This is the first of what I expect will be a few posts. I started a machine learning blog on WordPress in 2017 but abandoned it 2 posts in after finding that showing code without messing up the formatting was not possible — the narrow column format would wrap the code and render it unreadable.
For 2018, my new year’s resolution is to write 5 posts (as Github project pages). I wanted to play with PixelCNNs and finally try PyTorch (I use Tensorflow for my work at Envision.AI and previously used Theano at McGill) so this post will include my thoughts on both. In particular, I was curious if PixelCNNs conditioned on class labels could generate believable between-class examples.
PixelCNNs are the convolutional version of PixelRNNs, which treat the pixels in an image as a sequence and predict each pixel after seeing the preceding pixels (defined as above and to the left, though this is arbitrary). PixelRNNs are an autoregressive model of the joint prior distribution for images:
p(x) = p(x0) ∏ p(xi| xi<)
PixelRNNs are slow to train since the recurrence can’t be parallelized — even small images have hundreds or thousands of pixels, which is a relatively long sequence for RNNs. Replacing the recurrence with masked convolutions, such that the convolution filter only sees pixels above and to the left, allows for faster training (figure from conditional PixelCNN paper).
However, it’s worth noting that the original PixelCNN implementation produced worse results than the PixelRNN. One possible reason for the degraded results, conjectured in the follow-up paper (Conditional Image Generation with PixelCNN Decoders), is the relative simplicity of the ReLU activations in the PixelCNN compared to the gated connections in the LSTM. The Conditional PixelCNN paper subsequently replaced the ReLUs with gated activations:
y = tanh(Wf∗ x) • σ(Wg∗ x)
Another possible reason offered in the follow-up paper is that stacking masked convolutional filters results in blind spots, failing to capture all the pixels above the one being predicted (figure from paper):
PixelCNNs vs GANs
PixelCNNs and GANs are currently the two flavors of deep learning models for generating images. GANs are receiving a lot of attention recently, but in many ways I find their popularity unwarranted.
It’s unclear what objective GANs are actually trying to optimize as the minimum of the training objective (i.e. fooling the discriminator) would result in the generator recreating all the training images and/or generating adversarial examples that don’t necessarily resemble natural images. This is reflected in the notorious difficulty of training GANs and the myriad hacks to regularize them. The idea of pitting two nets against each other to produce training signals is interesting and has produced many good papers (notably cycleGAN) but I remain unconvinced that they’re useful for much beyond making flashy posts on social media.
On the other hand, PixelCNNs have a nice probabilistic underpinning. This allows them to not only generate images by sampling the distribution (left-to-right, top-to-bottom, following their autoregressive definition), but also means they can be used for other tasks. For example: as a pre-screening network to detect out-of-domain or adversarial examples; for detecting outliers in a training set; or estimating uncertainty at test. I’ll cover some of these extensions more in my next post.
I’d be interested in hearing if anyone has tried combining PixelCNNs and GANs. Perhaps the PixelCNN can be used as a prior or as a final stage of the decoder (conditioned on some higher-level learned representation) to avoid some of the training difficulties with GANs.
My implementation uses the gated blocks but for rapid implementation, I decided to forego the two-stream solution to the blind spot problem (separating the filters into horizontal and vertical components). There’s code available for solving the blind-spot problem in Tensorflow and it’d be fairly trivial to re-write it in PyTorch. This way the masking is simple: everything below and to the right of the current pixel is zeroed-out in the filter and in the first layer the current pixel is also set to zero in the filter.
class MaskedConv(nn.Conv2d): def __init__(self,mask_type,in_channels,out_channels,kernel_size,stride=1): """ mask_type: 'A' for first layer of network, 'B' for all others """ super(MaskedConv,self).__init__(in_channels,out_channels,kernel_size, stride,padding=kernel_size//2) assert mask_type in ('A','B') mask = torch.ones(1,1,kernel_size,kernel_size) mask[:,:,kernel_size//2,kernel_size//2+(mask_type=='B'):] = 0 mask[:,:,kernel_size//2+1:] = 0 self.register_buffer('mask',mask) def forward(self,x): self.weight.data *= self.mask return super(MaskedConv,self).forward(x)
The implementation for the gated ResNet blocks is slightly more complicated: the PixelCNN has shortcut connections between the two halfs of the network, like a U-Net; PyTorch allows the forward method of a Module to take multiple inputs only if they’re Variables; since the feature maps from the first half of the network are not Variables, they must be concatenated with the other input (the features from the preceding layer). This is avoided with the conditioning vector, since it is a Variable (in this case, the class label).
class GatedRes(nn.Module): def __init__(self,in_channels,out_channels,n_classes,kernel_size=3,stride=1, aux_channels=0): super(GatedRes,self).__init__() self.conv = MaskedConv('B',in_channels,2*out_channels,kernel_size, stride) self.y_embed = nn.Linear(n_classes,2*out_channels) self.out_channels = out_channels if aux_channels!=2*out_channels and aux_channels!=0: self.aux_shortcut = nn.Sequential( nn.Conv2d(aux_channels,2*out_channels,1), nn.BatchNorm2d(2*out_channels,momentum=0.1)) if in_channels!=out_channels: self.shortcut = nn.Sequential( nn.Conv2d(in_channels,out_channels,1), nn.BatchNorm2d(out_channels,momentum=0.1)) self.batchnorm = nn.BatchNorm2d(out_channels,momentum=0.1) def forward(self,x,y): # check for aux input from first half of net stacked into x if x.dim()==5: x,aux = torch.split(x,1,dim=0) x = torch.squeeze(x,0) aux = torch.squeeze(x,0) else: aux = None x1 = self.conv(x) y = torch.unsqueeze(torch.unsqueeze(self.y_embed(y),-1),-1) if aux is not None: if hasattr(self,'aux_shortcut'): aux = self.aux_shortcut(aux) x1 = (x1+aux)/2 # split for gate (note: pytorch dims are [n,c,h,w]) xf,xg = torch.split(x1,self.out_channels,dim=1) yf,yg = torch.split(y,self.out_channels,dim=1) f = torch.tanh(xf+yf) g = torch.sigmoid(xg+yg) if hasattr(self,'shortcut'): x = self.shortcut(x) return x+self.batchnorm(g*f)
I wasn’t sure where to put batch normalization from reading the original papers, so I placed it where I thought it made sense: prior to adding the residual connection.
With those two classes implemented, the full network was relatively easy.
The PyTorch scheme of defining everything as subclasses of
nn.Module, initializing all the layers/operations/etc. in the constructor and then connecting them together in the
forward method can be messy.
This is especially true if you have lots of shortcut connections and want to code your model with loops for arbitrary depth.
Note: to be able to save/restore the model, you have to store layers in a
ModuleList instead of a regular list.
Appending and indexing this list is otherwise the same though.
class PixelCNN(nn.Module): def __init__(self,in_channels,n_classes,n_features,n_layers,n_bins, dropout=0.5): super(PixelCNN,self).__init__() self.layers = nn.ModuleList() self.n_layers = n_layers # Up pass self.input_batchnorm = nn.BatchNorm2d(in_channels,momentum=0.1) for l in range(n_layers): if l==0: # start with normal conv block = nn.Sequential( MaskedConv('A',in_channels+1,n_features,kernel_size=7), nn.BatchNorm2d(n_features,momentum=0.1), nn.ReLU()) else: block = GatedRes(n_features, n_features, n_classes) self.layers.append(block) # Down pass for _ in range(n_layers): block = GatedRes(n_features, n_features,n_classes, aux_channels=n_features) self.layers.append(block) # Last layer: project to n_bins (output is [-1, n_bins, h, w]) self.layers.append( nn.Sequential(nn.Dropout2d(dropout), nn.Conv2d(n_features,n_bins,1), nn.LogSoftmax(dim=1))) def forward(self,x,y): # Add channel of ones so network can tell where padding is x = nn.functional.pad(x,(0,0,0,0,0,1,0,0),mode='constant',value=1) # Up pass features =  i = -1 for _ in range(self.n_layers): i += 1 if i>0: x = self.layers[i](x,y) else: x = self.layers[i](x) features.append(x) # Down pass for _ in range(self.n_layers): i += 1 x = self.layers[i](torch.stack((x,features.pop())),y) # Last layer i += 1 x = self.layers[i](x) assert i==len(self.layers)-1 assert len(features)==0 return x
MNIST is practically black and white, so I discretized the label to only 4 grayscale levels for the purposes of calculating cross-entropy loss. On natural images, the number of output levels would obviously need to be higher. All layers in the network have 200 features. For data augmentation I used random rotations of +/-5 degrees with nearest neighbour sampling. For training, I used Adam with a learning rate of 10-4 and dropout rate of 0.9.
The higher number of features (than is necessary for MNIST) and higher dropout is a trade-off of training time vs regularization. This is a trick that is rarely mentioned in papers but is helpful to avoid overfitting — I’ve only seen it mentioned in a paper for training on action recognition in video, where overfitting is a problem due to the high dimensionality vs the dataset sizes currently available.
I have a single GTX1070 GPU at home, so I didn’t run any kind of hyperparameter optimization: the ability to guess reasonable hyperparameters and have your model work says a lot about the robustness of Adam + batch normalization + dropout. The learning rate definitely could’ve been higher but this makes for a more interesting GIF.
The gif above shows a batch of 50 images (5 examples per class) generated after each epoch throughout training, from seemingly random scribbles to something resembling actual digits. Here’s the results at the best epoch:
The motivation for this work was to see if a Conditional PixelCNN could also generate reasonable examples between classes. This is done by conditioning on soft labels instead of one-hot encoded labels.
Let’s try what I’d expect are easily confused pairs of digits: (1,7), (3,8), (4,9), (5,6)
The generated between-class examples do not appear as realistic as the normal examples. It’s possible the model needs some additional training signal (e.g. teacher forcing from a classifier network) to interpolate along the image manifold like that. This is somewhat disappointing because I had hoped that generating between-class examples might allow for a learned form of mixup to be used (rather than averaging images). Obviously, testing this idea further would require many more GPUs to generate batches of inputs so it’s out of my range for now anyway.
Miscellaneous Thoughts on PyTorch
- Debugging something without a compiled computational graph (i.e. like in Tensorflow or Theano) is faster and more intuitive.
- Writing models where you have to initialize every operation/layer/etc. in the constructor and then call them in the forward method seems unnecessarily complicated. This is especially error-prone for models with shortcut connections if you write them with loops for arbitrary depth.
- Point 2 cancels out point 1 for me. I prefer Tensorflow.