If you were to survey researchers, and ask them to name the 5 most broadly influential ideas in Machine Learning from the last 5 years, I’d bet good money that Batch Normalization would be somewhere on everyone’s lists. Before Batch Norm, training meaningfully deep neural networks was an unstable process, and one that often took a long time to converge to success. When we added Batch Norm to models, it allowed us to increase our learning rates substantially (leading to quicker training) without the risk of activations either collapsing or blowing up in values. It had this effect because it addressed one of the key difficulties of deep networks: internal covariate shift. 

To understand this, imagine the smaller problem, of a one-layer model that’s trying to classify based on a set of input features. Now, imagine that, over the course of training, the input distribution of features moved around, so that, perhaps, a value that was at the 70th percentile of the data distribution initially is now at the 30th. We have an obvious intuition that this would make the model quite hard to train, because it would learn some mapping between feature values and class at the beginning of training, but that would become invalid by the end. This is, fundamentally, the problem faced by higher layers of deep networks, since, if the distribution of activations in a lower layer changed even by a small amount, that can cause a “butterfly effect” style outcome, where the activation distributions of higher layers change more dramatically. Batch Normalization - which takes each feature “channel” a network learns, and normalizes [normalize = subtract mean, divide by variance] it by the mean and variance of that feature over spatial locations and over all the observations in a given batch - helps solve this problem because it ensures that, throughout the course of training, the distribution of inputs that a given layer sees stays roughly constant, no matter what the lower layers get up to. 

On the whole, Batch Norm has been wildly successful at stabilizing training, and is now canonized - along with the likes of ReLU and Dropout - as one of the default sensible training procedures for any given network. However, it does have its difficulties and downsides. One salient one of these comes about when you train using very small batch sizes - in the range of 2-16 examples per batch. Under these circumstance, the mean and variance calculated off of that batch are noisy and high variance (for the general reason that statistics calculated off of small sample sizes are noisy and high variance), which takes away from the stability that Batch Norm is trying to provide. 

One proposed alternative to Batch Norm, that didn’t run into this problem of small sample sizes, is Layer Normalization. This operates under the assumption that the activations of all feature “channels” within a given layer hopefully have roughly similar distributions, and, so, you an normalize all of them by taking the aggregate mean over all channels, *for a given observation*, and use that as the mean and variance you normalize by. Because there are typically many channels in a given layer, this means that you have many “samples” that go into the mean and variance. However, this assumption - that the distributions for each feature channel are roughly the same - can be an incorrect one. 

A useful model I have for thinking about the distinction between these two approaches is the idea that both are calculating approximations of an underlying abstract notion: the in-the-limit mean and variance of a single feature channel, at a given point in time. Batch Normalization is an approximation of that insofar as it only has a small sample of points to work with, and so its estimate will tend to be high variance. Layer Normalization is an approximation insofar as it makes the assumption that feature distributions are aligned across channels: if this turns out not to be the case, individual channels will have normalizations that are biased, due to being pulled towards the mean and variance calculated over an aggregate of channels that are different than them. 

Group Norm tries to find a balance point between these two approaches, one that uses multiple channels, and normalizes within a given instance (to avoid the problems of small batch size), but, instead of calculating the mean and variance over all channels, calculates them over a group of channels that represents a subset. The inspiration for this idea comes from the fact that, in old school computer vision, it was typical to have parts of your feature vector that - for example - represented a histogram of some value (say: localized contrast) over the image. Since these multiple values all corresponded to a larger shared “group” feature. If a group of features all represent a similar idea, then their distributions will be more likely to be aligned, and therefore you have less of the bias issue. 

One confusing element of this paper for me was that the motivation part of the paper strongly implied that the reason group norm is sensible is that you are able to combine statistically dependent channels into a group together. However, as far as I an tell, there’s no actually clustering or similarity analysis of channels that is done to place certain channels into certain groups; it’s just done so semi-randomly based on the index location within the feature channel vector. So, under this implementation, it seems like the benefits of group norm are less because of any explicit seeking out of dependant channels, and more that just having fewer channels in each group means that each individual channel makes up more of the weight in its group, which does something to reduce the bias effect anyway.

The upshot of the Group Norm paper, results-wise, is that Group Norm performs better than both Batch Norm and Layer Norm at very low batch sizes. This is useful if you’re training on very dense data (e.g. high res video), where it might be difficult to store more than a few observations in memory at a time. However, once you get to batch sizes of ~24, Batch Norm starts to do better, presumably since that’s a large enough sample size to reduce variance, and you get to the point where the variance of BN is preferable to the bias of GN.  

This paper explores the use of convolutional (PixelCNN) and recurrent units (PixelRNN) for modeling the distribution of images, in the framework of autoregression distribution estimation. In this framework, the input distribution $p(x)$ is factorized into a product of conditionals $\Pi p(x_i | x_i-1)$. Previous work has shown that very good models can be obtained by using a neural network parametrization of the conditionals (e.g. see our work on NADE \cite{journals/jmlr/LarochelleM11}). Moreover, unlike other approaches based on latent stochastic units that are directed or undirected, the autoregressive approach is able to compute log-probabilities tractably. 

So in this paper, by considering the specific case of x being an image, they exploit the topology of pixels and investigate appropriate architectures for this.

Among the paper's contributions are:

1. They propose Diagonal BiLSTM units for the PixelRNN, which are efficient (thanks to the use of convolutions) while making it possible to, in effect, condition a pixel's distribution on all the pixels above it (see Figure 2 for an illustration).

2. They demonstrate that the use of residual connections (a form of skip connections, from hidden layer i-1 to layer $i+1$) are very effective at learning very deep distribution estimators (they go as deep as 12 layers).

3. They show that it is possible to successfully model the distribution over the pixel intensities (effectively an integer between 0 and 255) using a softmax of 256 units.

4. They propose a multi-scale extension of their model, that they apply to larger 64x64 images.

The experiments show that the PixelRNN model based on Diagonal BiLSTM units achieves state-of-the-art performance on the binarized MNIST benchmark, in terms of log-likelihood. They also report excellent log-likelihood on the CIFAR-10 dataset, comparing to previous work based on real-valued density models. Finally, they show that their model is able to generate high quality image samples.

This paper argues that, in semi-supervised learning, it's suboptimal to use the same weight for all examples (as happens implicitly, when the unsupervised component of the loss for each example is just added together directly. Instead, it tries to learn weights for each specific data example, through a meta-learning-esque process. 

The form of semi-supervised learning being discussed here is label-based consistency loss, where a labeled image is augmented and run through the current version of the model, and the model is optimized to try to induce the same loss for the augmented image as the unaugmented one. The premise of the authors argument for learning per-example weights is that, ideally, you would enforce consistency loss less on examples where a model was unconfident in its label prediction for an unlabeled example. 

As a way to solve this, the authors suggest learning a vector of parameters - one for each example in the dataset - where element i in the vector is a weight for element i of the dataset, in the summed-up unsupervised loss. They do this via a two-step process, where first they optimize the parameters of the network given the example weights, and then the optimize the example weights themselves. To optimize example weights, they calculate a gradient of those weights on the post-training validation loss, which requires backpropogating through the optimization process (to determine how different weights might have produced a different gradient, which might in turn have produced better validation loss). This requires calculating the inverse Hessian (second derivative matrix of the loss), which is, generally speaking, a quite costly operation for huge-parameter nets. To lessen this cost, they pretend that only the final layer of weights in the network are being optimized, and so only calculate the Hessian with respect to those weights. They also try to minimize cost by only updating the example weights for the examples that were used during the previous update step, since, presumably those were the only ones we have enough information to upweight or downweight. With this model, the authors achieve modest improvements - performance comparable to or within-error-bounds better than the current state of the art, FixMatch. 

Overall, I find this paper a little baffling. It's just a crazy amount of effort to throw into something that is a minor improvement. A few issues I have with the approach: 

- They don't seem to have benchmarked against the simpler baseline of some inverse of using Dropout-estimated uncertainty as the weight on examples, which would, presumably, more directly capture the property of "is my model unsure of its prediction on this unlabeled example"
- If the presumed need for this is the lack of certainty of the model, that's a non-stationary problem that's going to change throughout the course of training, and so I'd worry that you're basically taking steps in the direction of a moving target
- Despite using techniques rooted in meta-learning, it doesn't seem like this models learns anything generalizable - it's learning index-based weights on specific examples, which doesn't give it anything useful it can do with some new data point it finds that it wasn't specifically trained on

Given that, I think I'd need to see a much stronger case for dramatic performance benefits for something like this to seem like it was worth the increase in complexity (not to mention computation, even with the optimized Hessian scheme)

TLDR; The authors propose an importance-sampling approach to deal with large vocabularies in NMT models. During training, the corpus is partitioned, and for each partition only target words occurring in that partition are chosen. To improve decoding speed over the full vocabulary, the authors build a dictionary mapping from source sentence to potential target vocabulary. The authors evaluate their approach on standard MT tasks and perform better than the baseline models with smaller vocabulary.

#### Key Points:

- Computing partition function is the bottleneck. Use sampling-based approach.
- Dealing with large vocabulary during training is separate from dealing with large vocab during decoding. Training is handled with importance sampling. Decoding is handled with source-based candidate list.
- Decoding with candidate list takes around 0.12s (0.05) per token on CPU (GPU). Without target list 0.8s (0.25s).
- Issue: Candidate list is depended on source sentence, so it must be re-computed for each sentence. 
- Reshuffling the data set is expensive as new partitions need to be calculated (not necessary, but improved scores).

#### Notes:

- How is the corpus partitioned? What's the effect of the partitioning strategy?
- The authors say that they replace UNK tokens using "another word alignment model" but don't go into detail what this is. The results show that doing this results in much larger score bump than increasing the vocab does. (The authors do this for all comparison models though).
- Reshuffling the dataset also results in a significant performance bump, but this operation is expensive. IMO the authors should take all these into account when reporting performance numbers. A single training update may be a lot faster, but the setup time increases. I'd would've like to see the authors assign a global time budget to train/test and then compare the models based on that.
- The authors only briefly mentioned that re-building the target vocab for each source sentence is an issue and how they solve it, no details given.

This is an interestingly pragmatic paper that makes a super simple observation. Often, we may want a usable network with fewer parameters, to make our network more easily usable on small devices. It's been observed (by these same authors, in fact), that pruned networks can achieve comparable weights to their fully trained counterparts if you rewind and retrain from early in the training process, to compensate for the loss of the (not ultimately important) pruned weights. This observation has been dubbed the "Lottery Ticket Hypothesis", after the idea that there's some small effective subnetwork you can find if you sample enough networks. 

Given these two facts - the usefulness of pruning, and the success of weight rewinding - the authors explore the effectiveness of various ways to train after pruning. Current standard practice is to prune low-magnitude weights, and then continue training remaining weights from values they had at pruning time, keeping the final learning rate of the network constant. The authors find that: 
1. Weight rewinding, where you rewind weights to *near* their starting value, and then retrain using the learning rates of early in training, outperforms fine tuning from the place weights were when you pruned 

but, also 
2. Learning rate rewinding, where you keep weights as they are, but rewind learning rates to what they were early in training, are actually the most effective for a given amount of training time/search cost 

To me, this feels a little bit like burying the lede: the takeaway seems to be that when you prune, it's beneficial to make your network more "elastic" (in the metaphor-to-neuroscience sense) so it can more effectively learn to compensate for the removed neurons. So, what was really valuable in weight rewinding was the ability to "heat up" learning on a smaller set of weights, so they could adapt more quickly. And the fact that learning rate rewinding works better than weight rewinding suggests that there is value in the learned weights after all, that value is just outstripped by the benefit of rolling back to old learning rates. 

All in all, not a super radical conclusion, but a useful and practical one to have so clearly laid out in a paper.