Convolutional Networks. In ECCV,