Severely Theoretical

Machine learning, computational neuroscience, cognitive science

Thoughts on Image-GPT

The following are some short notes on OpenAI’s Image-GPT paper, which is in my opinion one of the most important papers that came out in recent years.

The motivating question behind this paper is this: can likelihood-based generative pre-training lead to strong transfer learning results in computer vision? This question is inspired by the success of this same technique in NLP (where it’s commonly known as language modeling). In computer vision on the other hand, successful transfer has so far been achieved mostly through other (non-generative) pre-training objectives, like supervised pre-training (on ImageNet etc.), or more recently self-supervised pre-training (MoCo, SimCLR, BYOL, etc.). This raises the interesting question of whether there might be some fundamental differences between language and vision tasks that make these different methods more appropriate for these two respective domains. The Image-GPT paper answers this question in the negative and shows for the first time that likelihood-based generative pre-training can also lead to very strong transfer learning results provided that we use the right kind of architecture (GPT) at the right scale.

The other main interesting result from this paper is that the very same GPT architecture is shown to perform well in both language prediction and image prediction tasks, suggesting that these (and similar) tasks may share a deep common computational core (something very general like: predict given as much context as possible) despite their many superficial differences and can be solved effectively by the same kind of computational architecture. I think this observation has important implications for the brain and evolution. To me, one of the things it suggests is that whatever inductive biases our brains (more specifically, our neocortex) might have, they’re probably not very domain-specific biases like many psychologists seem to believe (e.g. specific biases about visual objects, agents, or language). Rather, they’re probably much more generic biases, having to do with better information processing in some very general sense, like being able to integrate over a much longer context, or being able to do better deep credit assignment (i.e. dealing with vanishing/exploding gradients) etc. It is important to emphasize here that the GPT architecture itself embodies absolutely no language-specific or vision-specific inductive biases.

This idea also accords well with the sources of recent progress in machine learning. When I look at what drives significant architectural progress in machine learning today, most of the time it’s somebody proposing a solution to a very generic information processing problem: in ResNets, solving an optimization problem (vanishing/exploding gradients); in transformers, getting rid of the serial processing bottleneck of RNNs to make it feasible to integrate over a much longer context; in batch normalization, dealing with the covariate shift problem during training etc. Certainly, biological evolution doesn’t have to respect the same rules as human innovation, but at least to me this suggests that there’s maybe more bang for the buck in targeting these general information processing related problems than targeting more domain specific issues, which makes it more plausible that evolution may also be primarily targeting the same general issues.

One final interesting result in the Image-GPT paper is that even for the same validation loss in the generative pre-training task (i.e. ImageNet modeling), bigger models seem to show better transfer learning performance (Figure 3 in the paper). This is interesting in light of my criticism of the GPT-3 paper, where different sized models were not given the same amount of compute and it seemed likely that the smaller models would reach the same (or maybe even better) validation loss as the largest 175B-parameter model if they were given the same amount of compute. The results in the Image-GPT paper suggest that even in that case, the larger models might have had an advantage in terms of transfer performance, but it would have been much better if, just like the Image-GPT paper, the GPT-3 paper had actually carried out this important experiment to see if the larger models have a transfer advantage above and beyond what can be accounted for by validation loss (or compute) alone.

Thoughts on GPT-3

A couple of months ago, OpenAI released a paper describing their latest language model, GPT-3. GPT-3 is distinguished from its predecessors by nothing other than its sheer scale: compared to its previous incarnations, it’s just a bigger language model trained with a bigger dataset (~1-2 orders of magnitude bigger in both model size and training data size). So, the paper is essentially an exercise in scaling. The main novel result in the paper is an impressive demonstration of the (in-context) few-shot learning abilities of such large-scale language models (it can be argued that even this main result is not entirely novel, as it was foreshadowed in some earlier language modeling work, e.g. see this and this). The paper reminded me, once again, of Philip Anderson’s famous More Is Different paper, where Anderson argues that quantitative changes in nature can sometimes lead to qualitative changes and that people (even scientists) don’t always appreciate the consequences of this fact enough. It was also inspiring for me to see all the amazing demos people have quickly built with GPT-3 and shared with the world (here is a nice collection of such demos as a Twitter thread).

In this post, I’d like to briefly discuss a few criticisms I had of the GPT-3 paper.

Umm, yeah, did we really need that 175B-parameter model?

The first one is about the actual need for scale: i.e. whether they really needed to train a giant 175B-parameter model or not. Figure 3.1 in the paper (reproduced above) clearly shows that many of their smaller models were not trained to saturation; this figure also shows that the smaller models are actually more compute-efficient up to the total compute used for those smaller models. To me, this strongly suggests that they actually didn’t have to train a 175B-parameter model, a ~5B-parameter model would probably have performed just as well (if not better) if trained longer. This point was also noted by Graham Neubig on Twitter.

This renders all the figures in the paper showing model size on the x-axis and performance on the y-axis (which is most of the figures in the paper) a bit suspect in my mind, because the smaller models were not given the same amount of compute in those figures.

So why did they train a 175B-parameter model then? One possibility is just because they could; they perhaps wanted to prepare this kind of infrastructure for projects down the line that actually do require models at this scale. A more sinister interpretation is that they want to commercialize this product at some point (this would be consistent with their CEO’s expressly stated objective of “capturing the light cone of all future value in the universe”) and a giant model is more “controllable” for this purpose: a client can easily put a 5B-parameter model on a few GPUs of their own to do inference and fine-tuning as they wish, but they can’t do this with a 175B-parameter model, making them more reliant on OpenAI’s specialized hardware.

A second difficulty with the paper for me was my constant struggle to understand to what extent the model was doing abstraction (or generalization) vs. rote memorization. In other words, to what extent the impressive looking results from the model can be attributed to the sheer size of the training data vs. the abstraction capacity of the model. To understand this better, it would have been extremely useful if, for example, at least for a subset of the tasks and examples, the authors showed the embedding space nearest neighbors to a given query among the training data, but surprisingly they never do this in the paper (I don’t suppose this would be technically more challenging than running a search over the input space, which they do multiple times in the paper). If these nearest neighbors are intuitively highly similar to the query and the model’s outputs more or less resemble the actual continuations of these nearest neighbors (say, with simple substitutions), that would favor a dataset size based explanation for the performance of the model. They do try to rule out the rote memorization based explanation in some of their experiments, but these were not entirely convincing for me. For example, in the arithmetic tasks, they look for patterns of the form “<NUM1> + <NUM2> =” and “<NUM1> plus <NUM2>” in their training data to investigate if the model is just memorizing these arithmetic equations. They find only a small number of matches, concluding that a rote memorization strategy seems unlikely. But the problem here is that these are just two of the almost endless ways the same arithmetic equations could be encoded in the training data (note that their training data includes a snapshot of the entire world wide web, which is a really really big place!): e.g. “<NUM1> <NUM2>”, “<NUM1> & <NUM2>”, “<NUM1> | <NUM2>”, “<NUM1> p <NUM2>”, “<NUM1> pl. <NUM2>”, “<NUM1> || <NUM2>”, etc. Here, again, it would have been much more meaningful if they showed us some nearest neighbor retrievals instead.

So, where do we go from here? Is training ever bigger language models on ever bigger training data the way forward for an ever more general kind of intelligence? I don’t think so. One immediate difficulty is that unlike compute, it is hard to imagine how the training data can be increased another couple of orders of magnitude. As mentioned above, their training data already includes a snapshot of the entire web (and then some). Perhaps more book datasets can be added to the training data or some improvements can be made in data quality through better cleaning up of the web data (which is, in itself, a significant challenge), but I just don’t see how these can be combined into a few orders of magnitude increase in the effective data size.

In my view, a much more promising route would be to try to add some sort of grounding to these language models, e.g. through pictures or videos from the web. I think grounding is crucial for models to have a better understanding of the world; and anecdotal evidence from human experience suggests to me that these models perhaps wouldn’t need nearly as much grounding experience as they need text data to achieve a reasonably good grounded understanding of the world. This is because it seems to me that we humans acquire most of our grounding early in our development through interactions with a fairly limited environment, and acquire pretty much all the rest of our knowledge only indirectly, through social and cultural means, for example, by learning things from other people, or by reading about them in books, articles, web pages etc. (Anthropologist Joe Henrich makes a similar point in his book The Secret of Our Success). Current language models already seem to be highly efficient at extracting information from extremely large scale text data. To complement this already super-human ability, finding good grounding objectives and grounding data for training large-scale grounded language models would be a very promising and exciting direction, I think (see this, this, and this for some recent attempts in this direction).

Update (09/04/2020): I apparently missed this earlier, but OpenAI has made its intention to make GPT-3 a commercial product very clear right from the beginning (see here). They even mention the large size of the model as an excuse not to release it:

… many of the models underlying the API are very large, taking a lot of expertise to develop and deploy and making them very expensive to run.

So, it seems like my sinister interpretation above for OpenAI training a much larger model than was actually warranted was not too much off the mark!

Deep learning can make more use of available data

This is just a short post on something I’ve been thinking about lately. The argument is often made that deep learning needs stronger, better priors, usually in the form of architectural improvements. I’m not necessarily against this idea, however in this post I’d like to make the complementary case that even with the current architectures and training algorithms, deep learning can probably make more use of the available data, i.e. it can squeeze more juice out of available data. Why do I think so and how can deep learning achieve this? There are a couple of reasons that make me think so:

  1. Argument from cold posteriors: in Bayesian neural networks, it has been empirically observed that the best predictive performance is obtained not with the actual posterior, but with “cold posteriors”, which correspond to artificially manipulated posteriors that overcount the effect of the data and undercount the effect of the (usually generic) prior. Conversely, this suggests that current techniques in deep learning may be undercounting the potential of the data given that one has to resort to an artificial boosting of its effect in Bayesian neural networks.
  2. Argument from slow and brittle convergence to “natural” solutions: there is some interesting theoretical work suggesting that in some simplified problems, standard deep learning techniques will converge to what I would consider the “natural” solutions, but the convergence is painfully slow and brittle. Let me give two examples: Soudry et al. (2018) show that in logistic regression with separable data, gradient descent converges to the max margin solution (which can be considered as the natural solution for this type of problem), but convergence is extremely slow, i.e. something like O(\log \log t / \log t), and is brittle in the sense that it doesn’t hold for some popular adaptive gradient descent algorithms like Adam. Ji & Telgarsky (2019) show a similar result for logistic regression problems with non-separable data, but the convergence here is again extremely slow, i.e. the rate of convergence in direction to the max margin solution is again O(\log \log t / \log t). On the other hand, it is clear that the convergence to the max margin solution in these problems can be significantly sped up with simple data-dependent initialization schemes. In a similar vein, some prior works have suggested that important generalization properties of neural networks, such as their ability to generalize compositionally, is extremely sensitive to initialization, again implying that starting from a data-agnostic, generic initialization may not be optimal.
  3. Argument from empirical Bayes: how can deep learning make more use of the available data? A straightforward idea is the one I mentioned at the end of the last paragraph, i.e. using a data-dependent initialization scheme (I gave a simple example of this kind of scheme in a previous post). This approach is reminiscent of the empirical Bayes method in Bayesian statistics, which underlies a whole host of beautiful and surprising results like the Stein phenomenon. The basic idea in empirical Bayes is refusing to assume a non-informative, generic prior for the variables of interest (for a neural network, these could be the parameters, for instance), but estimating these priors from the data instead. You can see that this idea accords nicely with a data-dependent initialization scheme for neural network parameters. Empirical Bayes enjoys some appealing theoretical performance guarantees compared to common alternatives like maximum likelihood, which suggests that similar improvements may hold for data-dependent initialization schemes for neural networks as well.

In defense of pure learning

It is often claimed animals are more sample efficient, meaning that they can learn from many fewer examples (sometimes from single examples), than artificial neural networks and that one of the reasons for this is the useful innate inductive biases sculpted into the brains of animals through adaptations over evolutionary time scales. This claim is then usually followed by a prescriptive advice that we do something similar with artificial neural networks, namely, build in more innate structure into our models (or at least include an outer optimization loop in our models that would learn such useful inductive biases over longer time scales, as in meta-learning). Tony Zador and Gary Marcus made arguments of this sort recently.

In this post, I’d like to take issue with arguments of this sort. Actually, my objection to these kinds of arguments is already well-known. Zador has a section in his paper addressing precisely this objection (the section is titled “supervised learning or supervised evolution?”). So, what is the objection? The objection is that arguments of this sort conflate biology and simulation. They assume that the learning that happens in an artificial neural network is comparable to the learning that happens in a biological system over its individual lifespan. But there’s no good reason to think of artificial learning in this way. We should rather think of it as a combination of the learning that happens over an individual lifespan and the adaptations that take place over evolutionary time scales. When we think of artificial learning in this light, the sample efficiency argument in favor of animals falls by the wayside, because biological evolution has been running the most intense optimization algorithm in the biggest and the most detailed simulation environment ever (called “the real world”) for billions of years (so much for “one-shot” learning).

As I said, Zador is aware of this objection, so what is his response to it? As far as I can tell, he doesn’t really have a very convincing response. He correctly points out the differences between biological optimization and learning in artificial networks, but this doesn’t mean that they can’t generate functionally equivalent networks.

For example, Zador notes that biological optimization runs two nested optimization loops, the inner loop characterizing the learning processes in individual lifespans, the outer loop characterizing the adaptations over evolutionary time scales. This is similar to a learning paradigm called meta-learning in machine learning. And because of its similarity to biology, Zador is very much sympathetic to meta-learning. But in my mind the jury is still out on whether meta-learning has any significant advantages over other standard learning paradigms in machine learning. There are recent results suggesting that in practical problems one doesn’t really need the two separate optimization loops in meta-learning (one loop is all you need!). Moreover, if one trains one’s model in a sufficiently diverse range of problems (but crucially using a standard learning paradigm, such as supervised learning or reinforcement learning), “meta-learning” like effects emerge automatically without any need for two separate optimization loops (see also this beautiful new theory paper explaining some of these experimental results).

The core problem here, I think, is again conflating biology and simulation. Just because we see something in biology doesn’t mean we should emulate it blindly. Biology is constrained in many ways simulation isn’t (and vice versa). Of course it makes sense to use two separate optimization loops in biology, because individual lifespans are limited, but this isn’t true in simulation. We can run our models arbitrarily long on arbitrarily many tasks in simulation.

I think this (i.e. the mismatch between biology and simulation) is also why naive ways of emulating the brain’s innate inductive biases, like trying to directly replicate the concept of “cell types” in the brain is usually not very effective in artificial neural networks, because in my opinion these features are essentially consequences of the brain’s suboptimal learning algorithms (over developmental time scales), which means that it has to off-load a significant chunk of the optimization burden to evolution, which needs to craft these intricate cell types to compensate for the suboptimality of learning (over developmental time scales). Learning in artificial neural networks, on the other, is much more powerful, it is not constrained by all the things that biological learning is constrained by (for example, locality and limited individual lifespans), so it doesn’t really need to resort to these kinds of tricks (like different innate cell types) to easily learn something functionally equivalent over the individual lifespan.

Does the brain have to do deep credit assignment?

In other words, does the brain have to do something like backprop? For those who are already familiar with this literature, my short answer is that no, the brain doesn’t have to do, and it probably doesn’t do, deep credit assignment. In this post, I’d like to discuss two reasons that make me think so. I should note from the outset that these are not really “knock-out” arguments, but more like plausibility arguments.

I have to first clarify what exactly I mean by “deep credit assignment” or “something like backprop”. This is still not going to be very exact, but by this I basically mean a global credit assignment scheme that propagates precise “credit signals” from elsewhere in a deep network in order to compute a local credit signal. I thus include any gradient-based method (first-, second-, or higher-order) in this category, as well as imperfect, heuristic versions of it such as feedback alignment or weight mirrors. There are some borderline cases such as decoupled neural interfaces that compute credits locally, but also learn gradient estimates over longer timescales. I’m inclined to include these in the “deep credit assignment” category as well, but I would have to think a bit more carefully about it before doing so confidently.

Now moving on to the two reasons that make me think the brain probably doesn’t do “deep credit assignment”. The first reason is this. I think it is very natural to think that the brain should be doing something like backprop, because it is what works today! Deep neural networks trained with gradient descent have been enormously successful in a variety of important and challenging tasks, like object recognition, object detection, speech recognition, machine translation etc. But it is very important to also remember that these successes depend on the current hardware technology. The methods that work well today are methods that work well on current hardware. But the current hardware technology is constrained in a variety of ways that brains aren’t.

Take the example of memory. Fast memory (for example, on-chip memory) is extremely limited in size in the current chip technology (although this may be beginning to change with novel chip designs such as Graphcore’s IPU and Cerebras’s WSE). But, there is no reason to think that the brain is limited in the same way, because the brain has a completely different computational architecture!

What is the significance of this observation? Well, even today we know that there are promising alternatives to backprop training in deep nets that are currently intractable precisely because of memory constraints: I’m thinking, in particular, of the deep nets as Gaussian processes (GPs) perspective. Amazingly, this method doesn’t require any training in the usual sense. Inference is accomplished through a single forward pass just like in backprop-trained nets. The catch is that, unlike in backprop-trained nets, this forward pass doesn’t scale well with the data size: it requires manipulating huge matrices. To my knowledge, these methods currently remain completely intractable for, say, ImageNet scale data, where the said matrices become terabyte-sized (there’s this recent paper that carries out exact GP computations on a million data points, but they use low-dimensional datasets and they don’t use deep-net kernels in their experiments; exact GPs on high-dimensional data using deep-net kernels remain highly intractable, to the best of my knowledge).

As a side note here, I would like to express my gut feeling that there may be more tractable versions of this idea (i.e. training-free or almost training-free deep nets) that are not being explored thoroughly enough by the community. One simple idea that I have been thinking about recently is the following. Suppose we take the architecture of your favorite deep neural network and set the parameters of this network layer by layer using only information inherent in the training data, say, a large image dataset. This would work as follows. Suppose the network has k filters in its first (convolutional) layer. Then we can either crop k random patches of the appropriate size from the training images or maybe do something more intelligent, like exhaustively cropping all non-overlapping patches of the appropriate size and then doing something like k-means clustering to reduce the number of crops to k and then setting those to be the first layer filters. This then fixes the first layer parameters (assuming the biases are zero). We can iterate this process layer by layer, at each layer computing a number of clusters over the activations of the previous layer across the entire training data and then basically “pasting” those clusters to the layer weights. Note that in this scheme even though there is learning (after all we are using the training data), it is minimal and non-parametric (we’re doing only k-means, for example), and nothing like a gradient-based learning scheme. I think it would be interesting to find out how well one can do with a scheme like this that uses minimal learning and utilizes almost exclusively the prior information inherent in the network architecture and the training data instead.

So, this was the first reason why I think the brain probably doesn’t do something like backprop, to wit: backprop seems to me too closely wedded to the current hardware technology. My hunch is that there are many more interesting, novel (and probably more biologically plausible) ways of building intelligent systems that don’t require anything like backprop, but we’re currently not exploring or considering these because they remain intractable with current hardware (large scale GPs being one concrete example).

The second reason is that even with the current hardware we have some recent hints that purely local learning schemes that don’t require any deep credit assignment can rival the performance of backprop training in realistic tasks (and if the brain doesn’t have to do something, the chances are it’s not going to do it!). I’d like to mention two recent papers, in particular: Greedy layerwise learning can scale to ImageNet by Belilovsky et al. and Training neural networks with local error signals by Nokland and Eidnes. These papers both introduce completely local, layerwise training schemes and show that they can work as well as end-to-end backprop in standard image recognition problems.

Although these results are impressive, in a way I consider these as still initial efforts. I feel pretty confident that if more collective effort is put into this field, even better local training schemes will be discovered. So, to me these results suggest that the problems we typically solve with end-to-end deep learning these days may not be hard enough to require the full force of end-to-end backprop. Furthermore, with each new clever local learning trick, we will discover these problems to be even easier than we had imagined previously, so in the end coming full circle from a time when we had considered computer vision to be an easy problem, to discovering that it is in fact hard, to discovering that it isn’t that hard after all!

Update (01/16/2020): I just found out that the idea I described in this post for building a gradient descent-free image recognition model using k-means clustering was explored in some early work by Adam Coates and colleagues with promising results (for example, see this paper and this). They use relatively small datasets and simple models in these papers (early days of deep learning!), so maybe it is time to reconsider this idea again and try to scale it up.

Google’s new paper on large-scale weakly supervised learning

Last week, a group of researchers from Google posted a paper on arxiv describing an object recognition model claimed to achieve state-of-the-art (sota) results on three benchmarks measuring out-of-sample generalization performance of object recognition models (ImageNet-A, ImageNet-C, and ImageNet-P), as well as on ImageNet itself. These claims have yet to be independently verified (the trained models have not been released yet), but the reported gains on previous sota results are staggering:

The previous sota results on ImageNet-A, ImageNet-C, and ImageNet-P were reported in a paper I posted on arxiv in July this year, and they were achieved by a large model trained by Facebook AI researchers on ~1B images from Instagram using “weak” (i.e. noisy) labels, then fine-tuned on ImageNet (these models are called ResNeXt WSL models, WSL standing for weakly supervised learning). People who have worked on these benchmarks before will appreciate how impressive these numbers are. Particularly impressive for me are the ImageNet-A results. This benchmark itself was introduced in the summer this year and given the lackluster performance of even the best ResNeXt WSL models reported in my paper, I thought it would take a while to see reasonably high accuracies on this challenging benchmark. I was spectacularly wrong!

So, how did they do it? Their method relies on the old idea of co-training: starting from a model trained on a relatively small amount of high-quality labeled examples (in this case, ImageNet trained models), they infer labels on a much larger unlabeled dataset (in this case, the private JFT-300M dataset), then they train a model on the combined dataset (labeled + unlabeled) using random data augmentation during training, then they iterate this whole process several times.

In my arxiv paper posted back in July, I had confidently claimed that:

We find it unlikely that simply scaling up the standard object classification tasks and models to even more data will be sufficient to feasibly achieve genuinely human-like, general-purpose visual representations: adversarially robust, more shape-based and, in general, better able to handle out-of-sample generalization.

Although the Google paper doesn’t use a “standard” training paradigm, I would definitely consider it pretty close (after all, they simply find a much better way to make use of the large amount of unlabeled data, bootstrapping from a relatively small amount of labeled data, otherwise the setup is a pretty standard semi-supervised learning setup). So, I would happily admit that these results at least partially disprove my claim (it still remains to be seen to what extent this model behaves more “human-like”, I would love to investigate this thoroughly once the trained models are released).

This paper also highlights a conflict that I feel very often these days (also discussed in this earlier post). Whenever I feel pretty confident that standard deep learning models and methods have hit the wall and that there’s no way to make significant progress without introducing more priors, somebody comes along and shatters this idea by showing that there’s actually still a lot of room for progress by slightly improved (yet still very generic) versions of the standard models and methods (with no need for stronger priors). I guess the lesson is that we really don’t have very good intuitions about these things, so it’s best not to have very strong opinions about them. In my mind, the empirical spirit driving machine learning these days (“just try it and see how well it works”) is probably the best way forward at this point.

Another lesson from this paper is that bootstrapping, self-training type algorithms might be powerful beyond our (or at least my) paltry imagination. GANs and self-play type algorithms in RL are other examples of this class. We definitely have to better understand when and why these algorithms work as well as they seem to do.

Update (12/03/19): Another interesting paper from Google came out recently, proposing adversarial examples as a data augmentation strategy in training large scale image recognition models. Surprisingly, this doesn’t seem to lead to the usual clean accuracy drop if the BatchNorm statistics are handled separately for the adversarial examples vs. the clean examples and the perturbation size is kept small. Interestingly for me, the paper also reports non-trivial ImageNet-A results for the large baseline EfficientNet models. For example, the standard ImageNet-trained EfficientNet-B7 model has a reported top-1 accuracy of 37.7%. This is far better than the 16.6% top-1 accuracy achieved by the largest ResNeXt WSL model. These large EfficientNet models use higher resolution images as inputs, so it seems like just increasing the resolution gains us non-trivial improvements on ImageNet-A. This doesn’t diminish the impressiveness of the self-training results discussed in the main post above, but it suggests that part of the improvements there can simply be attributed to using higher resolution images.

kin2vec: learning kinship relations

I should preface this post by cautioning that it may contain some premature ideas, as I’m writing this mainly to clarify my own thoughts about the topic of this post.

In a reading group on program induction, we’ve recently discussed an interesting paper by Mollica & Piantadosi on learning kinship words, e.g. father, mother, sister, brother, uncle, aunt, wife, husband etc. In this paper, they are formalizing this as a probabilistic program induction problem. This approach comes with all the benefits of explicit program-like representations: compositionality, sample efficiency, systematic generalization etc. However, I’m always interested in neurally plausible ways of implementing these types of representations. The paper discussed an earlier work by Paccanaro & Hinton, which proposes a vector space embedding approach to the same problem. So, I decided to check out that paper.

Paccanaro & Hinton model people as vectors and relations between people as matrices (so \mathbf{y} = \mathbf{R} \mathbf{x} might mean “\mathbf{y} is the father of \mathbf{x}“). The idea is then to learn vector representations of the people in the domain, \mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n, and matrix representations of the relations between them, \mathbf{R}_1, \mathbf{R}_2, \ldots, \mathbf{R}_k, such that the distance between \mathbf{x}_i and \mathbf{Rx}_j is minimized if the corresponding relation holds between \mathbf{x}_i and \mathbf{x}_j, and maximized otherwise. This is (by now) a very standard approach to learning vector space embeddings of all sorts of objects. I have discussed this same approach in several other posts on this blog (e.g. see here and here).

Paccanaro & Hinton model each relation with a separate unconstrained matrix. Unfortunately, I think this is not really the best way to approach this problem, since it ignores a whole lot of symmetry and compositionality in relationships (which very likely negatively impacts the generalization performance and the sample efficiency of the model): for example, if \mathbf{y} is the father of \mathbf{x}, then \mathbf{x} is a son or a daughter of \mathbf{y}. Most primitive relations are actually identical up to inversion and gender. Other relations can be expressed as compositions of more primitive relations as in Mollica & Piantadosi.

So, I tried to come up with a more efficient scheme than Paccanaro & Hinton. My first attempt was to use only two primitive relations, \mathbf{A} (e.g. mother of) and \mathbf{W} (e.g. wife of) and to use matrix inversion and transpose to express the symmetric and opposite-gendered versions of a relationship. Here are some examples:

\mathbf{A}: mother

\mathbf{A}^\top: father

\mathbf{A}^{-1}: daughter

\mathbf{A}^{-\top}: son

\mathbf{A} \mathbf{A}^\top: father’s mother

\mathbf{A}^\top \mathbf{A}: mother’s father

\mathbf{A}^{-\top} \mathbf{A}: mother’s son (brother)

\mathbf{A}^{-1} \mathbf{A}^\top: father’s daughter (sister)

At this point, we run into a problem: mother’s daughter and father’s son always evaluate to self (the identity matrix), and this just doesn’t quite feel right. Intuitively, we feel that extensions of these concepts should include our sisters and brothers as well, not just us. The fundamental problem here is that we want at least some of these concepts to be able to pick out a set of vectors, not just a single vector; but this is simply impossible when we’re using matrices to represent these concepts (when applied to a vector, they will give back another vector). This seems like a fairly basic deficiency in the expressive power of this type of model. If anybody reading this has any idea about how to deal with this issue in the context of vector space models, I’d be interested to hear about it.

Another question is: assuming something like this is a reasonably good model of kinship relations (or similar relations), how do we learn the right concepts given some relationship data, e.g. (y, \; father, \; x), (z, \; mother, \; x) etc.? If we want to build an end-to-end differentiable model, one idea is to use something like a deep sparsely gated mixture of experts model where at each “layer” we pick one of our 7 primitive relations (indexed from 0 to 6):

\mathbf{I}, \mathbf{A}, \mathbf{A}^{-1}, \mathbf{A}^\top, \mathbf{W}, \mathbf{W}^{-1}, \mathbf{W}^\top

and the specific gating chosen depends on the input and output, \mathbf{x} and \mathbf{y}, g(\mathbf{x}, \mathbf{y}).

So, to give an example, if we allow up to 5 applications of the primitives, the output of the gating function for a particular input-output pair might be something like: g(\mathbf{x}, \mathbf{y})=[3, 0, 0, 0, 0] or a suitable continuous relaxation of this. This particular gating expresses the relationship, \mathbf{y} = \mathbf{A}^\top \mathbf{x}, whereas g(\mathbf{x}, \mathbf{y})=[3, 2, 0, 0, 0] would correspond to \mathbf{y} = \mathbf{A}^{-1} \mathbf{A}^\top \mathbf{x}. If we use a suitably chosen continuous relaxation for the discrete gating function, the whole model becomes end-to-end differentiable and can be trained in the same way as in Paccanaro & Hinton. We can also add a bias favoring the identity primitive over the others in order to learn simpler mappings (as in Mollica & Piantadosi). It would be interesting to test how well this model performs compared to the probabilistic program induction model of Mollica & Piantadosi and compared to less constrained end-to-end differentiable models.

Update (10/11/19): There’s some obvious redundancy in the scheme for representing compositional relations described in the last paragraph: applications of the identity don’t have any effect on the resulting matrix and successive applications of \mathbf{A} and \mathbf{A}^{-1} (or \mathbf{W} and \mathbf{W}^{-1}) cancel out each other. So, a leaner scheme might be to first decide on the number of non-identity primitives to be applied and generate a sequence of exactly that length using only the 6 non-identity primitives. The successive application of inverted pairs can be further eliminated by essentially hard-coding this constraint into g(\cdot, \cdot). These details may or may not turn out to be important.

The relative value of learning over memorizing

At the end of my last post, I mentioned the possibility that a large episodic memory might obviate the need for sophisticated learning algorithms. As a fun and potentially informative exercise, I decided to quantify this argument with a little experiment. Specifically, given a finite amount of data, I wanted to quantify the relative value of learning from that data (i.e. by updating the parameters of a model using that data) vs. just memorizing the data.

To do this, I compared models that employ a mixture of learning and memorizing strategies. Given a finite amount of “training” data, a k%-learner uses k% of this data for learning and memorizes the rest of the data using a simple key-value based cache memory. A 100%-learner is a pure learner that is typical in machine learning. For the learning model, I used a ResNet-32 and for the memory model, I used the cache model described in this paper. The predictions of a k%-learner are given by a linear combination of the predictions obtained from the learner (ResNet-32) and the predictions obtained from the cache memory:

prediction = w * prediction from the learning model + (1-w ) * prediction from the cache memory

where w is a hyper-parameter that is estimated separately for each k%-learner (I assume that the cost of learning a single hyper-parameter is negligible compared to the cost of learning the parameters of a model).

Suppose I already used up k% of the data for training my ResNet-32 model and this achieves a generalization accuracy of x. Now the question is: what should I do with the rest of the data? I can either use that data to continue to train my model, which leads to a 100% learner and let’s say this 100% learner achieves an accuracy of y; alternatively I can just memorize the remaining data by caching (with the help of my partially trained ResNet-32 model), which leads to a k%-learner and let’s say this k%-learner achieves an accuracy of z. Then, given that I have already used k% of the data for learning, the relative value of learning the remaining data over just memorizing it is defined by:

relative_value_of_learning(k) = (y-x) / (z-x)

that is, the improvement in accuracy achieved by a 100%-learner divided by the improvement in accuracy achieved by the k%-learner. A large value here indicates that learning is much more valuable than memorizing (i.e. it pays off to learn from the remaining data rather than just memorizing it) and a value of 1 would indicate that learning and memorizing are equally valuable. In the latter case, given that learning is usually computationally much more expensive than memorizing, we would probably be inclined to memorize rather than learn.

The following figure shows the relative_value_of_learning(k) as a function of k for the CIFAR-10 benchmark.

So, by this measure learning is ~10 times as valuable as memorizing in this task. There appears to be a decreasing trend in the value of learning as k becomes larger, but the data is a bit noisy (ideally, I should have run this simulation multiple times to get more reliable estimates).

Is this result surprising? It was surprising to me! I was expecting the relative value of learning to be smaller and the curve shown above to approach 1 much more quickly. So, now I am a bit less skeptical of the growing literature on biologically plausible analogues of backpropagation after this exercise. There is definitely a lot of value in learning good representations (much more value than I had initially thought).

Some caveats: this exercise is specific to a particular task and particular learning and memorizing models. The results might be different in different setups. Given that much of the effort in machine learning is directed toward coming up with better pure learning models (rather than better memory models), I expect that the relative value of learning estimated here is an overestimate, in the sense that one can improve the performance of memorizing models by using more sophisticated memory models than the simple key-value cache model assumed in this exercise.

Finally, an analysis like this should help us perform a cost-benefit analysis for learning vs. memorizing both in natural and artificial agents. Coming up with cost estimates is probably easier in artificial agents: for example, one can estimate the FLOPS involved in learning vs. memorizing a given amount of data; or one can include memory costs as well. Depending on our exact cost function, the optimal strategy would involve a specific mix, or a specific trajectory, of learning vs. memorizing during the lifetime of the agent.

A conjecture about “how the brain works”

I put “how the brain works” in quotes in the title, because it is in fact a misleading expression. There is no single way “the brain works.” There are different mechanisms the brain uses to solve different types of problems. My conjecture is specifically about object recognition type problems that current deep learning methods arguably excel at. As is well-known, the way current deep learning methods solve these types of problems is by training a very deep network with lots of labeled examples. The success of these methods has led many to think that the brain may be solving the same problem, or similar problems, in more or less the same way (same in terms of the final mechanism, not necessarily in terms of how the brain gets there). A manifestation of this way of thinking is the ongoing search for biologically plausible variants of the backpropagation algorithm, the “workhorse” of deep learning (see a recent review here), which is biologically patently unrealistic the way it is used in current deep learning models.

To be fair, there are good reasons to think like this. Deep learning models trained in this way are currently the best models of the ventral visual cortical areas in primates and just considering their zeroth-order performance, nothing else really even comes close to achieving near human or, in some cases, even super-human performance in sufficiently challenging object recognition tasks.

Of course, when we look a bit more closely, there are also very good reasons to be skeptical of the claims that these models are adequate models of the primate visual systems in general (and human visual system in particular). Chief among those reasons is the surprising (almost shocking) sensitivity of these models to adversarial and natural perturbations, very unlike human vision. Another reason to be skeptical is that when people actually do a more fine-grained analysis of how humans vs. deep vision models perform on realistic image recognition tasks, they find significant differences between how the two behave.

In this post, I would like to add one more reason to the skeptic’s arsenal and argue that current deep learning models for object recognition behave psychologically unrealistically and that our brains don’t seem to me to be solving object recognition type problems in the same way. My argument is exceedingly simple. It’s an argument from subjective experience and it goes as follows. When I recognize an object, it usually comes with a strong sense of novelty or familiarity. When I recognize a coffee mug, for instance, I don’t just recognize it as a mug, but as this particular mug that I have seen before (maybe even as my own mug) or as a novel mug that I haven’t seen before. This sense of familiarity/novelty comes automatically, involuntarily, even when we are not explicitly trying to judge the familiarity/novelty of an object we are seeing. More controlled psychological experiments also confirm this: humans have a phenomenally good memory for familiarity with a massive capacity even in difficult one-shot settings (see e.g. this classic study by Lionel Standing or this more recent study by Tim Brady and colleagues).

In other words, our recognitions have a strong and automatic episodic component. This episodic component is mostly lacking in current deep vision models. They don’t have a natural way of telling whether an object is novel or familiar at the same time as they are performing the recognition task.

There may be indirect ways of doing this in trained networks, for example, maybe novel and familiar –i.e. training and test– objects produce different activation distributions in a trained network. I actually don’t know if this is the case or not, but my point is just that current deep vision models do not perform this computation naturally and automatically as part of the computation they perform for recognizing the objects in the first place. This appears to me to be a big difference from the way we humans seem to do similar tasks.

So, how can we add this episodic component to the current generation of deep vision models? Shameless plug: I wrote a paper on this. The solution turns out to be really simple: just cache everything you can (ideally everything you have seen so far), using sufficiently high-level features (not too low-level stuff). And use the cache while making predictions. Retrieval from the cache is essentially a form of episodic memory. This is not even a novel solution. People have been proposing similar ideas in reinforcement learning and in language modeling (in fact, my paper was directly inspired by this last paper). In my paper, I showed that this cache-based model is incredibly robust to adversarial perturbations, so much so that when using only the cache memory to make predictions, I wasn’t able to generate any convincing adversarial examples, even with very strong attack methods (similar robustness results have been demonstrated in other papers as well). I strongly believe such cache-based models will also be much more adequate models of the human (and primate) visual system.

In a recent interview, Geoff Hinton said something quite similar to what I have tried to argue in this post about the difference between the current generation of deep learning models and the brain (if I interpret it correctly):

The brain is solving a very different problem from most of our neural nets… I think the brain isn’t concerned with squeezing a lot of knowledge into a few connections, it’s concerned with extracting knowledge quickly using lots of connections.

I think Hinton is fundamentally right here and I think a massive episodic memory is one of the basic mechanisms the brain uses to “extract knowledge quickly using lots of connections.” Among other things, I think one of the important implications of this point of view is that the current emphasis in some circles on trying to find sophisticated and powerful learning algorithms in the brain, which I alluded to above, may be misplaced. I actually think that backpropagation is probably much more sophisticated and powerful than anything we will find in the brain. Any learning algorithm in the brain is restricted in various ways machine learning algorithms don’t have to be (e.g. locality, respecting the rules governing different cell types etc.). On the other hand, in terms of the sheer number of neurons and the sheer number of connections, the human brain is massive compared to any model we have ever trained. It seems to me that we will soon find out that the algorithms relevant for the kind of machine the brain is are much more different than the machine learning algorithms relevant for today’s piddling models (piddling relative to the size of the human brain, of course). For example, I have always thought that hashing algorithms, essential for performing similarity search over very large sets of high-dimensional objects, should be at least as important and as relevant as backpropagation (and probably more) in our quest to understand the brain. And I have at least some corroborating evidence from the fly brain, of all places!

Surprising things you can learn from lavish data with no supervision

There is a well-known argument in psychology of language put forward by Noam Chomsky called the poverty of the stimulus argument. Roughly speaking, this is the claim that the linguistic data a child receives during language acquisition are vastly incomplete and insufficient to converge on the correct grammar. This claim is then used to bolster nativist arguments to the effect that large portions of grammar must be innate, already present in the child’s brain from birth in the form of a universal grammar.

There are several problematic aspects of this line of argument. The first and probably the most obvious one is that when people make a claim like this, they rarely, if ever, quantify how much linguistic data a child actually receives during the first few years of its life. Secondly, even if one does go ahead and quantify exactly the kind and amount of data a child receives during language acquisition, one still has to do the hard work and show that convergence to the correct grammar cannot happen (or is very unlikely to happen) with relatively weak, generic biases, but instead requires strong language-specific biases (i.e. that the biases have to be in the form of some kind of universal grammar). This can be tested either with architecture-agnostic methods such as Bayesian learners or with specific learning architectures like neural networks. Perfors et al., for example, show, through the Bayesian route, that linguistic input contains enough information to favor a hierarchical (as opposed to linear or flat) grammar with no prior bias favoring hierarchical grammars, directly refuting the often-made Chomskyan claim that language learners must have a strong innate prior bias in favor of hierarchical grammars. This just demonstrates how error-prone our intuitions  can be regarding the learnability or otherwise of certain structures from data without strong priors and the importance of actually checking what one can or cannot learn from given data.

As we are increasingly able to train very large models on very large datasets, I think we are beginning to grapple with a fundamental question about the nature of human-level or super-human intelligence: how far can we go with fairly generic, but very large architectures trained on very large datasets optimizing very generic objectives like prediction or curiosity? Is it possible to get all the way to human-level or super-human perceptual and cognitive abilities in this way, or alternatively is it necessary to incorporate strong inductive biases into the network architecture and we simply don’t know how to do this yet, both because we don’t know what the right inductive biases are and also because we don’t know how to implement them in our models? Personally, I would easily rate this as one of the most important outstanding questions in cognitive sciences and AI today.

My own thinking on this question has been evolving in the direction of the first possibility lately, i.e. that we can learn a lot more than we might naively imagine using fairly generic architectures and fairly generic unsupervised objectives. Part of the reason for this shift is a whole slew of recent work demonstrating that one can indeed learn highly non-trivial, surprising things even from relatively modest amounts of data using very generic network architectures and generic training objectives. In this post, I’d like to highlight a few of these recent results.

Building on earlier work demonstrating the power of transfer learning in language-related tasks, there has been a lot of progress this year in unsupervised pre-training of language models with large amounts of unlabeled data. For example, Radford et al. first pre-trained a large Transformer model on a language modeling task (i.e. given some prior context, predict the next token) with a relatively large dataset (see also this earlier paper by Howard & Ruder that implements essentially the same idea, but with a different model and different datasets). They then fine-tuned this pre-trained model on a variety of downstream supervised classification tasks and observed large gains in most downstream tasks over state of the art models specifically trained on those tasks. The dataset that they pre-trained the model on was a corpus of ~7000 books. Although this may seem like a big dataset (and it is big compared to the datasets typically used in NLP research), it is in fact a miniscule dataset relative to how large it could potentially be. For example, as of October 2015, Google Books contained ~25 million books, i.e. \sim O(10^7), which is about 4 orders of magnitude larger than the corpus used in this study. I’m not sure about the number of parameters in the Transformer model used in this paper, but my rough estimate would be that it must be \sim O(10^8). By comparison, the human brain has O(10^{15}) synapses. We’ve never even come close to running models this big. Now try to imagine the capabilities of a system with O(10^{15}) parameters trained on a corpus of O(10^{7}) or more books. It’s almost certain that such a system would shatter all state of the art results on pretty much any NLP benchmark that exists today. It would almost definitely lead to qualitatively recognizable improvements in natural language understanding and common-sense reasoning skills, just as today’s neural machine translation systems are recognizably better than earlier machine translation systems, due in large part to much bigger models trained on much bigger datasets.

Another conceptually similar model that has been shown to work even better is the more recent BERT model by Devlin et al. The two major innovations in this paper over the Radford et al. paper are (i) the use of a bidirectional attention model, instead of the unidirectional –strictly left-to-right– attention model used in Radford et al.; and (ii) the use of two novel unsupervised pre-training objectives. Specifically, they use a masked token prediction task, where the goal is to predict some masked word or words in a sequence, rather than the more standard left-to-right prediction task used in Radford et al. and other language modeling papers. This objective allows the bidirectional attention model to make use of both the left and the right context in order to predict the masked words. In addition, they also use a novel unsupervised next sentence prediction task, where the objective is to simply predict whether two given input sentences actually follow each other or not. Training examples for this objective can be easily generated from the corpus. The motivation behind this second objective is to force the model to learn the relationships between sentences, rather than relationships between lower-level units such as words. This second objective turns out to be crucial for significantly improved performance in question answering and natural language inference tasks. The datasets used for pre-training the model amount to the equivalent of some ~30000 books by my estimation. This is significantly bigger than the dataset used by Radford et al., however it’s still a few orders of magnitude smaller than the number of books that were available on Google Books as of October 2015.

The bidirectional BERT model significantly outperforms the Radford et al. model on the GLUE benchmark even after controlling for the model size. This suggests that although both the model architecture and the pre-training objectives in the paper are still quite generic, not all generic architectures and objectives are the same, and finding the “right” architectures and objectives for unsupervised pre-training requires careful thinking and ingenuity (not to mention a lot of trial and error).

Large-scale study of curiosity-driven learning is another paper that came out this year demonstrating the power of unsupervised learning in reinforcement learning problems. In this quite remarkable paper, the authors show that an agent receiving absolutely no extrinsic reward from the environment (not even the minimal “game over” type terminal reward signal) and instead learning entirely based on an internally generated prediction error signal can learn useful skills in a variety of highly complex environments. The prediction error signal here is the error of an internal model that predicts a representation of the next state of the environment given the current observation and the action taken by the agent. As the internal model is updated over training to minimize the prediction error, the agent takes actions that lead to more unpredictable or uncertain states. One of the important messages of this paper is, again, that not all prediction error signals, hence not all training objectives, are equal. For example, trying to predict the pixels or, in general, some low-level representation of the environment doesn’t really work. The representations have to be sufficiently high-level (i.e. compact or low-dimensional). This is consistent with the crucial importance of the high-level next sentence prediction task in the BERT paper reviewed above.

As the authors note, however, this kind of prediction error objective can suffer from a severe pathology, sometimes called the noisy TV problem (in the context of this paper, this problem can be more appropriately called a “pathological gambling” problem): if the agent itself is a source of stochasticity in the environment, it may choose to exploit this to always choose actions that lead to high-entropy “chancy” states. This strategy may in turn lead to pathological behaviors completely divorced from any external goals or objectives relevant to the task or tasks at hand. The authors illustrate this kind of behavior by introducing a “noisy TV” in one of their tasks and allowing the agent to change the channel on the TV. Predictably, the agent learns to just keep changing the channel, without making any progress in the actual external task, because this strategy produces high-entropy states that can be used to keep updating its internal model, i.e. an endless stream of “interesting”, unpredictable states (incidentally, this kind of pathological behavior seems to be common in humans as well).

Once more, this highlights the importance of choosing the right kind of unsupervised learning objective that would be less prone to such pathologies. One simple way to reduce this kind of pathology might be to yoke the intrinsic reward of prediction error to whatever extrinsic reward is available in the environment: for example, one may value the intrinsic reward only to the extent that it leads to an increase in the extrinsic reward after some number of actions.

To summarize the main points I’ve tried to make in this post and to conclude with a few final thoughts:

  • Unsupervised learning with generic architectures and generic training objectives can be much more powerful than we might naively think. This is why we should refrain from making a priori judgments about the learnability or otherwise of certain structures from given data without hard empirical evidence.
  • I predict that as we apply these approaches to ever larger models and datasets, the capabilities of the resulting systems will continue to surprise us.
  • Although fairly generic architectures and training objectives have so far worked quite well, not all generic training objectives (and architectures) are the same. Some work demonstrably better than others. Finding the right objectives (and architectures) requires careful thinking and a lot of trial and error.
  • One general principle, however, seems to be that one should choose objectives that force the model to learn high-level features or variables in the environment and the relationships between them. Understanding more rigorously why this is the case is an important question in my opinion: are low-level objectives fundamentally incapable of learning the kinds of things learnable through high-level objectives or is it more of a sample efficiency problem?
  • In addition to the examples given above, another great example of the importance of this general principle is the generative query network (GQN) paper by Deepmind, where the authors demonstrate the power of a novel objective that forces the model to learn the high-level latent variables in a visual scene and relationships between those variables. More specifically, the objective proposed in this paper is to predict what a scene would look like from different viewpoints given its appearance from a single viewpoint. This is a powerful objective, since it requires the model to figure out the 3d geometry of the scene, properties of the objects in the scene and their spatial relationships with each other etc. from a single image. Coming up with similar objectives in other domains (e.g. in language) is, I think, a very interesting problem.
  • Probing the capabilities of the resulting trained systems in detail to understand exactly what they can or cannot do is another important problem, I think. For example, do pre-trained language models like BERT display compositionality? Are they more or less compositional than the standard seq2seq models? Etc.

Update: Here‘s an accessible NY Times article on the recent progress in unsupervised pre-training of language models.

Update (01/07/19): Yoav Goldberg posted an interesting paper evaluating the syntactic abilities of the pre-trained BERT model discussed in this post on a variety of English syntactic phenomena such as subject-verb agreement and reflexive anaphora resolution, concluding that “BERT model performs remarkably well on all cases.”

Common initialization schemes for recurrent neural networks are likely suboptimal

Training of recurrent neural networks (RNNs) suffers from the same kind of degeneracy problem faced by deep feedforward networks. In fact, the degeneracy problem is likely compounded in RNNs, because empirically the spectral radius of W^k tends to be much larger than the spectral radius of  W_k W_{k-1} \ldots W_1 where W, W_1, \dots, W_k are random matrices drawn from the same ensemble (e.g. random Gaussian). I don’t know of a rigorous proof of this claim for random matrices (although, heuristically, it is easy to see that something like this should be true for random scalars: \sum_i^{k} w_i \sim \mathcal{N}(0, k), but kw \sim \mathcal{N}(0, k^2) for w, w_1, \ldots, w_k \sim \mathcal{N}(0,1) –this is essentially the difference between a true random walk vs. a biased random walk (I thank Xaq Pitkow for pointing this out to me)–; exponentiating both sides, we can then see that the product of k random scalars should be exponentially larger than the k-th power of a random scalar), but this empirical observation would explain why training linear RNNs would be harder than training deep feedforward networks and one can reasonably expect something like this to hold approximately in the nonlinear case as well.

Researchers have developed methods to deal with this degeneracy problem, hence to overcome training difficulties in RNNs. One of the most well-known of these methods is the identity initialization for the recurrent weight matrix. Others proposed constraining the weight matrix to always be orthogonal, instead of orthogonalizing it at initialization only. The logic behind both of these methods is that since orthogonal transformations are isometries of the Euclidean space, applying a bunch of these transformations in a cascade does not lead to a degeneration of the metric (by “degeneration” here, I mean the collapse of the metric along the overwhelming majority of the directions in the input space and the astronomical expansion of the metric along a very small number of remaining directions). This is guaranteed in the linear case and, again, one hopes and expects (with some justification) that things are not all that different in the nonlinear case as well. So, in other words, a sequence of orthogonal transformations propagate vectors in Euclidean space without distortion, i.e. without changing their norms or the distances between them.

This is all true and fine, however, this analysis ignores a crucial factor that is relevant in training neural networks, namely the effect of noise. Noise comes in both through the stochasticity of SGD and sometimes through direct noise injection (as in Dropout) for regularization purposes. It is a bit hard to precisely characterize the noise that arises due to SGD, but let us assume for the sake of simplicity that the noise is additive so that what we propagate in the end is some kind of “signal + noise”. Now, although it is true that orthogonal transformations propagate the signal without distortion, they also propagate the noise without distortion as well. But, ultimately, we probably want a transformation that maximizes something like the signal-to-noise ratio (SNR) of the propagated signal + noise. Then, it is no longer obvious that orthogonal transformations are optimal for this purpose, because, one can, for example, imagine transformations that would amplify the signal more than they would amplify the noise (hence distorting both the signal and the noise), thus yielding a better SNR than an orthogonal transformation.

And indeed it turns out that for linear systems with additive Gaussian noise, one can mathematically show that optimal transformations (in the sense of maximizing the total SNR of the propagated signal + noise) are not orthogonal. In fact, one can say something even stronger: any optimal transformation has to be non-normal (a normal matrix is a unitarily diagonalizable matrix; all orthogonal matrices are normal, but the reverse is not true). This is the main result of this beautiful and insightful paper by Surya Ganguli and colleagues. Perhaps the simplest example of an optimal transformation in this sense is a feedforward chain: W_{ij} = \alpha \delta_{i,j-1}, where \delta is the Kronecker delta function. This particular example maximizes the total SNR through a mechanism known as transient amplification: it exponentially amplifies the norm of its input transiently before the norm eventually decays to zero.

This brings me to the main message of this post: that the commonly used orthogonal initializations for recurrent neural networks are likely suboptimal because of the often neglected effect of noise. Another evidence for this claim comes from looking at the trained recurrent connectivity matrices in tasks that require memory. In this work (currently under review), we have shown that the trained recurrent connectivity matrices in such tasks always end up non-normal, with a feedforward structure hidden in the recurrent connectivity, even when they are initialized with an approximately normal matrix. How non-normal the trained matrices end up depend on a wide range of factors and investigating those factors was the main motivation for our paper. So, initializing RNNs with a non-normal matrix would potentially be a useful inductive bias for these networks.

In ongoing work, I have been investigating the merits of various non-normal initialization schemes for non-linear RNNs. One particular non-normal initialization scheme that seems to work quite well (and that is very easy to implement) is combining an identity matrix (or a scaled identity matrix) with a chain structure (which was shown by Ganguli et al. to be optimal in the case of a linear model with additive Gaussian noise). More details on these results will be forthcoming in the following weeks, I hope. Another open question at this point is whether non-normal initialization schemes are also useful for the more commonly used gated recurrent architectures like LSTMs or GRUs. These often behave very differently than vanilla recurrent networks, so I am not sure whether non-normal dynamics in these architectures will be as useful as it is in vanilla RNNs.

Update (06/15/19): Our work on a new non-normal initialization scheme for RNNs described in this post is now on arxiv. The accompanying code for reproducing some of the results reported in the paper is available in this public repository.

Simple inductive biases to make neural networks train faster and generalize better: two case studies

Perhaps the most important factor determining how quickly a neural network trains and how well it generalizes beyond the range of data it receives during training is the inductive biases inherent in its architecture. If the inductive biases embodied in the architecture match the kind of data the network receives, that can enable it to both train much faster and generalize much better. A well-known example in this regard is the convolutional architecture of the modern neural network models for vision tasks. The convolutional layers in these models implement the assumption (or the expectation) that the task that the model attempts to solve is more or less translation invariant (i.e. a given feature, of any complexity, can appear anywhere in the image). A more recent example is the relational inductive biases implemented in relational neural networks. Mechanistically, this is usually implemented with an inner-product like mechanism (sometimes also called attention) that computes an inner-product like measure between different parts of the input (e.g. as in this paper) or with a more complex MLP-like module with shared parameters (e.g. as in this paper). This inductive bias expresses the expectation that interactions between features (of any complexity) are likely to be important in solving the task that the model is being applied to. This is clearly the case for obviously relational VQA tasks such CLEVR, but may be true even in less obvious cases such as the standard ImageNet classification task (see the results in this paper).

Coming up with the right inductive biases for a particular type of task (or types of tasks) is not always straightforward and it is, in my mind, one of the things that make machine learning a creative enterprise. Here, by the “right inductive biases”, I mean inductive biases that (i) only exploit the structure in the problem (or problems) we are interested in and nothing more or less, but (ii) are also flexible enough that if the same model is applied to a problem that doesn’t display the relevant structure exactly, the model doesn’t break down disastrously (some “symbol”-based neural machines may suffer from such fragility).

In this post, I’d like to briefly highlight two really nice recent papers that introduce very simple inductive biases that enable neural networks to train faster and generalize better in particular types of problems.

The first one is from Uber AI: An intriguing failing of convolutional neural networks and the CoordConv solution. In this paper, the authors first observe that state of the art convolutional networks fail quite badly in tasks that require spatial coordinate transformations, for example, changing from Cartesian coordinates to image-based coordinates or vice versa (e.g. given the Cartesian coordinates (x,y), draw a square of a certain size centered at (x,y)). This may not be too surprising, since convolutional networks are explicitly designed to be translation-invariant, hence to ignore any spatial information, but the authors correctly note that ignoring spatial information completely (being rigidly translation-invariant) may not always be advisable (this may lead to failures of the type mentioned in (ii) above). It is rather much better to provide the model with the spatial information and let it figure out itself how much translation-invariant it needs to be in any particular task. This is exactly what the authors do. Specifically, they provide the spatial information in an explicit format through additional (fixed) channels that represent the Cartesian coordinates of each “pixel”. For image-based tasks, one thus needs only two additional channels, representing the x and y coordinates of each pixel. Pictorially, their scheme, which they call CoordConv, looks like this (Figure 3 in the paper):


That’s basically it. If the task at hand is highly translation-invariant, the model can learn to set the weights coming from those two Cartesian coordinate channels to small values; if the task at hand requires precise spatial information, on the other hand, the model can learn to utilize those channels appropriately. NLP people may recognize the conceptual similarity of this scheme to the positional encodings of items in sequence-based tasks. For the NLP people, we may thus summarize their contribution by saying that they extend the positional encoding idea from the temporal domain (in sequence-based tasks) to the spatial domain (in image-based tasks). It’s always a good idea to think about such exchanges between different domains!

The authors then go on to demonstrate that introducing a few of these CoordConv layers in standard architectures improves performance in a diverse range of tasks (but not in all tasks), including object detection, GAN training and Atari playing.

The second paper I’d like to highlight, called Neural Arithmetic Logic Units, starts from the observation that generic neural network architectures cannot generalize well in numerical tasks requiring arithmetic operations such addition, multiplication etc., even when they may successfully fit any given training data in such tasks (and sometimes they cannot even achieve that). The authors of this paper introduce very simple, elegant and easy-to-impement inductive biases that enable generic models (LSTMs and MLPs) to extrapolate from training data much better in such tasks. The basic idea is to “nudge” standard neural network operations (linear combination, pointwise nonlinearity etc.) to behave like arithmetic operators. For instance, for addition, they parametrize a dense weight matrix as:

\mathbf{W} = \tanh(\mathbf{V}) \circ \sigma(\mathbf{M})

where \circ denotes elementwise multiplication, and \sigma(\cdot) is the sigmoid nonlinearity. In the saturated regime, this parametrization encourages \mathbf{W} to have entries, -1, 0, 1, and so a linear combination using this kind of \mathbf{W}, i.e. \mathbf{W}\mathbf{x}, tends to behave like an addition or subtraction of its inputs (without scaling). In light of the preceding discussion, it is important to note here again that the model does not force this kind of behavior, but rather it merely facilitates it.

As an inductive bias for multiplication, they use the exponentiated sum of logs formulation:

\exp \mathbf{W} (\log (\mathbf{x} + \epsilon))

using the same matrix \mathbf{W} as above. This (approximately) expresses the multiplication of the elements in \mathbf{x}. A linear combination of these addition and multiplication operations gated by a sigmoid unit (called a NALU in the paper) then can function as either an addition or a multiplication operation (which can be learned as appropriate). One can then stack these operations to express, in principle, arbitrarily complex arithmetic operations.

This beautiful, simple idea apparently works fantastically well! I was quite impressed by the results in the paper. However, I would have liked to see (i) some results with more complex arithmetic operations than they report in the paper and also (ii) some results with tasks that do not have a strong arithmetic component to gauge how strong the introduced arithmetic inductive biases are. Again, the idea is to see whether, or how badly, the model fails when faced with a task without a strong arithmetic component. Ideally, we would hope that the model does not fail too badly in such cases.

Note: I will collect, and report here, examples of inductive biases, like the ones I discussed in this post, that I encounter in the literature, with brief descriptions of the bias introduced, how it is supposed to work and what kinds of problem it is intended to be applied to. To facilitate this, I tagged this post with the tag inductive biases and I will file similar posts under the same tag in the future.

Hyperreal numbers

Inspired by this thread from Ian Goodfellow, I recently finished reading a nice introductory textbook on nonstandard analysis, Infinitesimal Calculus, and I’d like to summarize here what I’ve learnt from the book.

There is a well-known construction of real numbers from rational numbers that defines real numbers as equivalence classes of Cauchy sequences of rational numbers. Two Cauchy sequences of rational numbers are counted as equal if their difference converges to zero in the Euclidean metric (choosing other metrics here gives rise to some interesting, alternative number systems, e.g. p-adic numbers).

In an analogous fashion, one can define a new number system starting from the reals this time. This new number system, called hyperreal numbers (\mathbb{HR}), is defined in terms of equivalence classes of sequences of real numbers. Two sequences of real numbers, (\alpha_n)_{n \in \mathbb{N}} and (\beta_n)_{n \in \mathbb{N}}, are considered equal (hence represent the same hyperreal number) iff the set of indices where they agree, \{ n| \alpha_n = \beta_n\}, is quasi-big. The technical definition of a quasi-big set is somewhat complicated, but the most important properties of quasi-big sets can be listed as follows:

  1. No finite set is quasi-big.
  2. If A and B are quasi-big sets, then so is A \cap B.
  3. If A is quasi-big and A \subseteq B, then B is also quasi-big.
  4. For any set A, either A or its complement is quasi-big.

Given this definition of hyperreal numbers, the following properties can be established relatively straightforwardly:

  1. \mathbb{R} \subset \mathbb{HR}, i.e. hyperreals strictly contain the reals (we map any real r to the equivalence class containing the sequence \alpha_n \equiv r).
  2. \mathbb{HR} contains an infinitesimal, i.e. a number \epsilon such that \epsilon > 0 , yet \epsilon < r for every positive real number r (to see this, consider the sequence \alpha_n \equiv \frac{1}{n+1}).
  3.  In a formal language L that is expressive enough to develop the entire calculus in, any given sentence is true with respect to \mathbb{R} iff it is true with respect to \mathbb{HR}. This is an extremely useful property that can be derived from Łoś’s Theorem.

At this point, you might be (and you should be!) wondering whether Properties 2 and 3 above are consistent with each other. Property 2 says there’s an infinitesimal number in \mathbb{HR}, but we know that there’s no infinitesimal number in \mathbb{R}. So, how is it possible that every sentence true in \mathbb{HR} is also true in \mathbb{R}? The answer is that the language L mentioned in Property 3, although powerful enough to allow us to do calculus, is a rather restricted language. In particular, it doesn’t allow us to define what a real number is. This is because the definition of a real number crucially relies on the completeness axiom. The completeness axiom cannot be expressed in L, because it requires talking about sets of numbers, something that turns out not to be possible in the language L. So, Property 2 cannot be expressed in the language L, hence L is not powerful enough to distinguish hyperreals from reals.

Here are some further useful properties of hyperreal numbers:

  • We mentioned above that there is at least one infinitesimal hyperreal. Are there more? Yes, definitely! There are in fact an infinite number of them: for, if \epsilon is an infinitesimal and r \neq 0 is a real number, then \epsilon r is also infinitesimal. Moreover, if \epsilon_1 and \epsilon_2 are infinitesimals, then so are \epsilon_1 + \epsilon_2 and \epsilon_1 \epsilon_2.
  • We say that a hyperreal is infinite iff it is either greater than all real numbers or smaller than all real numbers. A hyperreal is finite iff it is not infinite. It is easy to show that if \epsilon is infinitesimal, 1/\epsilon is an infinite hyperreal.
  • \mathbb{HR} contains infinite integers. The proof is very easy, using Property 3 above. We know that for any r in \mathbb{R}, there is an integer n greater than r (note that one can express this statement in the language L, because “x is an integer”, unlike “x is a real”, can be defined in L). Since this is true for \mathbb{R}, it must be true for \mathbb{HR} as well. In particular, there must be an integer greater than 1/\epsilon. But we just observed that 1/\epsilon is infinite, hence that integer must be infinite too! In fact, it immediately follows that there must be an infinite number of infinite integers.
  • We say that two hyperreals s and t are infinitely close, denoted by s \approx t, if s - t is infinitesimal or zero.
  • Let’s say that a hyperreal h is nonstandard if h is not real. Then, it’s easy to show that for any real r and nonstandard h, r + h (or r - h) is nonstandard.
  • If h is any finite hyperreal, then there exists a unique real r infinitely close to h.

One can develop the entire calculus using hyperreal numbers, instead of the reals. This is sometimes very useful, as some results turn out to be much easier to state and prove in \mathbb{HR} than in \mathbb{R} and we know, by Property 3 above, that we won’t ever be led astray by doing so. Just to give a few examples: in \mathbb{HR}, we can define a function f to be continuous at r iff p \approx r implies f(p) \approx f(r). In \mathbb{R}, on the other hand, we would have to invoke the notion of a limit to define continuity, which in turn is defined in terms of a well-known \epsilon, \delta-type argument.

Using hyperreals, derivatives are also very easy to define. The derivative of a function f at b is defined as:

f^\prime(b) \equiv \frac{f(b+\epsilon) - f(b)}{\epsilon}

where \epsilon is an infinitesimal (it’s easy to check that the definition doesn’t depend on which infinitesimal \epsilon is chosen). Again, the definition of a derivative in standard analysis would involve limits.

Taking the derivatives of specific functions is equally easy, since we can manipulate infinitesimals \epsilon just like we manipulate real numbers (again by Property 3 above).

One important difference between \mathbb{R} and \mathbb{HR} is that \mathbb{R} is complete (in fact, the construction of reals as equivalence classes of Cauchy sequences of rationals, which was mentioned at the beginning of this post, is known as the completion of rationals with respect to the Euclidean metric), whereas \mathbb{HR} is not complete. As mentioned before, this is not inconsistent with Property 3 above, because completeness cannot be expressed in the language L.

Bonus: I recently discovered this video of a great lecture by Terry Tao on ultraproducts. Ultraproduct is the required technical concept I glossed over above when I defined quasi-bigness. Most of the lecture is quite accessible and I recommend it to anybody who wants to learn more about this topic.

Why do neural networks generalize well?

One of the most important outstanding theoretical issues in machine learning today is explaining generalization in deep neural networks. Despite the fact that these models sometimes contain several orders of magnitude more parameters than the number of data points they are trained on, they do not seem to overfit the data easily and they are able to generalize successfully to new data. This highly over-parametrized regime makes the generalization error bounds derived with standard tools in statistical learning theory mostly vacuous. So, coming up with new theoretical ideas that can give non-vacuous quantitative bounds on the generalization performance of deep neural networks has been a top priority in theoretical machine learning in recent years.

There is a fair amount of empirical observation in the literature on what makes a neural network generalize better or worse. These empirical observations typically report significant correlations between the generalization performance of models on a test set and some measure of their noise robustness or degeneracy. For example, Morcos et al. (2018) observe that models that generalize better are less sensitive to perturbations in randomly selected directions in the network’s activity space. Similarly, Novak et al. (2018) observe that models that generalize better have smaller mean Jacobian norms, \langle \| \partial p(\mathbf{y}|\mathbf{x})/\partial \mathbf{x} \| \rangle, than models that don’t generalize well. The Jacobian norm at a given point can be considered as an estimate of the average sensitivity of the model to perturbations around that point. So, similar to Morcos et al.’s observation, this result amounts to a correlation between the model’s generalization performance and a particular measure of its noise robustness. Wu et al. (2017) observe that models that generalize better have smaller Hessian norms (Frobenius) than models that don’t generalize well. The Hessian norm can be considered as a measure of the total degeneracy of the model, i.e. the flatness of the loss landscape around the solution. Hence this result is in line with earlier hypotheses (by Hochreiter & Schmidhuber and by Hinton & Van Camp) about a possible connection between the generalization capacity of neural networks and the local flatness of the loss landscape. For two-layer shallow networks, Wu et al. also establish a link between the noise robustness of the models as measured by their mean Jacobian norm and the local flatness of the loss landscape as measured by the Hessian norm (in a nutshell, under some fairly plausible assumptions, the mean Jacobian norm can be upper-bounded by the Hessian norm -see Corollary 1 in the paper-).

Although these observations are all informative, they do not provide a rigorous quantitative explanation for why heavily over-parametrized neural networks generalize as well as they do in practice. Such an explanation would take the form of a rigorous quantitative bound on the generalization error of a model given what we know about the model’s various properties. As I mentioned above, standard theoretical tools have so far not been able to yield non-vacuous, i.e. meaningful, bounds on generalization error for models operating in the highly over-parametrized regime.

In this post, I’d like to highlight a recent paper by Wenda Zhou and colleagues that was able to derive non-vacuous generalization error bounds for ImageNet-scale models for the first time. Following prior work by Dziugaite & Roy, their starting point is PAC-Bayes theory. They leverage the compressibility of the trained deep neural nets to derive a non-vacuous (but still very loose) PAC-Bayes bound on the generalization error of large-scale image recognition models.

Roughly speaking, PAC-Bayes theory gives bounds on the generalization error that are of the following form: O(\sqrt{\mathrm{KL}(\rho(h|\mathcal{D}),\pi(h))/n}), where n is the number of training points, \rho(h|\mathcal{D}) is a data-dependent posterior distribution over the “hypotheses” (think of a distribution over neural network parameters) specific to a learning algorithm and training data \mathcal{D} and \pi(h) is a data-independent prior distribution over the hypotheses. The main challenge in coming up with useful bounds in this framework is finding a good prior \pi(h) that would minimize the \mathrm{KL}-divergence between the prior and the posterior in the bound. This basically requires guessing what kinds of structure are favored by the training algorithm, training data and model class we happen to use (previously, Dziugaite & Roy had directly minimized the PAC-Bayes bound with respect to the posterior distribution instead of guessing an appropriate prior, assuming a Gaussian form with diagonal covariance for the posterior, and came up with non-vacuous and fairly tight bounds for MNIST models. Although this is a clever idea, it obscures which properties of trained neural nets specifically lead to better generalization).

The main idea in the paper by Wenda Zhou and colleagues is to choose \pi(h) such that greater probability mass is assigned to models h with short code length, i.e. more compressible models. This implicitly expresses the hypothesis that neural networks trained with SGD on real-world datasets are more likely to yield compressible solutions. When we choose \pi(h) \propto 2^{-|h|_c} and \rho = \delta(h - \hat{h}), where |h|_c denotes the number of bits required to describe the model h according to some pre-specified compression scheme and \hat{h} denotes the compressed version of the trained model, we get a bound like the following (Theorem 4.1 simplified):

\mathrm{KL}(\rho,\pi) \leq O(|\hat{h}|_c)

This means that the more compressible the model is, the better the generalization error bound will be. The authors then refine this bound by making the compression scheme more explicit and also incorporating the noise robustness property of the trained networks into the bound. I won’t go into those details here, but definitely do check out the paper if you are interested. They then apply a state-of-the-art compression method (based on dynamic network surgery) to pre-trained deep networks to instantiate the bound. This gives non-vacuous but still very loose bounds for models trained on both MNIST and ImageNet. For example, for ImageNet, they obtain a top-1 error bound of 96.5\%, whereas their compressed model achieves an actual error of \sim 40\% on the validation data.

So, although it is encouraging to get a non-vacuous bound for an ImageNet model for the first time, there is still a frustratingly large gap between the actual generalization performance of the model and the derived bound, i.e. the bounds are nowhere near tight. Why is this? There are two possibilities as I see it:

  1. The authors’ approach of using a PAC-Bayes bound based on compression is on the right track, however the current compression algorithms are not as good as they can be, hence the bounds are not as tight as they can be. If true, this would be a practically exciting possibility, since it suggests that a few orders of magnitude better compression can be achieved with better compression methods currently not yet available.
  2. Just exploiting compression (or compression + noise robustness) will not be enough to construct a tight PAC-Bayes bound. We have to incorporate more specific properties of the trained neural networks that generalize well in order to come up with a better prior \pi(h).

The second possibility would imply the existence of models that have similar training errors and are compressible to a similar extent, yet have quite different generalization errors. I’m not aware of any reports of such models, but it would be interesting to test this hypothesis.

The softmax bottleneck is a special case of a more general phenomenon

One of my favorite papers this year so far has been this ICLR oral paper by Zhilin Yang, Zihang Dai and their colleagues at CMU. The paper is titled “Breaking the softmax bottleneck: a high-rank RNN language model” and uncovers an important deficiency in neural language models. These models typically use a softmax layer at the top of the network, mapping a relatively low dimensional embedding of the current context to a relatively high dimensional probability distribution over the vocabulary (the distribution represents the probability of each possible word in the vocabulary given the current context). The relatively low dimensional nature of the embedding space causes a potential degeneracy in the model. Mathematically, we can express this degeneracy problem as follows:

\mathbf{H} \mathbf{W}^\top = \mathbf{A}

Here, \mathbf{A} is an N \times M matrix containing the log-probability of each word in the vocabulary given each context in the dataset: \mathbf{A}_{ij} = \log p(x_j|c_i), where N is the number of all distinct contexts in the dataset and M is the vocabulary size; \mathbf{H} is an N \times d matrix containing the embedding space representations of all distinct contexts in the dataset, where d is the dimensionality of the embedding space; and \mathbf{W} is an M\times d matrix containing the softmax weights.

In typical applications, d \sim O(10^{2-3}) and M \sim O(10^5), so d is a few order orders of magnitude smaller than M. This means that while the right-hand side of the above equation can be full-rank, the left-hand side is rank-deficient: i.e. \mathrm{rank}(\mathbf{H} \mathbf{W}^\top) = d \ll M =\mathrm{rank}(\mathbf{A}) (we’re assuming that N is larger than M and d, which is usually the case). This means that the model is not expressive enough to capture \mathbf{A}.

I think it is actually more appropriate to frame the preceding discussion in terms of the distribution of singular values rather than the ranks of matrices. This is because \mathbf{A} can be full-rank, but it can have a large number of small singular values, in which case the softmax bottleneck would presumably not be a serious problem (there would be a good low-rank approximation to \mathbf{A}). So, the real question is: what is the proportion of near-zero, or small, singular values of \mathbf{A}? Similarly, the proportion of near-zero singular values of the left-hand side, that is, the degeneracy of the model, is equally important. It could be argued that as long as the model has enough non-zero singular values, it shouldn’t matter how many near-zero singular values it has, because it can just adjust those non-zero singular values appropriately during training. But this is not really the case. From an optimization perspective, near-zero singular values are as bad as zero singular values: they both restrict the effective capacity of the model (see this previous post for a discussion of this point).

Framed in this way, we can see that the softmax bottleneck is really a special case of a more general degeneracy problem that can arise even when we’re not mapping from a relatively low dimensional space to a relatively high dimensional space and even when the nonlinearity is not softmax. I will now illustrate this with a simple example from a fully-connected (dense) layer with relu units.

In this example, we assume that the input and output spaces have the same dimensionality (i.e. d=M=128, using the notation above), so there is no “bottleneck” due to a dimensionality difference between the input and output spaces. Mathematically, the layer is described by the equation: \mathbf{y}= f(\mathbf{W}\mathbf{x}), where f(\cdot) is relu and we ignore the biases for simplicity. The weights, \mathbf{W}, are drawn according to the standard Glorot uniform initialization scheme. We assume that the inputs x are standard normal variables and we calculate the average singular values of the Jacobian of the layer, \partial \mathbf{y}/\partial \mathbf{x}, over a bunch of inputs. The result is shown by the blue line (labeled “Dense”) below:


The singular values decrease roughly linearly up to the middle singular value, but drop sharply to zero after that. This is caused by the saturation of approximately half of the output units. Of course, the degeneracy here is not as dramatic as in the softmax bottleneck case, where more than 99\% of the singular values would have been degenerate (as opposed to roughly half in this example), but this is just a single layer and degeneracy can increase sharply in deeper models (again see this previous post for a discussion of this point).

The remaining lines in this figure show the average singular values of the Jacobian for a mixture-of-experts layer that’s directly inspired by, and closely related to, the mixture-of-softmaxes model proposed by the authors to deal with the softmax bottleneck problem. This mixture-of-experts layer is defined mathematically as follows:

\mathbf{y} = \sum_{k=1}^K g(\mathbf{v}_k^\top \mathbf{x})f(\mathbf{W}_k\mathbf{x})

where K denotes the number of experts,  g(\mathbf{v}_k^\top \mathbf{x}) represents the gating model for the k-th expert and f(\mathbf{W}_k\mathbf{x}) is the k-th expert model (a similar mixture-of-experts layer was recently used in this paper from Google Brain). The mixture-of-softmaxes model proposed in the paper roughly corresponds to the case where both f(\cdot) and g(\cdot) are softmax functions (with the additional difference that in their model the input \mathbf{x} is first transformed through a linear combination plus tanh nonlinearity).

The figure above shows that this mixture-of-experts model effectively deals with the degeneracy problem (just like the mixture-of-softmaxes model effectively deals with the softmax bottleneck problem). Intuitively, this is because when we add a number of matrices that are each individually degenerate, the resulting matrix is less likely to be degenerate (assuming, of course, that the degeneracies of the different matrices are not “correlated” in some sense, e.g. caused by the same columns). Consistent with this intuition, we see in the above figure that using more experts (larger K) makes the model better conditioned.

However, it should be emphasized that the mixture-of-experts layer (hence the mixture-of-softmaxes layer) likely has additional benefits other than just breaking the degeneracy in the model. This can be seen by observing that setting the gates to be constant, e.g. g(\cdot)= 1 / K, already effectively breaks the degeneracy:


I haven’t yet run detailed benchmarks, but it seems highly unlikely to me that this version with constant gates would perform as well as the full mixture-of-experts layer with input-dependent gating. This suggests that, in addition to breaking the degeneracy, the mixture-of-experts layer implements other useful inductive biases (e.g. input-dependent, differential weighting of the experts in different parts of the input space), so the success of this layer cannot be explained entirely in terms of degeneracy breaking. The same comment applies to the mixture-of-softmaxes model too.

Finally, I would like to point out that the recently introduced non-local neural networks can also be seen as a special case of the mixture-of-experts architecture. In that paper, a non-local layer was defined as follows:

\mathbf{y}_i = \frac{1}{C(\mathbf{x})} \sum_{j} g(\mathbf{x}_i,\mathbf{x}_j) f(\mathbf{x}_j)

with input-dependent gating function g(\cdot,\cdot) and expert model f(\cdot). The canonical applications for this model are image-based, so the layers are actually two-dimensional (a position dimension -represented by the indices i, j– and a feature dimension -implicitly represented by the vectors \mathbf{x}_j, \mathbf{y}_i etc.-), hence the inductive biases implemented by this model are not exactly the same as in the flat dense case above, but the analogy suggests that part of the success of this model may be explained by degeneracy breaking as well.

Note: I wrote a fairly general Keras layer implementing the mixture-of-experts layer discussed in this post. It is available here. Preliminary tests with small scale problems suggest that this layer actually works much better than a generic dense layer, so I am making the code available as a Keras layer in the hope that other people will be encouraged to explore its benefits (and its potential weaknesses). I am currently working on a convolutional version of the mixture-of-experts layer. The convolutional mixture of experts models have now been implemented and uploaded to the GithHub repository together with some examples illustrating how to use these dense and convolutional mixture of experts layers.

Comments on DeepMind’s grid cell paper

I just read the new Nature paper by DeepMind (& UCL) on grid cells and I’d like to share my thoughts about it. Just to briefly summarize the main points of the paper, they first show that grid cell like representations emerge naturally in fairly generic recurrent neural networks trained to do certain navigation tasks and secondly, they argue that this grid-like representation of space has benefits over alternative representations when artificial agents learn to perform relatively complex, realistic navigation tasks.

I really enjoyed reading the paper. For me, the most surprising result in the paper was the model’s apparent ability to reproduce some very subtle aspects of actual grid cell phenomenology in the brain, particularly the emergence of discrete spatial scales with geometric ordering. It is not often that one gets such discrete structures when one trains neural networks on any task. The default pattern I would have predicted would rather be continuous variation in the spatial scale of grid cells, i.e. something like a uniform or normal distribution over spatial scales.

Here are some other (more critical) thoughts on the paper:

1) I found the paper frustratingly short on explanations. Yes, hexagonal grids seem to emerge in the network, but why? Some form of regularization seems to be necessary to get grid-like representations in the network, but again why? Yes, grid-like representation of space seems to be more beneficial than a place field like representation in some navigation problems in RL, again why? Unfortunately, they don’t even put forward any plausible hypotheses as to why these results are observed.

2) An ICLR paper this year reported somewhat similar (but less detailed) results on the emergence of a grid like representation of space in trained recurrent nets. There are some important differences between the setups of these two papers; for instance, the ICLR paper trains the networks on the x, y coordinates of the agent’s position, whereas in the DeepMind paper, the training signal is a population code (place field) representation of space (plus a similar population code representation of head direction). The emergent grid in the ICLR paper was a square grid, I think; so they weren’t able to explain the hexagonal grid, nor the discrete geometrically-ordered spatial scales, as far as I know. So, this difference between the training signals seems to be important in explaining the different properties of the emergent grid. I think it is really important to understand which features of the training setup or the architecture are necessary to get the biologically more realistic hexagonal grids with geometrically spaced scales: is the linear read-out important? Is the network architecture important? What kind of training signal is necessary for the emergence of a hexagonal grid? Etc.

3) The paper also claims that when provided as input to a standard RL agent (A3C), periodic grid code representations of self-location and goal location are more beneficial than place field representations of the same variables. Again, not enough control experiments were run to determine why this is the case, but there are at least two possibilities, as far as I see: (i) it could be that a grid code affords a “better” representation of space in some sense, (ii) perhaps this effect has nothing to do with the representational power of a grid code, but it’s just a training effect (i.e. this kind of representation just happens to work better when training the policy network in A3C with SGD).

I find (i) unlikely: as the grid cells themselves were trained to match place cell responses with a linear read-out, they can’t really be a better representation of space than the place cells; it can’t be the case that grid cells provide more reliable information about space than place cells, for example, because place cells are the ground truth here. It may be argued that a grid code contains more easily decodable information than a place cell code, but I don’t see how this could be possible (if anything I would actually expect the opposite; this was, in fact, one of the messages of this paper by Sreenivasan & Fiete: simply put, although a grid code can encode a lot of information, decoding that information is not straightforward).

This leaves (ii) as the likely (and less exciting) explanation. One possible reason why a grid code would be better for training the policy network could be that the activations in the grid code are less sparse than activations in a place cell code (a straightforward prediction is that the performance of the place cell network should improve with wider place fields). A crucial control sorely missing in this section was using a smooth, random, non-periodic basis (sort of like the basis that emerges in the network when it’s trained without regularization). If that works as well as the grid code, that would suggest that there’s nothing special about periodic multi-scale codes like the grid code, similar results hold for any sufficiently dense and smooth code (a less exciting conclusion than what the paper currently seems to be concluding).

4) In the RL experiments, the grid-cell network wasn’t trained end-to-end, different components of the network were trained with different objectives (for example, see Extended Data Fig. 5). Do grid cells still emerge if the entire network is trained end-to-end with the reward maximization (RL) objective? Alternatively, let’s assume that we plug a pre-trained grid-cell module to a policy network and then fine tune it with the RL objective. Are the grid cell representations in the grid cell module stable in the face of this fine-tuning? If not, that would be problematic for the claims of the paper.

5) This is really a more minor point compared to the first four points above. In training the grid-like cells in the network, the training objective they use is to match the responses of some place cells (plus some head-direction cells) which are assumed to have developed already. I think it would be much nicer and more realistic to have an unsupervised objective that would lead to the emergence of both grid cells and place cells. For justification, the authors note that place cells seem to develop earlier than grid cells; so coming up with an unsupervised objective that would reproduce this developmental order would be very interesting.

A simple cache model for image recognition

I just posted a new paper on arxiv titled “A Simple Cache Model for Image Recognition”. In this paper, I make a very basic observation: in image recognition tasks, the layers of a deep network close to the output layer contain independent, easily extractable class-relevant information that is not already contained in the output layer itself. I then propose to read out this extra information using a simple continuous key-value cache memory that is directly inspired by Grave et al. (2017), who add a similar continuous cache memory to recurrent sequence-to-sequence models.

The really nice thing about this model is that by properly setting only two hyper-parameters, it is possible to significantly improve the accuracy of a pre-trained model at test time. For example, here are some error rates on the CIFAR-10 and CIFAR-100 test sets:


where ResNet32 (\lambda=0) is the baseline model without a cache component, ResNet32-Cache3 is a model that linearly combines the predictions of the cache component and the baseline model (as you may have guessed, \lambda represents the mixture weight of the cache component), and ResNet32-Cache3-CacheOnly (\lambda=1) is a model that uses the predictions of the cache component only.

I also found that a cache component substantially improves the robustness of baseline models against adversarial attacks. Here are the classification accuracies of the same three models above on adversarial images generated from the CIFAR-10 test set by four different attack methods applied to the baseline ResNet32 model:


Interestingly, the cache-only model is more robust against adversarial perturbations than the other models. It is intuitively easy to understand why cache models improve robustness against adversarial examples: they effectively extend the range in the input space over which the model behaves similarly to the way it behaves near training points and recent work has shown that neural networks behave more regularly near training points than elsewhere in the input space (for example, as measured by the Jacobian norm or the number of “linear regions” in the neighborhood of a given point).

Indeed, I confirmed this by looking at the Jacobian of the models at the test points:


Both cache models reduce the mean Jacobian norm over the baseline model (a). The two cache models, however, have somewhat different characteristics: the linear-combination model (ResNet32-Cache3) has the smallest first singular value but slightly increased lower-order singular values compared to the baseline model, whereas the cache-only model reduces the singular values (and the Jacobian norms per trial) more consistently. This pattern appears to be related to the differential generalization behavior of the two models: the linear-combination cache has the higher test accuracy, but the cache-only model is more robust against adversarial perturbations. So, the conjecture is that within-sample generalization performance is mostly determined by the first singular value (or the first few singular values), whereas the out-of-sample generalization performance, e.g. sensitivity of the models against adversarial perturbations, is also significantly affected by the lower-order singular values.

Short-horizon bias in meta-learning

In my last post, I mentioned meta-learning as a promising approach for learning useful inductive biases in neural networks. In meta-learning, several steps of a standard optimization problem are unrolled and “baked” into the computation graph. One then performs backpropagation over the entire computation graph to optimize particular parameters or hyper-parameters of interest, for example, hyper-parameters controlling the learning rate or momentum schedules or the initial parameters of the network.

One potential problem with this approach is that it becomes computationally prohibitive to unroll, and bake into the computation graph, more than a relatively small number of optimization steps, compared to the number of optimization steps or iterations one would normally perform for training the model. One might then worry that parameters or hyper-parameters optimized over this shorter horizon might not generalize well to longer horizons (the longer horizon problem is ultimately the problem we are interested in).

A new ICLR paper by Yuhuai Wu, Mengye Ren and colleagues argues that such worries are justified, at least for a certain class of meta-learning problems. They consider the problem of meta-learning the learning rate and momentum schedules in neural network training. In the first part of the paper, they approach this problem analytically by considering a noisy quadratic loss function. For this particular loss function, it turns out that one can estimate both the optimal learning rate and momentum schedules (using Theorem 1 in the paper), as well as the “greedy-optimal” learning rate and momentum schedules that minimize the one-step ahead loss function at each time step (using Theorem 2). The authors show that although the greedy-optimal strategy outperforms the optimal schedule initially during training, its final performance is substantially worse than that of the optimal strategy. The reason appears to be that the greedy-optimal strategy reduces the learning rate prematurely to be able to decrease the loss along high-curvature directions in the loss landscape quickly enough.

In the second part of the paper, a gradient-based hyper-parameter meta-learning method is shown to suffer from a similar short-horizon bias where meta-learning of the hyper-parameters controlling the learning rate schedule (i.e. the initial learning learning rate and a hyper-parameter controlling the learning rate decay over time) with a small number of unrolled optimization steps, i.e. over a short horizon, again leads to learning rates that are reduced much too quickly compared to the optimal schedule, resulting in severely suboptimal final performances.

Here are some thoughts and questions inspired by this work:

  • In the inner loop of their meta-learning problems, the authors only consider vanilla SGD with momentum. It is a bit unfortunate that they don’t consider the Adam algorithm (or similar algorithms) in the inner loop. I suspect that using Adam in the inner loop might ameliorate the short-horizon bias significantly, because Adam is supposed to be less sensitive to hyper-parameter choices (one might then object, with some justification, that this would be a case where meta-learning is not really needed).
  • Because the bias reported in the paper seems to be consistent, i.e. always in the direction of reduced learning rates, a quick fix one could imagine would be to introduce a prior that would counteract this bias by, for example, discouraging small learning rates. I’m not quite sure what the best prior to use would be in this context and I don’t know how well this would work in practice compared to the optimal learning rate schedules, but it seems like a straightforward fix that could be worth trying.
  • As the authors note, the short-horizon bias becomes less significant (or might even disappear) when the optimization problem is either deterministic or well-conditioned. We have previously argued that skip connections reduce degeneracies in neural networks, thereby making the optimization problem better-conditioned. It’s likely that other methods such as batch normalization have similar effects. This suggests that more realistic architectures using skip connections and batch normalization might be less prone to the short-horizon problem uncovered in this paper. It would be interesting to investigate this.
  • It would also be interesting to see if other kinds of meta-learning problems, for example, meta-learning a good initialization for the model parameters (as in the MAML algorithm) suffer from similar short-horizon biases.

Human priors: you can’t have your cake and eat it too

This is just a brief post to highlight a brilliant paper I have read recently and to share some thoughts sparked off by the paper. The paper in question is Investigating human priors for playing video games by Rachit Dubey and colleagues. In this paper, the authors address a very basic question: What kinds of specific prior knowledge do humans bring to the task of learning a new video game that make them much more efficient (in terms of the amount of training experience required) than neural networks learning a similar video game through deep RL algorithms.

Their approach to this question is thoroughly (and admirably) empirical. By designing various versions of the basic game that eliminate different kinds of structure in the game, they are able to not only identify, but also quantify the relative importance of, various different regularities or meaningful structures in the game for human players. The idea is that if the removal of a particular meaningful structure or regularity in the game severely affects the learning performance of human players, then that must be an important regularity that humans rely on heavily in learning the game. Conversely, if the removal of a meaningful structure or regularity does not affect the learning performance of human players, it is reasonable to assume that that regularity is not an important part of the prior knowledge that human players bring to the learning of the new game.

Using this logic, the authors show that (i) knowledge about how to interact with objects (e.g. one can climb a ladder by pressing the up key as opposed to zigzagging left and right, one can jump over monsters by pressing the up and right (or left) keys etc.), (ii) knowledge about object affordances (e.g. platforms support walking and ladders support climbing) and object semantics (e.g. one has to avoid fires and angry-looking monsters), (iii) the concept of an object as distinct from the background and a prior that expects visually similar things to behave similarly, are all parts of the prior knowledge that people utilize (in the order of increasing importance) in learning a new game.

One might naively think that it should be a great idea to build all this prior knowledge in our neural networks (if only we knew how to do that!). That would be expected to greatly increase the sample efficiency of neural networks when they encounter similar problems. However, one has to be careful here, since building in any kind of prior knowledge always comes at a cost: namely, if the problems encountered by the model do not conform to the assumed prior, the prior, rather than improving the sample efficiency, may in fact worsen it. The authors are (again, admirably) very clear about this potential drawback of prior knowledge (which they illustrate with a simple example video game):

However, being equipped with strong prior knowledge can sometimes lead to constrained exploration that might not be optimal in all environments… Thus, while incorporating prior knowledge in RL agents has many potential benefits, future work should also consider challenges regarding under-constrained exploration in certain kinds of settings.

Building in too many human priors could also make models vulnerable to a sort of “adversarial attack” where an adversary might design “unnatural” versions of a task that would be difficult to solve for a model with a lot of human priors (similar to the games created in the paper), a sort of inverse-CAPTCHA. A less constrained model with fewer human priors would be less vulnerable to such attacks. Besides, although human priors (especially more general ones such as the concept of an object or the visual similarity prior) are probably broadly useful in vision- or language-related problems, increasingly many problems we face today in applied sciences are not obviously amenable to any kind of useful human prior: problems in bioinformatics or molecular design come to mind, for example. Trying to prematurely incorporate human priors in our models (whatever they might be) may hinder performance in such cases. Furthermore, human priors (especially more specific ones such as those that arise in naive physics or in naive psychology) are often plain wrong, in which case incorporating them in our models doesn’t make any sense.

This is where the importance of meta-learning comes in, I think. Rather than trying to build in some assumptions a priori, in a lot of cases, meta-learning some prior knowledge or assumptions that would be broadly useful for a particular class of problems would be a more promising approach. Although there has recently been some interesting work in this direction, the prior knowledge or assumptions meta-learned in these works are quite primitive, e.g. meta-learning a good initialization. Meta-learning more substantial prior knowledge (for example, in the form of a model architecture) very much remains an interesting open problem.

Compositional embeddings, again

Continuing on the topic of the last post, I would like to briefly discuss yet another proposal for learning vector space embeddings for compositional objects. In the last post, I discussed a method that only used vector addition and circular convolution (which really boils down to elementwise multiplication, when everything is expressed in the Fourier domain) to construct vector space embeddings of compositional objects. Although this kind of embedding satisfies certain properties that we might want in a desirable vector space embedding (e.g. mapping objects with different structures onto the same space), it’s unlikely that it would be the optimal embedding when these vectors are used in the context of a particular task such as sentiment prediction. This is because this kind of embedding does not have any learnable parameters and hence is independent of the task.

At least in some cases, however, one might want to have more flexible embeddings that are optimized for a particular task, or set of tasks. Moreover, capturing the meaning of a word by a single vector may not be entirely appropriate either. Some words, for example, do not have a clear meaning on their own, but rather they have what we might call operator semantics: they act to modify the meanings of other words they are attached to in certain ways. Adverbs, for instance, usually function in this way. It seems more appropriate to model this aspect of meaning in terms of operators (e.g. matrices) rather than vectors. This is, in a nutshell, what is proposed in Socher et al. (2012).

Each word is assigned both a vector representation capturing its usual “content” semantics and a matrix representation capturing its operator semantics. Given two words represented by the tuples (a,A) and (b,B), their composition is represented by the pair (p,P) where:

p = g\Big(W \begin{bmatrix} Ba \\ Ab \end{bmatrix}\Big)      and      P = W_{\mathrm{M}} \begin{bmatrix} A \\ B \end{bmatrix}

Here, W and W_{\mathrm{M}} are parameters shared across all compositions and optimized for a particular task, e.g. sentiment prediction.

The assumption of parameter sharing across all compositions in this model is somewhat problematic, since Adv-Adj compositions do not really work the same way as NP-VP compositions, for instance. So, a more flexible semantic composition model that distinguishes between different types of composition could potentially be more powerful.

Secondly, it isn’t really necessary to capture the operator semantics of a word with a matrix. Although they use low-rank matrices for this purpose, it’s possible to do this even more efficiently. All that is needed is to have separate parts of a word vector represent the content vs. operator semantics (and possibly other kinds of semantics) and to have them interact in a particular, constrained way with the content and operator parts of another word vector. For example, we can have a vector representation of a word like: [a, \alpha] where the first part of the vector a captures the content semantics, and the second part \alpha captures the operator semantics. After composing two words, [a, \alpha] and [b, \beta], we get the following content and operator parts:

p = g\Big(W \begin{bmatrix} f(a,\beta) \\ f(b,\alpha) \end{bmatrix}\Big)      and      \pi = W_{\mathrm{M}} \begin{bmatrix} \alpha \\ \beta \end{bmatrix}

where we replaced the matrix-vector products Ab with a more general function f(a,\beta), which can be a shallow MLP, for instance. The important point is to note that f(\cdot) is encapsulated in the sense that it only allows the content part of a word vector to interact with the operator part of another vector (just as in the original matrix-vector product model above).