Last year, a machine translation paper came out, with an unfortunately un-memorable name (the Transformer network) and a dramatic proposal for sequence modeling that eschewed both Recurrent NNN and Convolutional NN structures, and, instead, used self-attention as its mechanism for “remembering” or aggregating information from across an input. Earlier this month, the same authors released an extension of that earlier paper, called Image Transformer, that applies the same attention-only approach to image generation, and also achieved state of the art performance there. 

The recent paper offers a framing of attention that I find valuable and compelling, and that I’ll try to explicate here. They describe attention as being a middle ground between the approaches of CNNs and RNNs, and one that, to use an over-abused cliche, gets the best of both worlds.  CNNs are explicitly local: each convolutional filter only gathers information from the cells that fall in specific locations along some predefined grid. And, because convolutional filters have a unique parameter for every relative location in the grid they’re applied to, increasing the size of any given filter’s receptive field would engender an exponential increase in parameters: to go from a 3x3 grid to a 4x4 one, you go from 9 parameters to 16.  Convolutional networks typically increase their receptive field through the mechanism of adding additional layers, but there is still this fundamental limitation that for a given number of layers, CNNs will be fairly constrained in their receptive field. 

On the other side of the receptive field balance, we have RNNs. RNNs have an effectively unlimited receptive field, because they just apply one operation again and again: take in a new input, and decide to incorporate that information into the hidden state. This gives us the theoretical ability to access things from the distant past, because they’re stored somewhere in the hidden state. However, each element is only seen once and needs to be stored in the hidden state in a way that sort of “averages over” all of the ways it’s useful for various points in the decoding/translation process. (My mental image basically views RNN hidden state as packing for a long trip in a small suitcase: you have to be very clever about what you decide to pack, averaging over all the possible situations you might need to be prepared for. You can’t go back and pull different things into your suitcase as a function of the situation you face; you had to have chosen to add them at the time you encountered them). All in all, RNNs are tricky both because they have difficulty storing information efficiently over long time frames, and also because they can be monstrously slow to train, since you have to run through the full sequence to built up hidden state, and can’t chop it into localized bits the way you can with CNNs. 

So, between CNN - with its locally-specific hidden state - and RNN - with its large receptive field but difficulty in information storage - the self-attention approach interposes itself. Attention works off of three main objects: a query, and a set of keys, each one is attached to a value. In general, all of these objects take the form of vectors. For a given query, you calculate its similarity with each key, and then normalize those into a distribution (a set of weights, all of which sum to 1) that is used as the weights in calculating a weighted average of the values. As a motivating example, think of a model that is “unrolling” or decoding a translated sentence. In order to translate a sentence properly, the model needs to “remember” not only the conceptual content of the sentence, but what it has already generated. So, at each given point in the unrolling, the model can “query” the past and get a weighted distribution over what’s relevant to it in its current context. In the original Transformer, and also in the new one, the models use “multi-headed attention”, which I think is best compared to convolution filters: in the same way that you learn different convolution filters, each with different parameters, to pick up on different features, you learn different “heads” of the attention apparatus for the same purpose. 

To go back to our CNN - Attention - RNN schematic from earlier: Attention makes it a lot easier to query a large receptive field, since you don’t need an additional set of learned parameters for each location you expand to; you just use the same query weights and key weights you use for every other key and query. And, it allows you to contextually extract information from the past, depending on the needs you have right now. That said, it’s still the case that it becomes infeasible to make the length of the past you calculate your attention distribution over excessively long, but that cost is in terms of computation, not additional parameters, and thus is a question of training time, rather than essential model complexity, the way additional parameters is. 

Jumping all the way back up the stack, to the actual most recent image paper, this question of how best to limit the receptive field is one of the more salient questions, since it still is the case that conducting attention over every prior pixel would be a very large number of calculations. The Image Transformer paper solves this in a slightly hacky way: by basically subdividing the image into chunks, and having each chunk operate over the same fixed memory region (rather than scrolling the memory region with each pixel shift) to take better advantage of the speed of batched big matrix multiplies. 

Overall, this paper showed an advantage for the Image Transformer approach relevative to PixelCNN autoregressive generation models, and cited the ability for a larger receptive field during generation - without explosion in number of parameters - as the most salient reason why. 

It’s a commonly understood problem in Reinforcement Learning: that it is difficult to fully specify your exact reward function for an agent you’re training, especially when that agent will need to operate in conditions potentially different than those it was trained in. The canonical example of this, used throughout the Inverse Rewards Design paper, is that of an agent trained on an environment of grass and dirt, that now encounters an environment with lava. In a typical problem setup, the agent would be indifferent to passing or not passing over the lava, because it was never disincentivized from doing so during training. 

The fundamental approach this paper takes is to explicitly assume that there exists a program designer who gave the agent some proxy reward, and that that proxy reward is a good approximation of the true reward on training data, but might not be so on testing. This framing, of the reward as a noisy signal, allows the model to formalize its uncertainty about scenarios where the proxy reward might be a poor mapping to the real one. 

The way the paper tests this is through a pretty simplified model. In the example, the agent is given a reward function expressed by a weighting of different squares it could navigate into: it has a strong positive weight on dirt, and a strong negative one on grass. The agent then enters an environment where there is lava, which, implicitly, it has a 0 penalty for in its rewards function. However, it’s the case that, if you integrate over all possible weight values for “lava”, none of them would have produced different behavior over the training trajectories. Thus, if you assume high uncertainty, and adopt a risk-averse policy where under cases of uncertainty you assume bad outcomes, this leads to avoiding values of the environment feature vector that you didn’t have data weighting against during training. 

Overall, the intuition of this paper makes sense to me, but it’s unclear to me if the formulation it uses generalizes outside of a very trivial setting, where your reward function is an explicit and given function of your feature vectors, rather than (as is typical) a scalar score not explicitly parametrized by the states of game prior to the very last one. It’s certainly possible that it might, but, I don’t feel like I quite have the confidence to say at this point. 

This paper has an unusual and interesting goal, compared to those I more typically read: it wants to develop a “translation” between the messages produced by a model, and natural language used by a human. More specifically, the paper seeks to do this in the context of an two-player game, where one player needs to communicate information to the other. A few examples of this are: 
- Being shown a color, and needing to communicate to your partner so they can choose that color 
- Driving, in an environment where you can’t see the other car, but you have to send a coordinating message so that you don’t collide 


Recently, people have started training multi-agent that play games like these, where they send “message” vectors back and forth, in a way fully integrated with the rest of the backpropogation procedure. From just observing the agents’ actions, it’s not necessarily clear which communication strategy they’re using. That’s why this paper poses as an explicit problem: how can we map between the communication vectors produced by the agents and the words that would be produced by a human in a similar environment? 

Interestingly, the paper highlights two different ways you could think about structuring a translation objective. The first is “pragmatic interpretation,” under which you optimize what you communicate about something according to the operation that needs to be performed afterwards. To make that more clear, take a look at the attached picture. Imagine that player one is shown a shape, and needs to use a phrase from the bottom language (based on how many sides the shape has) to describe it to player two, who then needs to guess the size of the shape (big or small), and is rewarded for guessing correctly. Because “many” corresponds to both a large and a small shape, the strategy that optimizes the action that player two takes, conditional on getting player one’s message, is to lie and describe a hexagon as “few”, since that will lead to correct inference about the size of the shape, which is what’s most salient here. This example shows how, if you optimize a translation mapping by trying to optimize the reward that the post-translation agent can get, you might get a semantically incorrect translation. That might be good for the task at hand, but, because it leaves you with incorrect beliefs about the true underlying mapping, it will generalize poorly to different tasks. 

The alternate approach, championed by the paper, is to train a translation such that the utterances in both languages are similar insofar as, conditional on hearing them, and having some value for their own current state, the listening player arrives at similar beliefs about the current state of the player sending the message. This is mathematically framed as by defining a metric q, representing the quality of the translation between two z vectors, as: “taking an expectation over all possible contextual states of (player 1, player 2), what is the difference between the distribution of beliefs about the state of player 1 (the sending player) induced in player 2 by hearing each of the z vectors. Because taking the full expectation over this joint distribution is intractable, the approach is instead done by sampling. 

These equations require that you have reasonable models of human language, and understanding of human language, in the context of games. To do this, the authors used two types of datasets: 
1. Linguistic descriptions of objects of things, like the xkcd color dataset. Here, the player’s hidden state is the color that they are trying to describe using some communication scheme. 
2. Mechanical turk game runs playing the aforementioned driver game, where they have to communicate to the other driver. Here, the player’s “hidden state” represents a combination of its current location and intentions. 
From these datasets, they can train simple emulator models that learn “what terms is a human most likely to use for a given color” [p(z|x)], and “what colors will a human guess, conditional on those terms”. 

The paper closes by providing a proof as to how much reward-based value is lost by optimizing for the true semantic meaning, rather than the most pragmatically useful translation. They find that there is a bound on the gap, and that, in many empirical cases, the observed gap is quite small. 

Overall, this paper was limited in scope, but provided an interesting conceptual framework for thinking about how you might structure a translation, and the different implications that structure might have on your results. 

At NIPS 2017, Ali Rahimi was invited on stage to give a keynote after a paper he was on received the “Test of Time” award. While there, in front of several thousand researchers, he gave an impassioned argument for more rigor: more small problems to validate our assumptions, more visibility into why our optimization algorithms work the way they do. The now-famous catchphrase of the talk was “alchemy”; he argued that the machine learning community has been effective at finding things that work, but less effective at understanding why the techniques we use work. A central example he used in his talk is that of Batch Normalization: a now nearly-universal step in optimizing deep nets, but one where our accepted explanation of “reducing internal covariate shift” is less rigorous than one might hope. 

With apologies for the long preamble, this is the context in which today’s paper is such a welcome push in the direction of what Rahimi was advocating for - small, focused experimentation that tries to build up knowledge from principles, and, specifically, asks the question: “Does Batch Norm really work via reducing covariate shift”. 

To answer the question of whether internal covariate shift is a likely mechanism of the - empirically very solid - improved performance of Batch Norm, the authors do a few simple experience. First, and most straightforwardly, they train a basic convolutional net with and without BatchNorm, pick a layer, and visualize the activation distribution of that layer over time, both in the Batch Norm and non-Batch Norm case. While they saw the expected performance boost, the Batch Norm case didn’t seem to be meaningfully more stable over time, relative to the normal case. Second, the authors tested what would happen if they added non-zero-mean random noise *after* Batch Norm in the network. The upshot of this was that they were explicitly engineering internal covariate shift, and, if control thereof was the primary useful purpose of Batch Norm, you would expect that to neutralize BN’s good performance. In this experiment, while the authors did indeed see noisier, less stable activation distributions in the noise + BN case (in particular: look at layer 13 activations in the attached image), but noisy BN performed nearly as well as non-noisy, and meaningfully better than the standard model without noise, but also without BN. 

As a final test, they approached the idea of “internal covariate shift” from a different definitional standpoint. Maybe a better way of thinking about it is in terms of stability of your gradients, in the face of updates made by lower layers of the network. That is to say: each parameter of the network pushes itself in the direction of lower loss all else held equal, but in practice, you change lower-level parameters simultaneously, which could cause the directional change the higher-layer parameter thought it needed to be off. So, the authors calculated the “gradient delta” between the gradient the model trains on, and what the gradient would be if you estimated it *after* all of the lower layers of the model had updated, such that the distribution of inputs to that layer has changed. Although the expectation would be that this gradient delta is smaller for batch norm, in fact, the authors found that, if anything, the opposite was true. 

So, in the face of none of these ideas panning out, the authors then introduce the best idea they’ve found for what motivates BN’s improved performance: a smoothing out of the loss function that SGD is optimizing. A smoother curve means, generally speaking, that the magnitudes of your gradients will be smaller, and also that the value of the gradient will change more slowly (i.e. low second derivative). As support for this idea, they show really different results for BN vs standard models in terms of, for example, how predictive a gradient at one point is of a gradient taken after you take a step in the direction of the first gradient. BN has meaningfully more predictive gradients, tied to lower variance in the values of the loss function in the direction of the gradient. The logic for why the mechanism of BN would cause this outcome is a bit tied up in math that’s hard to explain without LaTeX visuals, but basically comes from the idea that Batch Norm decreases the magnitude of the gradient of each layer output with respect to individual weight parameters, by averaging out those magnitudes over the batch. 

As Rahimi said in his initial talk, a lot of modern modeling is “applying brittle optimization techniques to loss surfaces we don’t understand.” And, by and large, that is in fact true: it’s devilishly difficult to get a good handle on what loss surfaces are doing when they’re doing it in several-million-dimensional space. But, it being hard doesn’t mean we should just give up on searching for principles we can build our understanding on, and I think this paper is a really fantastic example of how that can be done well. 

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.