Severely Theoretical

Computational neuroscience and machine learning

Tag: machine learning

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.


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.

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.

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 negative) 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) 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, which is what I suspect, 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 desirable.

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).

Holographic reduced representations and associative LSTMs

Vector space representations of words (also known as word embeddings) are familiar. By mapping words onto a vector space equipped with a metric, we capture the similarity relationships between them, reduce their dimensionality (in contrast to a high-dimensional representation such as one-hot encoding), and enable their integration into larger, end-to-end differentiable models. How can we achieve the same thing for compositional objects? There are various ways of doing this, each with its own strengths and weaknesses. In an earlier post, I discussed a particular method that relies on tensor products to construct representations of compositional objects. However, this method quickly runs into a dimensionality problem: the number of dimensions required to represent a compositional object with n constituents grows exponentially with n. Moreover, two objects with different numbers of constituents live in different spaces, hence are not directly comparable. In this post, I would like to discuss an alternative proposal first introduced by Tony Plate, called holographic reduced representations (HRR), that deals with both of these problems.

In this proposal, an association between two items, \mathbf{a} and \mathbf{b}, is represented by their circular convolution. Because convolution corresponds to product in the Fourier domain, we can express this as the inverse transform of the elementwise multiplication of the discrete Fourier transforms of the vectors:

\mathbf{a} \circledast \mathbf{b} = \mathcal{F}^{-1}([r_1s_1 \exp(i(\phi_1 + \chi_1)), \ldots, r_ns_n \exp(i(\phi_n + \chi_n))])

with \tilde{\mathbf{a}} = [r_1 \exp(i \phi_1), \ldots, r_n \exp(i \phi_n)] representing the discrete Fourier transform of \mathbf{a} and \tilde{\mathbf{b}} = [s_1 \exp(i \chi_1), \ldots, s_n \exp(i \chi_n)] representing the discrete Fourier transform of \mathbf{b}. It is easy to see that the circular convolution of two vectors, unlike their tensor product, successfully deals with the dimensionality problem: the circular convolution of two vectors has the same dimensionality as the vectors themselves. Moreover, circular convolution behaves very similarly to scalar multiplication, making the manipulation of the items extremely easy and efficient. For example, circular convolution is both commutative and associative, just like scalar multiplication.

If we want to represent multiple associations, we can just add them up via vector addition:

\mathbf{c} = \mathbf{a}_1 \circledast \mathbf{b}_1 + \ldots + \mathbf{a}_k\circledast \mathbf{b}_k

To retrieve an item \mathbf{b}_1 associated with an item \mathbf{a}_1 (which can be thought of as the “key” associated with \mathbf{b}_1), we take the circular convolution with the inverse of \mathbf{a}_1, where the inverse of a vector \mathbf{a} is defined through \tilde{\mathbf{a}^{-1}} = [s_1^{-1} \exp(-i \chi_1), \ldots, s_n^{-1} \exp(-i \chi_n)]:

\mathbf{a}_1^{-1} \circledast \mathbf{c} = \mathbf{a}_1^{-1} \circledast (\mathbf{a}_1 \circledast \mathbf{b}_1 + \ldots + \mathbf{a}_k\circledast \mathbf{b}_k) = \mathbf{b}_1 + noise

Here, the noise grows linearly with the number of items (or associations) in memory and will be small if the keys are chosen sufficiently orthogonally (in the paper, only random normal keys are considered; however, orthogonal keys would be better, since although any pair of random normal keys are likely to be orthogonal, linear dependencies between a number of such keys also become highly likely; orthogonal keys, on the other hand, eliminate such linear dependencies as well).

Various different types of compositional objects can be readily represented in this framework. Here are some examples given in the paper.

Representing a sequence

There are multiple ways to represent a sequence like \mathbf{a}\mathbf{b}\mathbf{c} in this scheme. We can, for instance, do something like:  \mathbf{a} + \mathbf{a} \circledast \mathbf{b} + \mathbf{a} \circledast\mathbf{b} \circledast \mathbf{c}.

Representing a stack

To represent a stack with items \mathbf{x}_1, \ldots, \mathbf{x}_k with \mathbf{x}_1 on top, we can do \mathbf{s} = \mathbf{x}_1 + \mathbf{p} \circledast \mathbf{x_2} + \ldots + \mathbf{p}^n \circledast \mathbf{x}_n where \mathbf{p} is an arbitrary vector. Then, we can define push and pop operations on this stack as follows:

push(\mathbf{s}, \mathbf{x}) = \mathbf{x} + \mathbf{p} \circledast \mathbf{s}                      (push item \mathbf{x} on top of stack \mathbf{s})

top(\mathbf{s}) = clean - up(\mathbf{s})                      (return the top item)

pop(\mathbf{s}) = (\mathbf{s} - top(\mathbf{s})) \circledast \mathbf{p}^{-1}                      (pop the top item)

Chunking of sequences

To represent a long sequence like \mathbf{abcdefgh}, we can chunk it into smaller sequences as follows:

\mathbf{s}_{abc} = \mathbf{a} + \mathbf{a} \circledast \mathbf{b} + \mathbf{a} \circledast \mathbf{b} \circledast \mathbf{c}

\mathbf{s}_{de} = \mathbf{d} + \mathbf{d} \circledast \mathbf{e}

\mathbf{s}_{fgh} = \mathbf{f} + \mathbf{f} \circledast \mathbf{g} + \mathbf{f} \circledast \mathbf{g} \circledast \mathbf{h}

\mathbf{s}_{abcdefgh} = \mathbf{s}_{abc} + \mathbf{s}_{abc} \circledast \mathbf{s}_{de} + \mathbf{s}_{abc} \circledast \mathbf{s}_{de} \circledast \mathbf{s}_{fgh}

Variable binding

Binding of \mathbf{a} to variable \mathbf{x} and binding of \mathbf{b} to variable \mathbf{y} can be achieved with: \mathbf{t} = \mathbf{x}\circledast \mathbf{a} + \mathbf{y} \circledast \mathbf{b}.

Frame (slot/filler) structures

The thematic structure of a sentence like “Mark ate the fish” can be represented by: \mathbf{s}_1 = \mathbf{eat} + \mathbf{agt}_{eat} \circledast \mathbf{mark} + \mathbf{obj}_{eat} \circledast \mathbf{thefish},

where \mathbf{agt}_{eat} and \mathbf{obj}_{eat} represent the thematic roles of “agent” and “object” for the eat frame, respectively.

Recursive frames

A sentence can fill a slot in another sentence. For instance, the thematic structure of the sentence “Hunger caused Mark to eat” can be represented by: \mathbf{s}_2= \mathbf{cause} + \mathbf{agt}_{cause}\circledast \mathbf{hunger} + \mathbf{obj}_{cause}\circledast \mathbf{s}_1.

Representing types, tokens, features

Each token of a given type can be assigned a unique identifier. For instance, \mathbf{mark} = \mathbf{being} + \mathbf{person} + \mathbf{id}_{mark}, where \mathbf{id}_{mark} denotes the unique identifier for “Mark”.

Associative LSTMs

Danihelka et al. (2016) use an HRR memory within a standard LSTM architecture to introduce an inductive bias in the model toward learning classic data structure algorithms like stacks and queues. They call this new model an “associative LSTM”. In the associative LSTM model, input and output key vectors, r_i and r_o, are introduced as linear functions of the inputs and hidden states (analogous to gates in standard LSTMs, but without the nonlinearity). The cell state is then updated as follows:

c_t = g_f \odot c_{t-1} + r_i \circledast (g_i \odot u)

and the output of the model is computed through:

h_t = g_o \odot \phi(r_o \circledast c_t)

Here \phi(\cdot) is a new nonlinearity they introduce. The only difference from the standard LSTM equations is the convolution with the input and output memory keys, r_i and r_o, in these two equations. They also use multiple copies of each item in memory (with different keys) to reduce the noise arising from interference, but I will not go into the details of how exactly this is done (the above equations are for a version that uses a single copy). They show that the associative LSTM works better than standard LSTMs and a few other LSTM variants in several tasks designed to be naturally solvable in terms of classic data structure algorithms such as queues and stacks. They claim that the associative LSTM performs better because it implements these data structure algorithms. For example, commenting on the superior performance of the associative LSTM in a task that requires the prediction of closing tags in a sequence of nested random XML-like tags, they write:

We hypothesise that Associative LSTM succeeds at this task by using associative lookup to implement multiple queues to hold tag names from different nesting levels.

However, they do not provide any evidence (or at least, nowhere near enough evidence) for this hypothesis. It is possible that the associative LSTM does not explicitly implement anything like a queue, but  solves the task in an opaque, black-boxy way just like a standard LSTM (for example, an alternative hypothesis would be that the extra multiplication introduced by the convolution operation makes the model effectively deeper, increasing its expressive capacity). I had a similar concern with the “fast weights” paper as well. I think, in general, when people introduce a new architecture, they should be more rigorous about the hypothesized mechanism through which the new model is supposed to improve performance over baseline models.

In addition to analyzing the behavior of the model in simple, well-controlled setups that require the implementation of a particular type of algorithm, another way to probe for a hypothesized mechanism would be to look at the generalization pattern of the model. For example, if the model genuinely learned an addition algorithm (another task considered in the paper), it should be able to generalize well beyond the training data it received: e.g. if the model is trained with at-most-10-digit numbers, it should be able to generalize to much longer numbers (ideally arbitrarily long numbers). This approach was beautifully exemplified in the original neural Turing machines paper, for instance.

Another clue that the model is indeed utilizing the intended inductive biases (in this case, toward implementing classic data structure algorithms) is the speed of training: such a model usually trains much faster than a more generic model without the right inductive biases. Although the associative LSTM indeed trains much faster than more generic LSTM variants, the number of iterations required for convergence still seems somewhat large (usually tens or sometimes hundreds of thousands of iterations).

Why is it hard to train deep neural networks? Degeneracy, not vanishing gradients, is the key

In this post, I will try to address a common misunderstanding about the difficulty of training deep neural networks. It seems to be a widely held belief that this difficulty is mainly, if not completely, due to the vanishing (and/or exploding) gradients problem. “Vanishing gradients” refers to the gradient norms becoming exponentially smaller for parameters deeper in the network. Smaller gradients mean parameters changing ever so slowly, and so learning gets stuck until the gradients become large enough, which could take exponentially long. This idea goes back, at least, to Bengio et al. (1994) and still seems to be everybody’s favorite explanation for why it is hard to train deep neural networks.

Let’s first consider a simple scenario: a deep linear network being trained to learn a linear mapping. Of course, deep linear networks aren’t interesting from a computational perspective, but Saxe et al. (2013) showed that learning dynamics in such networks can still be informative about the learning dynamics in nonlinear networks. So, let’s start  with this simple scenario. Here’s the learning curve and the initial gradient norms (before any training) for a 30-layer network (errorbars are standard errors over 10 independent runs).


I will shortly explain why these results are labeled “Fold 0” in the figure. The gradients here are with respect to layer activations (gradients with respect to parameters behave similarly). The network weights are initialized with the standard 1/\sqrt{n} initialization. The training loss decreases rapidly at first, but then quickly asymptotes at a suboptimal value. The gradients certainly don’t vanish (or explode), at least initially! They do become smaller as training progresses, but that is to be expected and it’s not clear at all that the gradients here are “too small” in any sense:


To show that convergence to a suboptimal solution here doesn’t have anything to do with the size of the gradient norms per se, I will now introduce a manipulation that will increase the gradient norms, but will worsen the performance. Here it is (in blue):


So, what did I do? I simply changed the initialization in a pretty minimal way. Each initial weight matrix in the original networks is a 64 \times 64 matrix (initialized with the standard 1/\sqrt{n} initialization). In the networks shown in blue here, I just copied the first half of each initial weight matrix to the second half (the initial weight matrix is “folded” once, so that’s why I denote these as “Fold 1” networks). This reduces the rank of the initial weight matrices, and makes them more degenerate. Note that this manipulation is done only on the initial weight matrices, so no other constraint is imposed on learning once the training gets started. Here’s how the gradient norms look after the first few epochs:


So, I introduced a manipulation that increased the size of the gradient norms overall, yet the performance got significantly worse. Conversely, I will now introduce another manipulation that will shrink the size of the gradient norms, yet will improve performance substantially. Here it is (in green):


As you may have guessed from the label, this new manipulation initializes the weight matrices to be orthogonal. Let’s recall that orthogonal matrices are the least degenerate matrices among matrices of a fixed (Frobenius) norm, where degeneracy can be measured in different ways, for example, as the fraction of singular values smaller than a given constant. Here’s how the gradients look after the first few epochs in this case:


In fact, I can even multiply the initial orthogonal matrices by some factor smaller than 1 (e.g. by 0.8 in the following example) and deliberately create vanishing gradient norms in the initial network without much effect on the training performance (magenta):


There’s another, slightly subtler, reason for why the size of the gradient norms can’t explain the bulk of the training difficulties with deep neural networks: modern adaptive SGD variants typically used in practice these days, such as Adam, apply updates that are approximately invariant to the size of the gradient norms. So, if the only problem with training deep networks were a simple scaling problem, these methods would have easily dealt with that issue.

So, what’s going on? If the size of the gradient norms per se isn’t responsible for training difficulties, then what is? The answer is that it’s the degeneracy of the model that, by and large, determines the training performance. Why is degeneracy harmful for training performance? Intuitively, the reason is that learning slows down substantially along degenerate directions in the parameter space, hence degeneracies reduce the effective dimensionality of the model. So, you might think that you’re fitting a model with \sim 130\mathrm{K} parameters, as in the above examples, but in reality you’re effectively fitting a model with substantially fewer degrees of freedom because of the degeneracies. So, the problem with the “Fold 0” and “Fold 1” networks above was that although the gradient norms are fine, the network’s available degrees of freedom contribute extremely unevenly to those norms: while a handful of degrees of freedom (non-degenerate ones) explain almost all of it, the vast majority (degenerate ones) don’t contribute anything at all (a conceptually helpful, but not strictly accurate, way of thinking about this may be to imagine only a few of the hidden units in each layer changing their activity in response to different inputs, while the vast majority just responding the same way independently of the input).

This reduction in the effective degrees of freedom can be quite substantial, as in the 1/\sqrt{n}-initialized random matrices above. As shown by Saxe et al. (2013), the product of such matrices becomes increasingly degenerate as the number of matrices multiplied (i.e. network depth) increases. Here’s an example with 1-layer, 10-layer and 100-layer networks respectively (adapted from Saxe et al. (2013)):

saxe_plotAs depth increases, the singular values of the product matrix become increasingly concentrated around 0, except for a vanishingly small fraction of singular values that become arbitrarily large. This result isn’t just relevant for linear networks. A similar thing happens in nonlinear networks too: as depth increases, the responses of the hidden units in a given layer become increasingly lower-dimensional, i.e. increasingly degenerate. In fact, this “degeneration process” can proceed much more quickly with depth in nonlinear networks with hard-saturating boundaries as in ReLU networks.

A nice visualization of this degeneration process is presented in a paper by Duvenaud et al. (2014):

drawing-1As depth increases, the input space (shown on the top left) gets distorted into increasingly thin filaments with only a single direction (orthogonal to the filament) at each point in the input space affecting the network’s response (it may be a bit hard to extrapolate from this figure how input spaces of more than two dimensions will behave under a similar mapping, but it turns out they become “hyper-pancakey” locally, i.e. there is a single direction at each point, orthogonal to the pancake surface, the network is sensitive to). Along that sensitive direction, the network in fact becomes hyper-sensitive to variations.

Finally, I can’t resist mentioning my own paper (with Xaq Pitkow) at this point. In this paper, through a series of experiments, we argue that the degeneracy problem I discussed in this post severely afflicts training in deep nonlinear networks, and that one of the ways (and quite possibly the single most important way) in which skip connections help training in deep networks is through breaking such degeneracies. I suspect that other methods like batch normalization or layer normalization that also help training in deep networks also work at least partly through a similar degeneracy-breaking mechanism, in addition to any other potentially independent mechanisms such as reducing the internal covariate shift as originally proposed. It is well-known, for example, that divisive normalization is an effective way of decorrelating the responses of hidden units, which in turn can be seen as a degeneracy-breaking mechanism.

Update (1/5/18): I should also mention this important recent paper by Pennington, Schoenholz & Ganguli. Orthogonal initialization completely eliminates degeneracies in linear networks, but not in nonlinear networks. In this paper, they provide a method to calculate the entire singular value distribution of the Jacobian of a nonlinear network and show that a depth-independent, non-degenerate singular value distribution can be achieved, with careful initialization, in networks with hard-tanh nonlinearities, but not in ReLU networks. The empirical results show that networks with depth-independent, non-degenerate singular value distributions train orders of magnitude faster than networks whose singular value distributions become wider (higher variance) and more degenerate with depth. This is a powerful demonstration of the importance of eliminating degeneracies and controlling the entire singular value distribution of a network, not just its mean.

A note on fast weights: do they really work as advertised?

In fields where one cannot prove one’s claims mathematically, it is immensely important to be as rigorous as possible. Among other things, this implies considering all plausible alternative hypotheses to one’s own hypothesis and making sure that the results cannot be explained by those alternative hypotheses.  People working in experimental fields are usually quite careful about this. Machine learning is also, by and large, an experimental field (in the sense that usually one cannot prove one’s claims with mathematical rigour), so similar standards of experimental rigour should apply there as well.

I was reminded of this when I was re-reading the “fast weights” paper by Ba, Hinton et al. In this paper, they extend the standard vanilla recurrent neural network architecture with some form of Hebbian short-term synaptic plasticity. This Hebbian connectivity maintains a dynamically changing short-term memory of the recent history of the activities of the units in the network. They call this Hebbian connectivity “fast weights” as opposed to the standard “slow” recurrent connectivity.

If I haven’t made a mistake, the equation governing the model can be condensed into the following form (I’m assuming that their inner loop of Hebbian plasticity is unrolled for one step only, i.e. S=1):

\mathbf{h}_{t+1} = f\Big( \mathcal{LN}\Big[\mathbf{W}_h \mathbf{h}_t + \mathbf{W}_x \mathbf{x}_t + \big(\eta \sum_{\tau=1}^{\tau=t-1}\lambda^{t-\tau-1}\mathbf{h}_\tau \mathbf{h}_\tau^{\top}\big) f(\mathbf{W}_h \mathbf{h}_t + \mathbf{W}_x \mathbf{x}_t) \Big] \Big)

where \mathcal{LN}[\cdot] refers to layer normalization. The fast weight matrix here is given by the Hebbian matrix \eta \sum_{\tau=1}^{\tau=t-1}\lambda^{t-\tau-1}\mathbf{h}_\tau \mathbf{h}_\tau^{\top}.

The authors show that this model performs better than a number of other recurrent models in a couple of tasks. They seem to think that the Hebbian fast weight matrix is crucial (and mainly responsible) for the superior performance of the model. However, there are two obvious controls that are sorely missing in the paper and that makes a fair assessment of the contribution of fast weights difficult. First, the model has more depth than the standard architectures (note the nested f(\cdot)s in the equation above), so maybe there’s nothing special about the Hebbian fast weight matrix, rather the advantage is mainly due to the added depth. This could be tested by replacing the Hebbian matrix by another matrix, e.g. a random, fixed matrix or the identity matrix. In this connection, it’s interesting to observe that the authors chose this fairly complicated architecture rather than simpler ways of incorporating a Hebbian component to the recurrent connectivity, for example something like this:

\mathbf{h}_{t+1} = f\Big( \mathcal{LN} \Big[ \big[\mathbf{W}_{h} + \eta \sum_{\tau=1}^{\tau=t-1}\lambda^{t-\tau-1}\mathbf{h}_\tau \mathbf{h}_\tau^{\top} \big] \mathbf{h}_{t} + \mathbf{W}_{x} \mathbf{x}_{t} \Big] \Big)

which doesn’t increase the effective depth and would in fact be closer to the biological idea of short-term synaptic plasticity (this, in fact, appears to be how fast weights were originally defined in Hinton’s old “fast weights” paper). This may be because they tried something simple like this and it didn’t work well, so they found the added complexity necessary. But then it would be important to understand why fast weights don’t work in this simple context and require the more complicated machinery.

Secondly, layer normalization by itself improves the performance of vanilla recurrent networks, so we know that part of the improvement is due to normalization, again the question is how much. The necessary control to run here is a vanilla recurrent network with layer normalization.

I’m currently working on running these (and some other) controls, so I should hopefully have some answers to these questions soon.