In November last year, I co-presented a tutorial on waveform-based music processing with deep learning with Jordi Pons and Jongpil Lee at ISMIR 2019. Jongpil and Jordi talked about music classification and source separation respectively, and I presented the last part of the tutorial, on music generation in the waveform domain. It was very well received, so I’ve decided to write it up in the form of a blog post.

ISMIR used to be my home conference when I was a PhD student working on music information retrieval, so it was great to be back for the first time in five years. With about 450 attendees (the largest edition yet), it made for a very different experience than what I’m used to with machine learning conferences like ICML, NeurIPS and ICLR, whose audiences tend to number in the thousands these days.

Our tutorial on the first day of the conference gave rise to plenty of interesting questions and discussions throughout, which inspired me to write some of these things down and hopefully provide a basis to continue these discussions online. Note that I will only be covering music generation in this post, but Jordi and Jongpil are working on blog posts about their respective parts. I will share them here when they are published. In the meantime, the slide deck we used includes all three parts and is now available on Zenodo (PDF) and on Google slides. I’ve also added a few things to this post that I’ve thought of since giving the tutorial, and some new work that has come out since.

This is also an excellent opportunity to revive my blog, which has lain dormant for the past four years. I have taken the time to update the blog software, so if anything looks odd, that may be why. Please let me know so I can fix it!

Overview

This blog post is divided into a few different sections. I’ll try to motivate why modelling music in the waveform domain is an interesting problem. Then I’ll give an overview of generative models, the various flavours that exist, and some important ways in which they differ from each other. In the next two sections I’ll attempt to cover the state of the art in both likelihood-based and adversarial models of raw music audio. Finally, I’ll raise some observations and discussion points. If you want to skip ahead, just click the section title below to go there.

• Motivation
• Generative models
• Likelihood-based models of waveforms
• Adversarial models of waveforms
• Discussion
• Conclusion
• References

Note that this blog post is not intended to provide an exhaustive overview of all the published research in this domain – I have tried to make a selection and I’ve inevitably left out some great work. Please don’t hesitate to suggest relevant work in the comments section!

Motivation

Why audio?

Music generation has traditionally been studied in the symbolic domain: the output of the generative process could be a musical score, a sequence of MIDI events, a simple melody, a sequence of chords, a textual representation¹ or some other higher-level representation. The physical process through which sound is produced is abstracted away. This dramatically reduces the amount of information that the models are required to produce, which makes the modelling problem more tractable and allows for lower-capacity models to be used effectively.

A very popular representation is the so-called piano roll, which dates back to the player pianos of the early 20th century. Holes were punched into a roll of paper to indicate which notes should be played at which time. This representation survives in digital form today and is commonly used in music production. Much of the work on music generation using machine learning has made use of (some variant of) this representation, because it allows for capturing performance-specific aspects of the music without having to model the sound.

Piano rolls are great for piano performances, because they are able to exactly capture the timing, pitch and velocity (i.e. how hard a piano key is pressed, which is correlated with loudness, but not equivalent to it) of the notes. They are able to very accurately represent piano music, because they cover all the “degrees of freedom” that a performer has at their disposal. However, most other instruments have many more degrees of freedom: think about all the various ways you can play a note on the guitar, for example. You can decide which string to use, where to pick, whether to bend the string or not, play vibrato, … you could even play harmonics, or use two-hand tapping. Such a vast array of different playing techniques endows the performer with a lot more freedom to vary the sound that the instrument produces, and coming up with a high-level representation that can accurately capture all this variety is much more challenging. In practice, a lot of this detail is ignored and a simpler representation is often used when generating music for these instruments.

Modelling the sound that an instrument produces is much more difficult than modelling (some of) the parameters that are controlled by the performer, but it frees us from having to manually design high-level representations that accurately capture all these parameters. Furthermore, it allows our models to capture variability that is beyond the performer’s control: the idiosyncracies of individual instruments, for example (no two violins sound exactly the same!), or the parameters of the recording setup used to obtain the training data for our models. It also makes it possible to model ensembles of instruments, or other sound sources altogether, without having to fundamentally change anything about the model apart from the data it is trained on.

Digital audio representations require a reasonably high bit rate to achieve acceptable fidelity however, and modelling all these bits comes with a cost. Music audio models will necessarily have to have a much higher capacity than their symbolic counterparts, which implies higher computational requirements for model training.

Why waveforms?

Digital representations of sound come in many shapes and forms. For reproduction, sound is usually stored by encoding the shape of the waveform as it changes over time. For analysis however, we often make use of spectrograms, both for computational methods and for visual inspection by humans. A spectrogram can be obtained from a waveform by computing the Fourier transform of overlapping windows of the signal, and stacking the results into a 2D array. This shows the local frequency content of the signal over time.

Spectrograms are complex-valued: they represent both the amplitude and the phase of different frequency components at each point in time. Below is a visualisation of a magnitude spectrogram and its corresponding phase spectrogram. While the magnitude spectrogram clearly exhibits a lot of structure, with sustained frequencies manifesting as horizontal lines and harmonics showing up as parallel horizontal lines, the phase spectrogram looks a lot more random.

When extracting information from audio signals, it turns out that we can often just discard the phase component, because it is not informative for most of the things we could be interested in. In fact, this is why the magnitude spectrogram is often referred to simply as “the spectrogram”. When generating sound however, phase is very important because it meaningfully affects our perception. Listen below to an original excerpt of a piano piece, and a corresponding excerpt where the original phase has been replaced by random uniform phase information. Note how the harmony is preserved, but the timbre changes completely.

The phase component of a spectrogram is tricky to model for a number of reasons:

• it is an angle: \(\phi \in [0, 2 \pi)\) and it wraps around;
• it becomes effectively random as the magnitude tends towards 0, because noise starts to dominate;
• absolute phase is less meaningful, but relative phase differences over time matter perceptually.

If we model waveforms directly, we are implicitly modelling their phase as well, but we don’t run into these issues that make modelling phase so cumbersome. There are other strategies to avoid these issues, some of which I will discuss later, but waveform modelling currently seems to be the dominant approach in the generative setting. This is particularly interesting because magnitude spectrograms are by far the most common representation used for discriminative models of audio.

Discretising waveforms

When representing a waveform digitally, we need to discretise it in both time and amplitude. This is referred to as pulse code modulation (PCM). Because audio waveforms are effectively band-limited (humans cannot perceive frequencies above ~20 kHz), the sampling theorem tells us that we can discretise the waveform in time without any loss of information, as long as the sample rate is high enough (twice the highest frequency). This is why CD quality audio has a sample rate of 44.1 kHz. Much lower sample rates result in an audible loss of fidelity, but since the resulting discrete sequences also end up being much shorter, a compromise is often struck in the context of generative modelling to reduce computational requirements. Most models from literature use sample rates of 16 or 24 kHz.

When we also quantise the amplitude, some loss of fidelity is inevitable. CD quality uses 16 bits per sample, representing 2¹⁶ equally spaced quantisation levels. If we want to use fewer bits, we can use logarithmically spaced quantisation levels instead to account for our nonlinear perception of loudness. This “mu-law companding” will result in a smaller perceived loss of fidelity than if the levels were equally spaced.

Generative models

Given a dataset \(X\) of examples \(x \in X\), which we assume to have been drawn independently from some underlying distribution \(p_X(x)\), a generative model can learn to approximate this distribution \(p_X(x)\). Such a model could be used to generate new samples that look like they could have been part of the original dataset. We distinguish implicit and explicit generative models: an implicit model can produce new samples \(x \sim p_X(x)\), but cannot be used to infer the likelihood of an example (i.e. we cannot tractably compute \(p_X(x)\) given \(x\)). If we have an explicit model, we can do this, though sometimes only up to an unknown normalising constant.

Conditional generative models

Generative models become more practically useful when we can exert some influence over the samples we draw from them. We can do this by providing a conditioning signal \(c\), which contains side information about the kind of samples we want to generate. The model is then fit to the conditional distribution \(p_X(x \vert c)\) instead of \(p_X(x)\).

Conditioning signals can take many shapes or forms, and it is useful to distinguish different levels of information content. The generative modelling problem becomes easier if the conditioning signal \(c\) is richer, because it reduces uncertainty about \(x\). We will refer to conditioning signals with low information content as sparse conditioning, and those with high information content as dense conditioning. Examples of conditioning signals in the image domain and the music audio domain are shown below, ordered according to density.

Note that the density of a conditioning signal is often correlated with its level of abstraction: high-level side information tends to be more sparse. Low-level side information isn’t necessarily dense, though. For example, we could condition a generative model of music audio on a low-dimensional vector that captures the overall timbre of an instrument. This is a low-level aspect of the audio signal, but it constitutes a sparse conditioning signal.

Likelihood-based models

Likelihood-based models directly parameterise \(p_X(x)\). The parameters \(\theta\) are then fit by maximising the likelihood of the data under the model:

Note that this is typically done in the log-domain because it simplifies computations and improves numerical stability. Because the model directly parameterises \(p_X(x)\), we can easily infer the likelihood of any \(x\), so we get an explicit model. Three popular flavours of likelihood-based models are autoregressive models, flow-based models and variational autoencoders. The following three subsections provide a brief overview of each.

Autoregressive models

In an autoregressive model, we assume that our examples \(x \in X\) can be treated as sequences \(\{x_i\}\). We then factorise the distribution into a product of conditionals, using the chain rule of probability:

These conditional distributions are typically scalar-valued and much easier to model. Because we further assume that the distribution of the sequence elements is stationary, we can share parameters and use the same model for all the factors in this product.

For audio signals, this is a very natural thing to do, but we can also do this for other types of structured data by arbitrarily choosing an order (e.g. raster scan order for images, as in PixelRNN² and PixelCNN³).

Autoregressive models are attractive because they are able to accurately capture correlations between the different elements \(x_i\) in a sequence, and they allow for fast inference (i.e. computing \(p_X(x)\) given \(x\)). Unfortunately they tend to be slow to sample from, because samples need to be drawn sequentially from the conditionals for each position in the sequence.

Flow-based models

Another strategy for constructing a likelihood-based model is to use the change of variables theorem to transform \(p_X(x)\) into a simple, factorised distribution \(p_Z(z)\) (standard Gaussian is a popular choice) using an invertible mapping \(x = g(z)\):

Here, \(J\) is the Jacobian of \(g(z)\). Models that use this approach are referred to as normalising flows or flow-based models⁴⁵. They are fast both for inference and sampling, but the requirement for \(g(z)\) to be invertible significantly constrains the model architecture, and it makes them less parameter-efficient. In other words: flow-based models need to be quite large to be effective.

For an in-depth treatment of flow-based models, I recommend Eric Jang’s two-part blog post on the subject, and Papamakarios et al.’s excellent review paper.

Variational autoencoders (VAEs)

By far the most popular class of likelihood-based generative models, I can’t avoid mentioning variational⁶ autoencoders⁷ – but in the context of waveform modelling, they are probably the least popular approach. In a VAE, we jointly learn two neural networks: an inference network \(q(z \vert x)\) learns to probabilistically map examples \(x\) into a latent space, and a generative network \(p(x \vert z)\) learns the distribution of the data conditioned on a latent representation \(z\). These are trained to maximise a lower bound on \(p_X(x)\), called the ELBO (Evidence Lower BOund), because computing \(p_X(x)\) given \(x\) (exact inference) is not tractable.

Typical VAEs assume a factorised distribution for \(p(x \vert z)\), which limits the extent to which they can capture dependencies in the data. While this is often an acceptable trade-off, in the case of waveform modelling it turns out to be a problematic restriction in practice. I believe this is why not a lot of work has been published that takes this approach (if you know of any, please point me to it). VAEs can also have more powerful decoders with fewer assumptions (autoregressive decoders, for example), but this may introduce other issues such as posterior collapse⁸.

To learn more about VAEs, check out Jaan Altosaar’s tutorial.

Adversarial models

Generative Adversarial Networks⁹ (GANs) take a very different approach to capturing the data distribution. Two networks are trained simultaneously: a generator \(G\) attempts to produce examples according to the data distribution \(p_X(x)\), given latent vectors \(z\), while a discriminator \(D\) attempts to tell apart generated examples and real examples. In doing so, the discriminator provides a learning signal for the generator which enables it to better match the data distribution. In the original formulation, the loss function is as follows:

The generator is trained to minimise this loss, whereas the discriminator attempts to maximise it. This means the training procedure is a two-player minimax game, rather than an optimisation process, as it is for most machine learning models. Balancing this game and keeping training stable has been one of the main challenges for this class of models. Many alternative formulations have been proposed to address this.

While adversarial and likelihood-based models are both ultimately trying to model \(p_X(x)\), they approach this target from very different angles. As a result, GANs tend to be better at producing realistic examples, but worse at capturing the full diversity of the data distribution, compared to likelihood-based models.

More exotic flavours

Many other strategies to learn models of complicated distributions have been proposed in literature. While research on waveform generation has chiefly focused on the two dominant paradigms of likelihood-based and adversarial models, some of these alternatives may hold promise in this area as well, so I want to mention a few that I’ve come across.

• Energy-based models measure the “energy” of examples, and are trained by fitting the model parameters so that examples coming from the dataset have low energy, whereas all other configurations of inputs have high energy. This amounts to fitting an unnormalised density. A nice recent example is the work by Du & Mordatch at OpenAI¹⁰. Energy-based models have been around for a very long time though, and one could argue that likelihood-based models are a special case.

• Optimal transport is another approach to measure the discrepancy between probability distributions, which has served as inspiration for new variants of generative adversarial networks¹¹ and autoencoders¹².

• Autoregressive implicit quantile networks¹³ use a similar network architecture as likelihood-based autoregressive models, but they are trained using the quantile regression loss, rather than maximimum likelihood.

• Two continuous distributions can be matched by minimising the L2 distance between the gradients of the density functions with respect to their inputs: \(\mathcal{L}(x) = \mathbb{E} [\vert\vert \nabla_x \log p_X(x) - \nabla_y \log p_Y(y) \vert\vert ^2]\). This is called score matching¹⁴ and some recent works have revisited this idea for density estimation¹⁵ and generative modelling¹⁶.

• Please share any others that I haven’t mentioned in the comments!

Mode-covering vs. mode-seeking behaviour

An important consideration when determining which type of generative model is appropriate for a particular application, is the degree to which it is mode-covering or mode-seeking. When a model does not have enough capacity to capture all the variability in the data, different compromises can be made. If all examples should be reasonably likely under the model, it will have to overgeneralise and put probability mass on interpolations of examples that may not be meaningful (mode-covering). If there is no such requirement, the probability mass can be focused on a subset of examples, but then some parts of the distribution will be ignored by the model (mode-seeking).

Likelihood-based models are usually mode-covering. This is a consequence of the fact that they are fit by maximising the joint likelihood of the data. Adversarial models on the other hand are typically mode-seeking. A lot of ongoing research is focused on making it possible to control the trade-off between these two behaviours directly, without necessarily having to switch the class of models that are used.

In general, mode-covering behaviour is desirable in sparsely conditioned applications, where we want diversity or we expect a certain degree of “creativity” from the model. Mode-seeking behaviour is more useful in densely-conditioned settings, where most of the variability we care about is captured in the conditioning signal, and we favour realism of the generated output over diversity.

Likelihood-based models of waveforms

In this section, I’ll try to summarise some of the key results from the past four years obtained with likelihood-based models of waveforms. While this blog post is supposed to be about music, note that many of these developments were initially targeted at generating speech, so inevitably I will also be talking about some work in the text-to-speech (TTS) domain. I recommend reading the associated papers and/or blog posts to find out more about each of these works.

WaveNet & SampleRNN

WaveNet¹⁷ and SampleRNN¹⁸ are autoregressive models of raw waveforms. While WaveNet is a convolutional neural network, SampleRNN uses a stack of recurrent neural networks. Both papers appeared on arXiv in late 2016 with only a few months in between, signalling that autoregressive waveform-based audio modelling was an idea whose time had come. Before then, this idea had not been seriously considered, as modelling long-term correlations in sequences across thousands of timesteps did not seem feasible with the tools that were available at that point. Furthermore, discriminative models of audio all used spectral input representations, with only a few works investigating the use of raw waveforms in this setting (and usually with worse results).

Although these models have their flaws (including slow sampling due to autoregressivity, and a lack of interpretability w.r.t. what actually happens inside the network), I think they constituted an important existence proof that encouraged further research into waveform-based models.

WaveNet’s strategy to deal with long-term correlations is to use dilated convolutions: successive convolutional layers use filters with gaps between their inputs, so that the connectivity pattern across many layers forms a tree structure (see figure above). This enables rapid growth of the receptive field, which means that a WaveNet with only a few layers can learn dependencies across many timesteps. Note that the convolutions used in WaveNet are causal (no connectivity from future to past), which forces the model to learn to predict what values the signal could take at each position in time.

SampleRNN’s strategy is a bit different: multiple RNNs are stacked on top of each other, with each running at a different frequency. Higher-level RNNs update less frequently, which means they can more easily capture long-range correlations and learn high-level features.

Both models demonstrated excellent text-to-speech results, surpassing the state of the art at the time (concatenative synthesis, for most languages) in terms of naturalness. Both models were also applied to (piano) music generation, which constituted a nice demonstration of the promise of music generation in the waveform domain, but they were clearly limited in their ability to capture longer-term musical structure.

WaveNet: paper - blog post
SampleRNN: paper - samples

Parallel WaveNet & ClariNet

Sampling from autoregressive models of raw audio can be quite slow and impractical. To address this issue, Parallel WaveNet¹⁹ uses probability density distillation to train a model from which samples can be drawn in a single feed-forward pass. This requires a trained autoregressive WaveNet, which functions as a teacher, and an inverse autoregressive flow (IAF) model which acts as the student and learns to mimic the teacher’s predictions.

While an autoregressive model is slow to sample from, inferring the likelihood of a given example (and thus, maximum-likelihood training) can be done in parallel. For an inverse autoregressive flow, it’s the other way around: sampling is fast, but inference is slow. Since most practical applications rely on sampling rather than inference, such a model is often better suited. IAFs are hard to train from scratch though (because that requires inference), and the probability density distillation approach makes training them tractable.

Due to the nature of the probability density distillation objective, the student will end up matching the teacher’s predictions in a way that minimises the reverse KL divergence. This is quite unusual: likelihood-based models are typically trained to minimise the forward KL divergence instead, which is equivalent to maximising the likelihood (and minimising the reverse KL is usually intractable). While minimising the forward KL leads to mode-covering behaviour, minimising the reverse KL will instead lead to mode-seeking behaviour, which means that the model may end up ignoring certain modes in the data distribution.

In the text-to-speech (TTS) setting, this may actually be exactly what we want: given an excerpt of text, we want the model to generate a realistic utterance corresponding to that excerpt, but we aren’t particularly fussed about being able to generate every possible variation – one good-sounding utterance will do. This is a setting where realism is clearly more important than diversity, because all the diversity that we care about is already captured in the conditioning signal that we provide. This is usually the setting where adversarial models excel, because of their inherent mode-seeking behaviour, but using probability density distillation we can also train likelihood-based models this way.

To prevent the model from collapsing, parallel WaveNet uses a few additional loss terms to encourage the produced waveforms to resemble speech (such as a loss on the average power spectrum).

If we want to do music generation, we will typically care more about diversity because the conditioning signals we provide to the model are weaker. I believe this is why we haven’t really seen the Parallel WaveNet approach catch on outside of TTS.

ClariNet²⁰ was introduced as a variant of Parallel WaveNet which uses a Gaussian inverse autoregressive flow. The Gaussian assumption makes it possible to compute the reverse KL in closed form, rather than having to approximate it by sampling, which stabilises training.

Parallel WaveNet: paper - blog post 1 - blog post 2
ClariNet: paper - samples

Flow-based models: WaveGlow, FloWaveNet, WaveFlow, Blow

Training an IAF with probability density distillation isn’t the only way to train a flow-based model: most can be trained by maximum likelihood instead. In that case, the models will be encouraged to capture all the modes of the data distribution. This, in combination with their relatively low parameter efficiency (due to the invertibility requirement), means that they might need to be a bit larger to be effective. On the other hand, they allow for very fast sampling because all timesteps can be generated in parallel, so while the computational cost may be higher, sampling will still be faster in practice. Another advantage is that no additional loss terms are required to prevent collapse.

WaveGlow²¹ and FloWaveNet²², both originally published in late 2018, are flow-based models of raw audio conditioned on mel-spectrograms, which means they can be used as vocoders. Because of the limited parameter efficiency of flow-based models, I suspect that it would be difficult to use them for music generation in the waveform domain, where conditioning signals are much more sparse – but they could of course be used to render mel-spectrograms generated by some other model into waveforms (more on that later).

WaveFlow²³ (with an F instead of a G) is a more recent model that improves parameter efficiency by combining the flow-based modelling approach with partial autoregressivity to model local signal structure. This allows for a trade-off between sampling speed and model size. Blow²⁴ is a flow-based model of waveforms for non-parallel voice conversion.

WaveGlow: paper - code - samples
FloWaveNet: paper - code - samples
WaveFlow: paper - samples
Blow: paper - code - samples

Hierarchical WaveNets

For the purpose of music generation, WaveNet is limited by its ability to capture longer-term signal structure, as previously stated. In other words: while it is clearly able to capture local signal structure very well (i.e. the timbre of an instrument), it isn’t able to model the evolution of chord progressions and melodies over longer time periods. This makes the outputs produced by this model sound rather improvisational, to put it nicely.

This may seem counterintuitive at first: the tree structure of the connectivity between the layers of the model should allow for a very rapid growth of its receptive field. So if you have a WaveNet model that captures up to a second of audio at a time (more than sufficient for TTS), stacking a few more dilated convolutional layers on top should suffice to grow the receptive field by several orders of magnitude (up to many minutes). At that point, the model should be able to capture any kind of meaningful musical structure.

In practice, however, we need to train models on excerpts of audio that are at least as long as the longest-range correlations that we want to model. So while the depth of the model has to grow only logarithmically as we increase the desired receptive field, the computational and memory requirements for training do in fact grow linearly. If we want to train a model that can learn about musical structure across tens of seconds, that will necessarily be an order of magnitude more expensive – and WaveNets that generate music already have to be quite large as it is, even with a receptive field of just one second, because music is harder to model than speech. Note also that one second of audio corresponds to a sequence of 16000 timesteps at 16 kHz, so even at a scale of seconds, we are already modelling very long sequences.

In 10 years, the hardware we would need to train a WaveNet with a receptive field of 30 seconds (or almost half a million timesteps at 16 kHz) may just fit in a desktop computer, so we could just wait until then to give it a try. But if we want to train such models today, we need a different strategy. If we could train separate models to capture structure at different timescales, we could have a dedicated model that focuses on capturing longer-range correlations, without having to also model local signal structure. This seems feasible, seeing as models of high-level representations of music (i.e. scores or MIDI) clearly do a much better job of capturing long-range musical structure already.

We can approach this as a representation learning problem: to decouple learning of local and large-scale structure, we need to extract a more compact, high-level representation \(h\) from the audio signals \(x\), that makes abstraction of local detail and has a much lower sample rate. Ideally, we would learn a model \(h = f(x)\) to extract such a representation from data (although using existing high-level representations like MIDI is also possible, as we’ll discuss later).

Then we can split up the task by training two separate models: a WaveNet that models the high-level representation: \(p_H(h)\), and another that models the local signal structure, conditioned on the high-level representation: \(p_{X \vert H}(x \vert h)\). The former model can focus on learning about long-range correlations, as local signal structure is not present in the representation it operates on. The latter model, on the other hand, can focus on learning about local signal structure, as relevant information about large-scale structure is readily available in its conditioning signal. Combined together, these models can be used to sample new audio signals by first sampling \(\hat{h} \sim p_H(h)\) and then \(\hat{x} \sim p_{X \vert H}(x \vert \hat{h})\).

We can learn both \(f(x)\) and \(p_{X \vert H}(x \vert h)\) together by training an autoencoder: \(f(x)\) is the encoder, a feed-forward neural network, and \(p_{X \vert H}(x \vert h)\) is the decoder, a conditional WaveNet. Learning these jointly will enable \(f(x)\) to adapt to the WaveNet, so that it extracts information that the WaveNet cannot easily model itself.

To make the subsequent modelling of \(h = f(x)\) with another WaveNet easier, we use a VQ-VAE²⁵: an autoencoder with a discrete bottleneck. This has two important consequences:

• Autoregressive models seem to be more effective on discrete sequences than on continuous ones. Making the high-level representation discrete makes the hierarchical modelling task much easier, as we don’t need to adapt the WaveNet model to work with continuous data.
• The discreteness of the representation also limits its information capacity, forcing the autoencoder to encode only the most important information in \(h\), and to use the autoregressive connections in the WaveNet decoder to capture any local structure that wasn’t encoded in \(h\).

To split the task into more than two parts, we can apply this procedure again to the high-level representation \(h\) produced by the first application, and repeat this until we get a hierarchy with as many levels as desired. Higher levels in the hierarchy make abstraction of more and more of the low-level details of the signal, and have progressively lower sample rates (yielding shorter sequences). a three-level hierarchy is shown in the diagram below. Note that each level can be trained separately and in sequence, thus greatly reducing the computational requirements of training a model with a very large receptive field.

My colleagues and I explored this idea and trained hierachical WaveNet models on piano music²⁶. We found that there was a trade-off between audio fidelity and long-range coherence of the generated samples. When more model capacity was repurposed to focus on long-range correlations, this reduced the capability of the model to capture local structure, resulting in lower perceived audio quality. We also conducted a human evaluation study where we asked several listeners to rate both the fidelity and the musicality of some generated samples, to demonstrate that hierarchical models produce samples which sound more musical.

Hierarchical WaveNet: paper - samples

Wave2Midi2Wave and the MAESTRO dataset

As alluded to earlier, rather than learning high-level representations of music audio from data, we could also use existing high-level representations such as MIDI to construct a hierarchical model. We can use a powerful language model to model music in the symbolic domain, and also construct a conditional WaveNet model that generates audio, given a MIDI representation. Together with my colleagues from the Magenta team at Google AI, we trained such models on a new dataset called MAESTRO, which features 172 hours of virtuosic piano performances, captured with fine alignment between note labels and audio waveforms²⁷. This dataset is available to download for research purposes.

Compared to hierarchical WaveNets with learnt intermediate representations, this approach yields much better samples in terms of musical structure, but it is limited to instruments and styles of music that MIDI can accurately represent. Manzelli et al. have demonstrated this approach for a few instruments other than piano²⁸, but the lack of available aligned data could pose a problem.

Wave2Midi2Wave: paper - blog post - samples - dataset
Manzelli et al. model: paper - samples

Sparse transformers

OpenAI introduced the Sparse Transformer model²⁹, a large transformer³⁰ with a sparse attention mechanism that scales better to long sequences than traditional attention (which is quadratic in the length of the modelled sequence). They demonstrated impressive results autoregressively modelling language, images, and music audio using this architecture, with sparse attention enabling their model to cope with waveforms of up to 65k timesteps (about 5 seconds at 12 kHz). The sparse attention mechanism seems like a good alternative to the stacked dilated convolutions of WaveNets, provided that an efficient implementation is available.

Sparse Transformer: paper - blog post - samples

Universal music translation network

An interesting conditional waveform modelling problem is that of “music translation” or “music style transfer”: given a waveform, render a new waveform where the same music is played by a different instrument. The Universal Music Translation Network³¹ tackles this by training an autoencoder with multiple WaveNet decoders, where the encoded representation is encouraged to be agnostic to the instrument of the input (using an adversarial loss). A separate decoder is trained for each target instrument, so once this representation is extracted from a waveform, it can be synthesised in an instrument of choice. The separation is not perfect, but it works surprisingly well in practice. I think this is a nice example of a model that combines ideas from both likelihood-based models and the adversarial learning paradigm.

Universal music translation network: paper - code - samples

Dadabots

Dadabots are a researcher / artist duo who have trained SampleRNN models on various albums (primarily metal) in order to produce more music in the same vein. These models aren’t great at capturing long-range correlations, so it works best for artists whose style is naturally a bit disjointed. Below is a 24 hour livestream they’ve set up with a model generating infinite technical death metal in the style of ‘Relentless Mutation’ by Archspire.

Adversarial models of waveforms

Adversarial modelling of audio has only recently started to see some successes, which is why this section is going to be a lot shorter than the previous one on likelihood-based models. The adversarial paradigm has been extremely successful in the image domain, but researchers have had a harder time translating that success to other domains and modalities, compared to likelihood-based models. As a result, published work so far has primarily focused on speech generation and the generation of individual notes or very short clips of music. As a field, we are still very much in the process of figuring out how to make GANs work well for audio at scale.

WaveGAN

One of the first works to attempt using GANs for modelling raw audio signals is WaveGAN³². They trained a GAN on single-word speech recordings, bird vocalisations, individual drum hits and short excerpts of piano music. They also compared their raw audio-based model with a spectrogram-level model called SpecGAN. Although the fidelity of the resulting samples is far from perfect in some cases, this work undoubtedly inspired a lot of researchers to take audio modelling with GANs more seriously.

WaveGAN: paper - code - samples - demo - colab

GANSynth

So far in this blog post, we have focused on generating audio waveforms directly. However, I don’t want to omit GANSynth³³, even though technically speaking it does not operate directly in the waveform domain. This is because the spectral representation it uses is exactly invertible – no other models or phase reconstruction algorithms are used to turn the spectograms it generates into waveforms, which means it shares a lot of the advantages of models that operate directly in the waveform domain.

As discussed before, modelling the phase component of a complex spectrogram is challenging, because the phase of real audio signals can seem essentially random. However, using some of its unique characteristics, we can transform the phase into a quantity that is easier to model and reason about: the instantaneous frequency. This is obtained by computing the temporal difference of the unwrapped phase between subsequent frames. “Unwrapping” means that we shift the phase component by a multiple of \(2 \pi\) for each frame as needed to make it monotonic over time, as shown in the diagram below (because phase is an angle, all values modulo \(2 \pi\) are equivalent).

The instantaneous frequency captures how much the phase of a signal moves from one spectrogram frame to the next. For harmonic sounds, this quantity is expected to be constant over time, as the phase rotates at a constant velocity. This makes this representation particularly suitable to model musical sounds, which have a lot of harmonic content (and in fact, it might also make the representation less suitable for modelling more general classes of audio signals, though I don’t know if anyone has tried). For harmonic sounds, the instantaneous frequency is almost trivial to predict.

GANSynth is an adversarial model trained to produce the magnitude and instantaneous frequency spectrograms of recordings of individual musical notes. The trained model is also able to generalise to sequences of notes to some degree. Check out the blog post for sound examples and more information.

GANSynth: paper - code - samples - blog post - colab

MelGAN & GAN-TTS

Two recent papers demonstrate excellent results using GANs for text-to-speech: MelGAN³⁴ and GAN-TTS³⁵. The former also includes some music synthesis results, although fidelity is still an issue in that domain. The focus of MelGAN is inversion of magnitude spectrograms (potentially generated by other models), whereas as GAN-TTS is conditioned on the same “linguistic features” as the original WaveNet for TTS.

The architectures of both models share some interesting similarities, which shed light on the right inductive biases for raw waveform discriminators. Both models use multiple discriminators at different scales, each of which operates on a random window of audio extracted from the full sequence produced by the generator. This is similar to the patch-based discriminators that have occasionally been used in GANs for image generation. This windowing strategy seems to dramatically improve the capability of the generator to correctly model high frequency content in the audio signals, which is much more crucial to get right for audio than for images because it more strongly affects perceptual quality. The fact that both models benefited from this particular discriminator design indicates that we may be on the way to figuring out how to best design discriminator architectures for raw audio.

There are also some interesting differences: where GAN-TTS uses a combination of conditional and unconditional discriminators, MelGAN uses only unconditional discriminators and instead encourages the generator output to match the ground truth audio by adding an additional feature matching loss: the L1 distance between discriminator feature maps of real and generated audio. Both approaches seem to be effective.

Adversarial waveform synthesis is particularly useful for TTS, because it enables the use of highly parallelisable feed-forward models, which tend to have relatively low capacity requirements because they are trained with a mode-seeking loss. This means the models can more easily be deployed on low-power hardware while still performing audio synthesis in real-time, compared to autoregressive or flow-based models.

MelGAN: paper - code - samples
GAN-TTS: paper - code (FDSD) - sample

Discussion

To wrap up this blog post, I want to summarise a few thoughts about the current state of this area of research, and where things could be moving next.

Why the emphasis on likelihood in music modelling?

Clearly, the dominant paradigm for generative models of music in the waveform domain is likelihood-based. This stands in stark contrast to the image domain, where adversarial approaches greatly outnumber likelihood-based ones. I suspect there are a few reasons for this (let me know if you think of any others):

• Compared to likelihood-based models, it seems like it has been harder to translate the successes of adversarial models in the image domain to other domains, and to the audio domain in particular. I think this is because in a GAN, the discriminator fulfills the role of a domain-specific loss function, and important prior knowledge that guides learning is encoded in its architecture. We have known about good architectural priors for images for a long time (stacks of convolutions), as evidenced by work on e.g. style transfer³⁶ and the deep image prior³⁷. For other modalities, we don’t know as much yet. It seems we are now starting to figure out what kind of architectures work for waveforms (see MelGAN and GAN-TTS, some relevant work has also been done in the discriminative setting³⁸).

• Adversarial losses are mode-seeking, which makes them more suitable for settings where realism is more important than diversity (for example, because the conditioning signal contains most of the required diversity, as in TTS). In music generation, which is primarily a creative application, diversity is very important. Improving diversity of GAN samples is the subject of intense study right now, but I think it could be a while before they catch up with likelihood-based models in this sense.

• The current disparity could also simply be a consequence of the fact that likelihood-based models got a head start in waveform modelling, with WaveNet and SampleRNN appearing on the scene in 2016 and WaveGAN in 2018.

Another domain where likelihood-based models dominate is language modelling. I believe the underlying reasons for this might be a bit different though: language is inherently discrete, and extending GANs to modelling discrete data at scale is very much a work in progress. This is also more likely to be the reason why likelihood-based models are dominant for symbolic music generation as well: most symbolic representations of music are discrete.

Alternatives to modelling waveforms directly

Instead of modelling music in the waveform domain, there are many possible alternative approaches. We could model other representations of audio signals, such as spectrograms, as long as we have a way to obtain waveforms from such representations. We have quite a few options for this:

• We could use invertible spectrograms (i.e. phase information is not discarded), but in this case modelling the phase poses a considerable challenge. There are ways to make this easier, such as the instantaneous frequency representation used by GANSynth.

• We could also use magnitude spectrograms (as is typically done in discriminative models of audio), and then use a phase reconstruction algorithm such as the Griffin-Lim algorithm³⁹ to infer a plausible phase component, based only on the generated magnitude. This approach was used for the original Tacotron model for TTS⁴⁰, and for MelNet⁴¹, which models music audio autoregressively in the spectrogram domain.

• Instead of a traditional phase reconstruction algorithm, we could also use a vocoder to go from spectrograms to waveforms. A vocoder, in this context, is simply a generative model in the waveform domain, conditioned on spectrograms. Vocoding is a densely conditioned generation task, and many of the models discussed before can and have been used as vocoders (e.g. WaveNet in Tacotron 2⁴², flow-based models of waveforms, or MelGAN). This approach has some advantages: generated magnitude spectrograms are often imperfect, and vocoder models can learn to account for these imperfections. Vocoders can also work with inherently lossy spectrogram representations such as mel-spectrograms and constant-Q spectrograms⁴³.

• If we are generating audio conditioned on an existing audio signal, we could also simply reuse the phase of the input signal, rather than reconstructing or generating it. This is commonly done in source separation, and the approach could also be used for music style transfer.

That said, modelling spectrograms isn’t always easier than modelling waveforms. Although spectrograms have a much lower temporal resolution, they contain much more information per timestep. In autoregressive models of spectrograms, one would have to condition along both the time and frequency axes to capture all dependencies, which means we end up with roughly as many sequential sampling steps as in the raw waveform case. This is the approach taken by MelNet.

An alternative is to make an assumption of independence between different frequency bands at each timestep, given previous timesteps. This enables autoregressive models to produce entire spectrogram frames at a time. This partial independence assumption turns out to be an acceptable compromise in the text-to-speech domain, and is used in Tacotron and Tacotron 2. Vocoder models are particularly useful here as they can attempt to fix the imperfections resulting from this simplification of the model. I’m not sure if anybody has tried, but I would suspect that this independence assumption would cause more problems for music generation.

An interesting new approach combining traditional signal processing ideas with neural networks is Differentiable Digital Signal Processing (DDSP)⁴⁴. By creating learnable versions of existing DSP components and incorporating them directly into neural networks, these models are endowed with much stronger inductive biases about sound and music, and can learn to produce realistic audio with fewer trainable parameters, while also being more interpretable. I suspect that this research direction may gain a lot of traction in the near future, not in the least because the authors have made their code publicly available, and also because of its modularity and lower computational requirements.

Finally, we could train symbolic models of music instead: for many instruments, we already have realistic synthesisers, and we can even train them given enough data (see Wave2Midi2Wave). If we are able to craft symbolic representations that capture the aspects of music we care about, then this is an attractive approach as it is much less computationally intensive. Magenta’s Music Transformer⁴⁵ and OpenAI’s MuseNet are two models that have recently shown impressive results in this domain, and it is likely that other ideas from the language modelling community could bring further improvements.

DDSP: paper - code - samples - blog post - colab
Music Transformer: paper - blog post
MuseNet: blog post

What’s next?

Generative models of music in the waveform domain have seen substantial progress over the past few years, but the best results so far are still relatively easy to distinguish from real recordings, even at fairly short time scales. There is still a lot of room for improvement, but I believe a lot of this will be driven by better availability of computational resources, and not necessarily by radical innovation on the modelling front – we have great tools already, they are simply a bit expensive to use due to substantial computational requirements. As time goes on and computers get faster, hopefully this task will garner interest as it becomes accessible to more researchers.

One interesting question is whether adversarial models are going to catch up with likelihood-based models in this domain. I think it is quite likely that GANs, having recently made in-roads in the densely conditioned setting, will gradually be made to work for more sparsely conditioned audio generation tasks as well. Fully unconditional generation with long-term coherence seems very challenging however, and I suspect that the mode-seeking behaviour of the adversarial loss will make this much harder to achieve. A hybrid model, where a GAN captures local signal structure and another model with a different objective function captures high-level structure and long-term correlations, seems like a sensible thing to build.

Hierarchy is a very important prior for music (and, come to think of it, for pretty much anything else we like to model), so models that explicitly incorporate this are going to have a leg up on models that don’t – at the cost of some additional complexity. Whether this additional complexity will always be worth it remains to be seen, but at the moment, this definitely seems to be the case.

At any rate, splitting up the problem into multiple stages that can be solved separately has been fruitful, and I think it will continue to be. So far, hierarchical models (with learnt or handcrafted intermediate representations) and spectrogram-based models with vocoders have worked well, but perhaps there are other ways to “divide and conquer”. A nice example of a different kind of split in the image domain is the one used in Subscale Pixel Networks⁴⁶, where separate networks model the most and least significant bits of the image data.

Conclusion

If you made it to the end of this post, congratulations! I hope I’ve convinced you that music modelling in the waveform domain is an interesting research problem. It is also very far from a solved problem, so there are lots of opportunities for interesting new work. I have probably missed a lot of relevant references, especially when it comes to more recent work. If you know about relevant work that isn’t discussed here, feel free to share it in the comments! Questions about this blog post and this line of research are very welcome as well.

References

This post is another collaboration with Jan Schlüter from the OFAI (@f0k on GitHub), a fellow MIR researcher and one of the lead developers of Lasagne. He recently added a cool new feature that we wanted to highlight: enabling the use of arbitrary Theano expressions as layer parameters.

As many of you probably know, Jan Schlüter and I are part of the team that develops Lasagne, a lightweight neural network library built on top of Theano.

One of the key design principles of Lasagne is transparency: we try not to hide Theano or numpy behind an additional layer of abstractions and encapsulation, but rather expose their functionality and data types and try to follow their conventions. This makes it very easy to learn how to use Lasagne if you already know how to use Theano – there just isn’t all that much extra to learn. But most importantly, it allows you to easily mix and match parts of Lasagne with vanilla Theano code. This is the way Lasagne is meant to be used.

In keeping with this philosophy, Jan recently added a feature that we’ve been discussing early on in designing the API (#11): it allows any learnable layer parameter to be specified as a mathematical expression evaluating to a correctly-shaped tensor. Previously, layer parameters had to be Theano shared variables, i.e., naked tensors to be learned directly. This new feature makes it possible to constrain network parameters in various, potentially creative ways. Below, we’ll go through a few examples of what is now possible that wasn’t before.

Default case

Let’s create a simple fully-connected layer of 500 units on top of an input layer of 784 units.

from lasagne.layers import InputLayer, DenseLayer
batch_size = 64
l1 = InputLayer((batch_size, 784))
l2 = DenseLayer(l1, num_units=500)

Autoencoder with tied weights

Autoencoders with tied weights are a common use case, and until now implementing them in Lasagne was a bit tricky. Weight sharing in Lasagne has always been easy and intuitive:

l2 = DenseLayer(l1, num_units=500)
l3 = DenseLayer(l1, num_units=500, W=l2.W)
# l2 and l3 now share the same weight matrix!

… but in an autoencoder, you want the weights of the decoding layer to be the transpose of the weights of the encoding layer. So you would do:

l2 = DenseLayer(l1, num_units=500)
l3 = DenseLayer(l2, num_units=784, W=l2.W.T)

… but that didn’t work before: l2.W.T is a Theano expression, but not a Theano shared variable as was expected. This is counter-intuitive, and indeed, people expected it to work and were disappointed to find out that it didn’t. With the new feature this is no longer true. The above will work just fine. Yay!

Factorized weights

To reduce the number of parameters in your network (e.g. to prevent overfitting), you could force large parameter matrices to be low-rank by factorizing them. In our example from before, we could factorize the 784x500 weight matrix into the product of a 784x100 and a 100x500 matrix. The number of weights of the layer then goes down from 392000 to 128400 (not including the biases).

import theano
import theano.tensor as T
from lasagne.init import GlorotUniform
from lasagne.utils import floatX
w_init = GlorotUniform()
w1 = theano.shared(floatX(w_init((784, 100))))
w2 = theano.shared(floatX(w_init((100, 500))))
l2 = DenseLayer(l1, num_units=500, W=T.dot(w1, w2))

Granted, this was possible before by inserting a biasless linear layer:

l2_a = DenseLayer(l1, num_units=100, b=None, nonlinearity=None)
l2 = DenseLayer(l2_a, num_units=500)

Other types of factorizations may also be worth investigating!

Positive weights

If you want to force the weights of a layer to be positive, you can learn their logarithm:

from lasagne.init import Normal
w = theano.shared(floatX(Normal(0.01, mean=-10)((784, 500))))
l2 = DenseLayer(l1, num_units=500, W=T.exp(w))

You could also use T.softplus(w) instead of T.exp(w). You might also be tempted to try sticking a ReLU in there (T.maximum(w, 0)), but note that applying the linear rectifier to the weight matrix would lead to many of the underlying weights getting stuck at negative values, as the linear rectifier has zero gradient for negative inputs!

Positive semi-definite weights

There are plenty of other creative uses, such as constraining weights to be positive semi-definite (for whatever reason):

l2 = DenseLayer(l1, num_units=500)
w = theano.shared(floatX(w_init((500, 500))))
w_psd = T.dot(w, w.T)
l3 = DenseLayer(l2, num_units=500, W=w_psd)

Limitations

There are only a couple of limitations to using Theano expressions as layer parameters. One is that Lasagne functions and methods such as Layer.get_params() will implicitly assume that any shared variable featuring in these Theano expressions is to be treated as a parameter. In practice that means you can’t mix learnable and non-learnable parameter variables in a single expression. Also, the same tags will apply to all shared variables in an expression. More information about parameter tags can be found in the documentation.

For almost all use cases, these limitations should not be an issue. If they are, your best bet is to implement a custom layer class. Luckily, this is also very easy in Lasagne.

Why it works

All of this is made possible because Lasagne builds on Theano, which takes care of backpropagating through the parameter expression to any underlying learned tensors. In frameworks building on hard-coded layer implementations rather than an automatic expression compiler, all these examples would require writing custom backpropagation code.

If you want to play around with this yourself, try the bleeding-edge version of Lasagne. You can find installation instructions here.

Have fun experimenting! If you’ve done something cool that you’d like to share, feel free to send us a pull request on our Recipes repository.

UPDATE (April 27th): the paper is now available on the journal website: http://mnras.oxfordjournals.org/content/450/2/1441

Together with Kyle Willett, one of the organizers of the Galaxy Challenge, I’ve written a paper about my winning solution for this competition. It is available on ArXiv.

The paper has been accepted for publication in MNRAS, a journal on astronomy and astrophysics, but is also aimed at people with a machine learning background. Due to this dual audience, it contains both an in-depth overview of deep learning and convolutional networks, and a thorough analysis of the resulting model and its potential impact for astronomy research.

There is some overlap with the blog post I wrote after the competition ended, but there is a lot more detail and background information, and the ‘results’ and ‘analysis’ sections are entirely new (although those of you who have seen one of my talks on the subject may have seen some of the images before).

I am very grateful to Kyle Willett for helping me write the manuscript. Without his help, writing a paper for an audience of astronomers would have been an impossible task for me. I believe it’s crucially important that applications of deep learning and machine learning in general get communicated to the people that could benefit from them, in such a way that they might actually consider using them.

I am also grateful to current and former supervisors, Joni Dambre and Benjamin Schrauwen, for supporting me when I was working on this competition and this paper, even though it is only tangentially related to the subject of my PhD.

Original arxiv link: http://arxiv.org/abs/1503.07077

The National Data Science Bowl, a data science competition where the goal was to classify images of plankton, has just ended. I participated with six other members of my research lab, the Reservoir lab of prof. Joni Dambre at Ghent University in Belgium. Our team finished 1st! In this post, we’ll explain our approach.

The ≋ Deep Sea ≋ team consisted of Aäron van den Oord, Ira Korshunova, Jeroen Burms, Jonas Degrave, Lionel Pigou, Pieter Buteneers and myself. We are all master students, PhD students and post-docs at Ghent University. We decided to participate together because we are all very interested in deep learning, and a collaborative effort to solve a practical problem is a great way to learn.

There were seven of us, so over the course of three months, we were able to try a plethora of different things, including a bunch of recently published techniques, and a couple of novelties. This blog post was written jointly by the team and will cover all the different ingredients that went into our solution in some detail.

Overview

This blog post is going to be pretty long! Here’s an overview of the different sections. If you want to skip ahead, just click the section title to go there.

• Introduction
• Pre-processing and data augmentation
• Network architecture
• Training
• Unsupervised and semi-supervised approaches
• Model averaging
• Miscellany
• Conclusion

Introduction

The problem

The goal of the competition was to classify grayscale images of plankton into one of 121 classes. They were created using an underwater camera that is towed through an area. The resulting images are then used by scientists to determine which species occur in this area, and how common they are. There are typically a lot of these images, and they need to be annotated before any conclusions can be drawn. Automating this process as much as possible should save a lot of time!

The images obtained using the camera were already processed by a segmentation algorithm to identify and isolate individual organisms, and then cropped accordingly. Interestingly, the size of an organism in the resulting images is proportional to its actual size, and does not depend on the distance to the lens of the camera. This means that size carries useful information for the task of identifying the species. In practice it also means that all the images in the dataset have different sizes.

Participants were expected to build a model that produces a probability distribution across the 121 classes for each image. These predicted distributions were scored using the log loss (which corresponds to the negative log likelihood or equivalently the cross-entropy loss).

This loss function has some interesting properties: for one, it is extremely sensitive to overconfident predictions. If your model predicts a probability of 1 for a certain class, and it happens to be wrong, the loss becomes infinite. It is also differentiable, which means that models trained with gradient-based methods (such as neural networks) can optimize it directly - it is unnecessary to use a surrogate loss function.

Interestingly, optimizing the log loss is not quite the same as optimizing classification accuracy. Although the two are obviously correlated, we paid special attention to this because it was often the case that significant improvements to the log loss would barely affect the classification accuracy of the models.

The solution: convnets!

Image classification problems are often approached using convolutional neural networks these days, and with good reason: they achieve record-breaking performance on some really difficult tasks.

A challenge with this competition was the size of the dataset: about 30000 examples for 121 classes. Several classes had fewer than 20 examples in total. Deep learning approaches are often said to require enormous amounts of data to work well, but recently this notion has been challenged, and our results in this competition also indicate that this is not necessarily true. Judicious use of techniques to prevent overfitting such as dropout, weight decay, data augmentation, pre-training, pseudo-labeling and parameter sharing, has enabled us to train very large models with up to 27 million parameters on this dataset.

Some of you may remember that I participated in another Kaggle competition last year: the Galaxy Challenge. The goal of that competition was to classify images of galaxies. It turns out that a lot of the things I learned during that competition were also applicable here. Most importantly, just like images of galaxies, images of plankton are (mostly) rotation invariant. I used this property for data augmentation, and incorporated it into the model architecture.

Software and hardware

We used Python, NumPy and Theano to implement our solution, in combination with the cuDNN library. We also used PyCUDA to implement a few custom kernels.

Our code is mostly based on the Lasagne library, which provides a bunch of layer classes and some utilities that make it easier to build neural nets in Theano. This is currently being developed by a group of researchers with different affiliations, including Aäron and myself. We hope to release the first version soon!

We also used scikit-image for pre-processing and augmentation, and ghalton for quasi-random number generation. During the competition, we kept track of all of our results in a Google Drive spreadsheet. Our code was hosted on a private GitHub repository, with everyone in charge of their own branch.

We trained our models on the NVIDIA GPUs that we have in the lab, which include GTX 980, GTX 680 and Tesla K40 cards.

Pre-processing and data augmentation

We performed very little pre-processing, other than rescaling the images in various ways and then performing global zero mean unit variance (ZMUV) normalization, to improve the stability of training and increase the convergence speed.

Rescaling the images was necessary because they vary in size a lot: the smallest ones are less than 40 by 40 pixels, whereas the largest ones are up to 400 by 400 pixels. We experimented with various (combinations of) rescaling strategies. For most networks, we simply rescaled the largest side of each image to a fixed length.

We also tried estimating the size of the creatures using image moments. Unfortunately, centering and rescaling the images based on image moments did not improve results, but they turned out to be useful as additional features for classification (see below).

Data augmentation

We augmented the data to artificially increase the size of the dataset. We used various affine transforms, and gradually increased the intensity of the augmentation as our models started to overfit more. We ended up with some pretty extreme augmentation parameters:

• rotation: random with angle between 0° and 360° (uniform)
• translation: random with shift between -10 and 10 pixels (uniform)
• rescaling: random with scale factor between 1/1.6 and 1.6 (log-uniform)
• flipping: yes or no (bernoulli)
• shearing: random with angle between -20° and 20° (uniform)
• stretching: random with stretch factor between 1/1.3 and 1.3 (log-uniform)

We augmented the data on-demand during training (realtime augmentation), which allowed us to combine the image rescaling and augmentation into a single affine transform. The augmentation was all done on the CPU while the GPU was training on the previous chunk of data.

We experimented with elastic distortions at some point, but this did not improve performance although it reduced overfitting slightly. We also tried sampling the augmentation transform parameters from gaussian instead of uniform distributions, but this did not improve results either.

Network architecture

Most of our convnet architectures were strongly inspired by OxfordNet: they consist of lots of convolutional layers with 3x3 filters. We used ‘same’ convolutions (i.e. the output feature maps are the same size as the input feature maps) and overlapping pooling with window size 3 and stride 2.

We started with a fairly shallow models by modern standards (~ 6 layers) and gradually added more layers when we noticed it improved performance (it usually did). Near the end of the competition, we were training models with up to 16 layers. The challenge, as always, was balancing improved performance with increased overfitting.

We experimented with strided convolutions with 7x7 filters in the first two layers for a while, inspired by the work of He et al., but we were unable to achieve the same performance with this in our networks.

Cyclic pooling

When I participated in the Galaxy Challenge, one of the things I did differently from other competitors was to exploit the rotational symmetry of the images to share parameters in the network. I applied the same stack of convolutional layers to several rotated and flipped versions of the same input image, concatenated the resulting feature representations, and fed those into a stack of dense layers. This allowed the network to use the same feature extraction pipeline to “look at” the input from different angles.

Here, we took this a step further. Rather than concatenating the feature representations, we decided to pool across them to get rotation invariance. Here’s how it worked in practice: the images in a minibatch occur 4 times, in 4 different orientations. They are processed by the network in parallel, and at the top, the feature maps are pooled together. We decided to call this cyclic pooling, after cyclic groups.

The nice thing about 4-way cyclic pooling is that it can be implemented very efficiently: the images are rotated by 0, 90, 180 and 270 degrees. All of these rotations can be achieved simply by transposing and flipping image axes. That means no interpolation is required.

Cyclic pooling also allowed us to reduce the batch size by a factor of 4: instead of having batches of 128 images, each batch now contained 32 images and was then turned into a batch with an effective size of 128 again inside the network, by stacking the original batch in 4 orientations. After the pooling step, the batch size was reduced to 32 again.

We tried several pooling functions over the course of the competition, as well as different positions in the network for the pooling operation (just before the output layer, between hidden layers, …). It turned out that root-mean-square pooling gave much better results than mean pooling or max pooling. We weren’t able to find a good explanation for this, but we suspect it may have something to do with rotational phase invariance.

One of our models pooled over 8 rotations, spaced apart 45 degrees. This required generating the input images at two angles (0 and 45 degrees). We also considered having the model do 8-way pooling by including flipped versions of each rotated image copy (dihedral pooling, after dihedral groups). Unfortunately this did not work better.

‘Rolling’ feature maps

Cyclic pooling modestly improved our results, but it can be taken a step further. A cyclic pooling convnet extracts features from input images in four different orientations. An alternative interpretation is that its filters are applied to the input images in four different orientations. That means we can combine the stacks of feature maps from the different orientations into one big stack, and then learn the next layer of features on this combined input. As a result, the network then appears to have 4 times more filters than it actually has!

This is cheap to do, since the feature maps are already being computed anyway. We just have to combine them together in the right order and orientation. We named the operation that combines feature maps from different orientations a roll.

Roll operations can be inserted after dense layers or after convolutional layers. In the latter case, care has to be taken to rotate the feature maps appropriately, so that they are all aligned.

We originally implemented the operations with a few lines of Theano code. This is a nice demonstration of Theano’s effectiveness for rapid prototyping of new ideas. Later on we spent some time implementing CUDA kernels for the roll operations and their gradients, because networks with many rolled layers were getting pretty slow to train. Using your own CUDA kernels with Theano turns out to be relatively easy in combination with PyCUDA. No additional C-code is required.

In most of the models we evaluated, we only inserted convolutional roll operations after the pooling layers, because this reduced the size of the feature maps that needed to be copied and stacked together.

Note that it is perfectly possible to build a cyclic pooling convnet without any roll operations, but it’s not possible to have roll operations in a network without cyclic pooling. The roll operation is only made possible because the cyclic pooling requires that each input image is processed in four different orientations to begin with.

Nonlinearities

We experimented with various variants of rectified linear units (ReLUs), as well as maxout units (only in the dense layers). We also tried out smooth non-linearities and the ‘parameterized ReLUs’ that were recently introduced by He et al., but found networks with these units to be very prone to overfitting.

However, we had great success with (very) leaky ReLUs. Instead of taking the maximum of the input and zero, y = max(x, 0), leaky ReLUs take the maximum of the input and a scaled version of the input, y = max(x, a*x). Here, a is a tunable scale parameter. Setting it to zero yields regular ReLUs, and making it trainable yields parameterized ReLUs.

For fairly deep networks (10+ layers), we found that varying this parameter between 0 and 1/2 did not really affect the predictive performance. However, larger values in this range significantly reduced the level of overfitting. This in turn allowed us to scale up our models further. We eventually settled on a = 1/3.

Spatial pooling

We started out using networks with 2 or 3 spatial pooling layers, and we initially had some trouble getting networks with more pooling stages to work well. Most of our final models have 4 pooling stages though.

We started out with the traditional approach of 2x2 max-pooling, but eventually switched to 3x3 max-pooling with stride 2 (which we’ll refer to as 3x3s2), mainly because it allowed us to use a larger input size while keeping the same feature map size at the topmost convolutional layer, and without increasing the computational cost significantly.

As an example, a network with 80x80 input and 4 2x2 pooling stages will have feature maps of size 5x5 at the topmost convolutional layer. If we use 3x3s2 pooling instead, we can feed 95x95 input and get feature maps with the same 5x5 shape. This improved performance and only slowed down training slightly.

Multiscale architectures

As mentioned before, the images vary widely in size, so we usually rescaled them using the largest dimension of the image as a size estimate. This is clearly suboptimal, because some species of plankton are larger than others. Size carries valuable information.

To allow the network to learn this, we experimented with combinations of different rescaling strategies within the same network, by combining multiple networks with different rescaled inputs together into ‘multiscale’ networks.

What worked best was to combine a network with inputs rescaled based on image size, and a smaller network with inputs rescaled by a fixed factor. Of course this slowed down training quite a bit, but it allowed us to squeeze out a bit more performance.

Additional image features

We experimented with training small neural nets on extracted image features to ‘correct’ the predictions of our convnets. We referred to this as ‘late fusing’ because the feature network and the convnet were joined only at the output layer (before the softmax). We also tried joining them at earlier layers, but consistently found this to work worse, because of overfitting.

We thought this could be useful, because the features can be extracted from the raw (i.e. non-rescaled) images, so this procedure could provide additional information that is missed by the convnets. Here are some examples of types of features we evaluated (the ones we ended up using are in bold):

• Image size in pixels
• Size and shape estimates based on image moments
• Hu moments
• Zernike moments
• Parameter Free Threshold Adjacency Statistics
• Linear Binary Patterns
• Haralick texture features
• Features from the competition tutorial
• Combinations of the above

The image size, the features based on image moments and the Haralick texture features were the ones that stood out the most in terms of performance. The features were fed to a neural net with two dense layers of 80 units. The final layer of the model was fused with previously generated predictions of our best convnet-based models. Using this approach, we didn’t have to retrain the convnets nor did we have to regenerate predictions (which saved us a lot of time).

To deal with variance due to the random weight initialization, we trained each feature network 10 times and blended the copies with uniform weights. This resulted in a consistent validation loss decrease of 0.01 (or 1.81%) on average, which was quite significant near the end of the competition.

Interestingly, late fusion with image size and features based on image moments seems to help just as much for multiscale models as for regular convnets. This is a bit counterintuitive: we expected both approaches to help because they could extract information about the size of the creatures, so the obtained performance improvements would overlap. The fact they were fully orthogonal was a nice surprise.

Example convnet architecture

Here’s an example of an architecture that works well. It has 13 layers with parameters (10 convolutional, 3 fully connected) and 4 spatial pooling layers. The input shape is (32, 1, 95, 95), in bc01 order (batch size, number of channels, height, width). The output shape is (32, 121). For a given input, the network outputs 121 probabilities that sum to 1, one for each class.

Note how the ‘cyclic slice’ layer increases the batch size fourfold. The ‘cyclic pooling’ layer reduces it back to 32 again near the end. The ‘cyclic roll’ layers increase the number of feature maps fourfold.

Training

Validation

We split off 10% of the labeled data as a validation set using stratified sampling. Due to the small size of this set, our validation estimates were relatively noisy and we periodically validated some models on the leaderboard as well.

Training algorithm

We trained all of our models with stochastic gradient descent (SGD) with Nesterov momentum. We set the momentum parameter to 0.9 and did not tune it further. Most models took between 24 and 48 hours to train to convergence.

We trained most of the models with about 215000 gradient steps and eventually settled on a discrete learning rate schedule with two 10-fold decreases (following Krizhevsky et al.), after about 180000 and 205000 gradient steps respectively. For most models we used an initial learning rate of 0.003.

We briefly experimented with the Adam update rule proposed by Kingma and Ba, as an alternative to Nesterov momentum. We used the version of the algorithm described in the first version of the paper, without the lambda parameter. Although this seemed to speed up convergence by a factor of almost 2x, the results were always slightly worse than those achieved with Nesterov momentum, so we eventually abandoned this idea.

Initialization

We used a variant of the orthogonal initialization strategy proposed by Saxe et al. everywhere. This allowed us to add as many layers as we wanted without running into any convergence problems.

Regularization

For most models, we used dropout in the fully connected layers of the network, with a dropout probability of 0.5. We experimented with dropout in the convolutional layers as well for some models.

We also tried Gaussian dropout (using multiplicative Gaussian noise instead of multiplicative Bernoulli noise) and found this to work about as well as traditional dropout.

We discovered near the end of the competition that it was useful to have a small amount of weight decay to stabilize training of larger models (so not just for its regularizing effect). Models with large fully connected layers and without weight decay would often diverge unless the learning rate was decreased considerably, which slowed things down too much.

Unsupervised and semi-supervised approaches

Unsupervised pre-training

Since the test set was much larger than the training set, we experimented with using unsupervised pre-training on the test set to initialize the networks. We only pre-trained the convolutional layers, using convolutional auto-encoders (CAE, Masci. et al.). This approach consists of building a stack of layers implementing the reverse operations (i.e. deconvolution and unpooling) of the layers that are to be pre-trained. These can then be used to try and reconstruct the input of those layers.

In line with the literature, we found that pre-training a network serves as an excellent regularizer (much higher train error, slightly better validation score), but the validation results with test-time augmentation (see below) were consistently slightly worse for some reason.

Pre-training might allow us to scale our models up further, but because they already took a long time to train, and because the pre-training itself was time-consuming as well, we did not end up doing this for any of our final models.

To learn useful features with unsupervised pre-training, we relied on the max-pooling and unpooling layers to serve as a sparsification of the features. We did not try a denoising autoencoder approach for two reasons: first of all, according to the results described by Masci et al., the max- and unpooling approach produces way better filters than the denoising approach, and the further improvement of combining these approaches is negligible. Secondly, due to how the networks were implemented, it would slow things down a lot.

We tried different setups for this pre-training stage:

• greedy layerwise training vs. training the full deconvolutional stack jointly: we obtained the best results when pre-training the full stack jointly. Sometimes it was necessary to initialize this stack using the greedy approach to get it to work.
• using tied weights vs. using untied weights: Having the weights in the deconvolutional layers be the transpose of those in the corresponding convolutional layers made the (full) autoencoder easier and much faster to train. Because of this, we never got the CAE with untied weights to reconstruct the data as well as the CAE with tied weights, despite having more trainable parameters.

We also tried different approaches for the supervised finetuning stage. We observed that without some modifications to our supervised training setup, there was no difference in performance between a pre-trained network and a randomly initialized one. Possibly, by the time the randomly initialized dense layers are in a suitable parameter range, the network has already forgotten a substantial amount of the information it acquired during the pre-training phase.

We found two ways to overcome this:

• keeping the pre-trained layers fixed for a while: before training the full networks, we only train the (randomly initialized) dense layers. This is quite fast since we only need to backpropagate through the top few layers. The idea is that we put the network more firmly in the basin of attraction the pre-training led us to.

• Halving the learning rate in the convolutional layers: By having the dense layers adapt faster to the (pre-trained) convolutional layers, the network is less likely to make large changes to the pre-trained parameters before the dense layers are in a good parameter range.

Both approaches produced similar results.

Pseudo-labeling

Another way we exploited the information in the test set was by a combination of pseudo-labeling and knowledge distillation (Hinton et al.). The initial results from models trained with pseudo-labeling were significantly better than we anticipated, so we ended up investigating this approach quite thoroughly.

Pseudo-labeling entails adding test data to the training set to create a much larger dataset. The labels of the test datapoints (so called pseudo-labels) are based on predictions from a previously trained model or an ensemble of models. This mostly had a regularizing effect, which allowed us to train bigger networks.

We experimented both with hard targets (one-hot coded) and soft targets (predicted probabilities), but quickly settled on soft targets as these gave much better results.

Another important detail is the balance between original data and pseudo-labeled data in the resulting dataset. In most of our experiments 33% of the minibatch was sampled from the pseudolabeled dataset and 67% from the real training set.

It is also possible to use more pseudo-labeled data points (e.g. 67%). In this case the model is regularized a lot more, but the results will be more similar to the pseudolabels. As mentioned before, this allowed us to train bigger networks, but in fact this is necessary to make pseudo-labeling work well. When using 67% of the pseudo-labeled dataset we even had to reduce or disable dropout, or the models would underfit.

Our pseudo-labeling approach differs from knowledge distillation in the sense that we use the test set instead of the training set to transfer knowledge between models. Another notable difference is that knowledge distillation is mainly intended for training smaller and faster networks that work nearly as well as bigger models, whereas we used it to train bigger models that perform better than the original model(s).

We think pseudo-labeling helped to improve our results because of the large test set and the combination of data-augmentation and test-time augmentation (see below). When pseudo-labeled test data is added to the training set, the network is optimized (or constrained) to generate predictions similar to the pseudo-labels for all possible variations and transformations of the data resulting from augmentation. This makes the network more invariant to these transformations, and forces the network to make more meaningful predictions.

We saw the biggest gains in the beginning (up to 0.015 improvement on the leaderboard), but even in the end we were able to improve on very large ensembles of (bagged) models (between 0.003 - 0.009).

Model averaging

We combined several forms of model averaging in our final submissions.

Test-time augmentation

For each individual model, we computed predictions across various augmented versions of the input images and averaged them. This improved performance by quite a large margin. When we started doing this, our leaderboard score dropped from 0.7875 to 0.7081. We used the acronym TTA to refer to this operation.

Initially, we used a manually created set of affine transformations which were applied to each image to augment it. This worked better than using a set of transformations with randomly sampled parameters. After a while, we looked for better ways to tile the augmentation parameter space, and settled on a quasi-random set of 70 transformations, using slightly more modest augmentation parameter ranges than those used for training.

Computing model predictions for the test set using TTA could take up to 12 hours, depending on the model.

Finding the optimal transformation instead of averaging

Since the TTA procedure improved the score considerably, we considered the possibility of optimizing the augmentation parameters at prediction time. This is possible because affine transformations are differentiable with respect to their parameters.

In order to do so, we implemented affine transformations as layers in a network, so that we could backpropagate through them. After the transformation is applied to an image, a pixel can land in between two positions of the pixel grid, which makes interpolation necessary. This makes finding these derivatives quite complex.

We tried various approaches to find the optimal augmentation, including the following:

• Optimizing the transformation parameters to maximize (or minimize) the confidence of the predictions.
• Training a convnet to predict the optimal transformation parameters for another convnet to use.

Unfortunately we were not able to improve our results with any of these approaches. This may be because selecting an optimal input augmentation as opposed to averaging across augmentations removes the regularizing effect of the averaging operation. As a consequence we did not use this technique in our final submissions, but we plan to explore this idea further.

Combining different models

In total we trained over 300 models, so we had to select how many and which models to use in the final blend. For this, we used cross-validation on our validation set. On each fold, we optimized the weights of all models to minimize the loss of the ensemble on the training part.

We regularly created new ensembles from a different number of top-weighted models, which we further evaluated on the testing part. In the end, this could give an approximate idea of suitable models for ensembling.

Once the models were selected, they were blended uniformly or with weights optimized on the validation set. Both approaches gave comparable results.

The models selected by this process were not necessarily the ones with the lowest TTA score. Some models with relatively poor scores were selected because they make very different predictions than our other models. A few models had poor scores due to overfitting, but were selected nevertheless because the averaging reduces the effect of overfitting.

Bagging

To improve the score of the ensemble further, we replaced some of the models by an average of 5 models (including the original one), where each model was trained on a different subset of the data.

Miscellany

Here are a few other things we tried, with varying levels of success:

• untied biases: having separate biases for each spatial location in the convolutional layer seemed to improve results very slightly.
• winner take all nonlinearity (WTA, also known as channel-out) in the fully connected layers instead of ReLUs / maxout.
• smooth nonlinearities: to increase the amount of variance in our blends we tried replacing the leaky rectified linear units with a smoothed version. Unfortunately this worsened our public leaderboard score.
• specialist models: we tried training special models for highly confusable classes of chaetognaths, some protists, etc. using the knowledge distillation approach described by Hinton et al.. We also tried a self-informed neural network structure learning (Warde-Farley et al.), but in both cases the improvements were negligible.
• batch normalization: unfortunately we were unable to reproduce the spectacular improvements in convergence speed described by Ioffe and Szegedy for our models.
• Using FaMe regularization as described by Rudy et al. instead of dropout increased overfitting a lot. The regularizing effect seemed to be considerably weaker.
• Semi-supervised learning with soft and hard bootstrapping as described by Reed et al. did not improve performance or reduce overfitting.

Here’s a non-exhaustive list of things that we found to reduce overfitting (including the obvious ones):

• dropout (various forms)
• aggressive data augmentation
• suitable model architectures (depth and width of the layers influence overfitting in complicated ways)
• weight decay
• unsupervised pre-training
• cyclic pooling (especially with root-mean-square pooling)
• leaky ReLUs
• pseudo-labeling

We also monitored the classification accuracy of our models during the competition. Our best models achieved an accuracy of over 82% on the validation set, and a top-5 accuracy of over 98%. This makes it possible to use the model as a tool for speeding up manual annotation.

Conclusion

We had a lot of fun working on this problem together and learned a lot! If this problem interests you, be sure to check out the competition forum. Many of the participants will be posting overviews of their approaches in the coming days.

Congratulations to the other winners, and our thanks to the competition organizers and sponsors. We would also like to thank our supervisor Joni Dambre for letting us work on this problem together.

We will clean up our code and put it on GitHub soon. If you have any questions or feedback about this post, feel free to leave a comment.

One of our team, Ira Korshunova, is currently looking for a good research lab to start her PhD next semester. She can be contacted at irene.korshunova@gmail.com.

UPDATE (March 25th): the code is now available on GitHub: https://github.com/benanne/kaggle-ndsb

Guest post: Jan Schlüter from the OFAI, a fellow MIR researcher I have met at several conferences, recently added a feature to Theano that fits so well with my previous two posts on fast convolutions that we decided to include his writeup on my blog. So enjoy the third part of the series, written by Jan!

Over the past year, Theano has accumulated several alternative implementations for 2D convolution, the most costly operation in Convolutional Neural Networks. There is no single implementation that is the fastest for all possible image and kernel shapes, but with Theano you can mix and match them at will. Now mixing and matching is something that can be easily automated: Meet meta-optimization!

The idea is to automatically select the fastest available implementation for each individual convolution operation in a Theano function, simply by timing them. The feature is already available in Theano: If you install the latest version from github, you can activate it by setting the environment variable THEANO_FLAGS=optimizer_including=conv_meta,metaopt.verbose=1.

In the following, I will explain what it does, how it works, and demonstrate that it can outperform all existing convnet libraries.

Batched convolution

Before we begin, note that the convolution operation in Convolutional Neural Networks (CNNs) as used for Computer Vision is not just a convolution of a single 2D input image with a single 2D filter kernel. For one, the input image can have multiple channels, such as a color image composed of three values per pixel. It can thus be expressed as a 3D tensor. To match this, the filter kernel has as many values per pixel as the input image, which makes it a 3D tensor as well. When computing the output, each channel is convolved separately with its corresponding kernel, and the resulting images are added up to produce a single 2D output image. But usually, each convolutional layer returns a multi-channel output (a 3D tensor), which is achieved by learning multiple sets of kernels (a 4D tensor). Finally, images are often propagated through the network in mini-batches of maybe 64 or 256 items to be processed independently, so the input and output become 4D tensors.

Putting everything together, the batched convolution operation convolves a 4D input tensor with a 4D kernel tensor to produce a 4D output tensor. Obviously, this gives ample of opportunities for parallelization. Add to this the different possible ways of computing a 2D convolution, and you can see why there are so many competing implementations.

The repertoire

As an actively maintained open-source project with several external contributors, Theano has grown to have access to five convolution implementations:

• a “legacy” implementation that has been created for Theano
• Alex Krizhevsky’s cuda-convnet, via a wrapper already described by Sander
• an FFT-based convolution started by Sander and finished by Arnaud Bergeron
• the gemm-based convolution from Caffe, started by Arjun Jain and Frédéric Bastién and finished by me
• Nvidia’s new cuDNN library, via a wrapper done by Arnaud and subsequently improved by Frédéric and me

All of these have their strengths and weaknesses. cuda-convnet only supports square kernels and places several restrictions on the number of input and output channels and the batch size. The FFT-based based convolution is applicable to any configuration, but requires a lot of extra memory that practically limits it to small batch and image sizes or very potent graphics cards. cuDNN requires a GPU of Compute Capability 3.0 or above, and the convolution ported from Caffe needs some extra memory again. Finally, the legacy implementation comes free of limitations, but is usually the slowest of the pack.

Depending on the configuration – that is, the batch size, image shape, filter count and kernel shape –, any of these five implementations can be the fastest.

Three convolutions per layer

To complicate matters, each convolutional layer in a convnet actually results in three batched convolution operations to be performed in training:

1. The forward pass, a valid convolution of images and kernels
2. The gradient wrt. weights, a valid convolution of images and the gradient wrt. output
3. The gradient wrt. input, a full convolution of the kernels and the gradient wrt. output

For a valid convolution, the kernel is applied wherever it completely overlaps with the input (i.e., it only touches valid data). For a full convolution, it is applied wherever it overlaps with the input by at least one pixel – this is equivalent to padding the input with a suitably-sized symmetric border of zeros and applying a valid convolution.

(For the eager ones: The third one in the list above is actually a correlation, because the kernels are not flipped as in the forward pass. And the second one requires the batch size and channels of the input, kernel and output tensors to be swapped. Still all of these can be expressed using the batched convolution operation described in the beginning.)

The “big libraries” (cuda-convnet, Caffe and cuDNN) each come with three algorithms specialized for these three cases, while the FFT-based convolution just distinguishes between valid and full convolutions.

Cherry-picking

A lot of my work on Theano’s convolution was triggered by following Soumith Chintala’s convnet-benchmarks initiative, which set out to compare all freely available Convolutional Neural Network libraries in terms of their performance. When looking at some of the first results posted, the first thing I noticed was that it would pay off to use a different library for each of the five configurations tested. This has quickly been included as a hypothetical “cherry-picking” row into the result tables.

I took over maintenance of Soumith’s Theano benchmark script and evolved it into a handy little tool to compare its convolution implementations for different configurations. Feel free to download the script and follow along.

So let’s see what we could gain with cherry-picking in Theano:

$ SKIP=meta python pylearn2_benchmark.py i3x64x64,k128x7x7,b64
Using gpu device 0: GeForce GTX 780 Ti

CONFIG: input = 3 x 64 x 64 * ker = 3 x 128 x 7 x 7 ( bs = 64 , stride = 1 )
theano.tensor.nnet.conv.conv2d                     ==> fprop         ==>      43
theano.tensor.nnet.conv.conv2d                     ==> bprop inputs  ==>      44
theano.tensor.nnet.conv.conv2d                     ==> bprop weights ==>     185

theano.sandbox.cuda.fftconv.conv2d_fft             ==> fprop         ==>      19
theano.sandbox.cuda.fftconv.conv2d_fft             ==> bprop inputs  ==>      26
theano.sandbox.cuda.fftconv.conv2d_fft             ==> bprop weights ==>      20

(auto) theano.sandbox.cuda.dnn.GpuDnnConv          ==> fprop         ==>       4
(auto) theano.sandbox.cuda.dnn.GpuDnnConv          ==> bprop inputs  ==>       7
(auto) theano.sandbox.cuda.dnn.GpuDnnConv          ==> bprop weights ==>       6

(auto) theano.sandbox.cuda.blas.GpuCorrMM          ==> fprop         ==>       6
(auto) theano.sandbox.cuda.blas.GpuCorrMM          ==> bprop inputs  ==>       7
(auto) theano.sandbox.cuda.blas.GpuCorrMM          ==> bprop weights ==>      10

pylearn2.sandbox.cuda_convnet(partial_sum=None)    ==> fprop         ==>       7
pylearn2.sandbox.cuda_convnet(partial_sum=None)    ==> bprop inputs  ==>      11
pylearn2.sandbox.cuda_convnet(partial_sum=None)    ==> bprop weights ==>      47

pylearn2.sandbox.cuda_convnet(partial_sum=1)       ==> fprop         ==>       7
pylearn2.sandbox.cuda_convnet(partial_sum=1)       ==> bprop inputs  ==>      11
pylearn2.sandbox.cuda_convnet(partial_sum=1)       ==> bprop weights ==>      13

What we see here are the respective computation times in milliseconds for a particular configuration (tensor shapes) for the legacy implementation, FFT-based convolution, cuDNN, gemm-based convolution and cuda-convnet (with two different values for a tuning parameter). For this layer, cuDNN would be the optimal choice.

Let’s try a second configuration:

$ SKIP=meta python pylearn2_benchmark.py i32x15x80,k64x5x5,b256
Using gpu device 0: GeForce GTX 780 Ti

CONFIG: input = 32 x 15 x 80 * ker = 32 x 64 x 5 x 5 ( bs = 256 , stride = 1 )
theano.tensor.nnet.conv.conv2d                     ==> fprop         ==>     146
theano.tensor.nnet.conv.conv2d                     ==> bprop inputs  ==>     182
theano.tensor.nnet.conv.conv2d                     ==> bprop weights ==>     162

theano.sandbox.cuda.fftconv.conv2d_fft             ==> fprop         ==>      20
theano.sandbox.cuda.fftconv.conv2d_fft             ==> bprop inputs  ==>      24
theano.sandbox.cuda.fftconv.conv2d_fft             ==> bprop weights ==>      15

(auto) theano.sandbox.cuda.dnn.GpuDnnConv          ==> fprop         ==>      18
(auto) theano.sandbox.cuda.dnn.GpuDnnConv          ==> bprop inputs  ==>      23
(auto) theano.sandbox.cuda.dnn.GpuDnnConv          ==> bprop weights ==>      25

(auto) theano.sandbox.cuda.blas.GpuCorrMM          ==> fprop         ==>      22
(auto) theano.sandbox.cuda.blas.GpuCorrMM          ==> bprop inputs  ==>      29
(auto) theano.sandbox.cuda.blas.GpuCorrMM          ==> bprop weights ==>      30

pylearn2.sandbox.cuda_convnet(partial_sum=None)    ==> fprop         ==>      16
pylearn2.sandbox.cuda_convnet(partial_sum=None)    ==> bprop inputs  ==>      20
pylearn2.sandbox.cuda_convnet(partial_sum=None)    ==> bprop weights ==>      40

pylearn2.sandbox.cuda_convnet(partial_sum=1)       ==> fprop         ==>      16
pylearn2.sandbox.cuda_convnet(partial_sum=1)       ==> bprop inputs  ==>      21
pylearn2.sandbox.cuda_convnet(partial_sum=1)       ==> bprop weights ==>      28

This time, the FFT-based convolution is faster, but the truly optimal choice would be combining it with cuda-convnet.

We see that the meta-optimizer should not just cherry-pick a different implementation per convolutional layer, but even a different implementation for each of the three convolutions in a layer – something that was not possible in Theano before (nor in any other library I am aware of).

The “swapping trick”

As you recall, cuda-convnet, Caffe and cuDNN come with specialized algorithms for the three convolutions per layer. Interestingly, when porting the gemm-based convolution from Caffe to Theano, I noticed that the effort I put in properly using its two backward pass algorithms when applicable did not always pay off: For some configurations, it was faster to just use the forward pass algorithm instead, transposing tensors as needed. I thus added a shape-based heuristic to select the fastest algorithm for the gemm-based convolution (making Theano’s port faster than Caffe for some configurations).

When adding support for Nvidia’s cuDNN library, Arnaud understandably assumed that it would hide this complexity from the user and select the optimal algorithm internally. So at first, Theano did not tell cuDNN whether a particular convolution’s purpose was a forward pass or one of the backward passes. When I changed the implementation accordingly, I again noticed that while performance generally improved a lot, for some configurations, using the “wrong” algorithm was actually faster.

Just as for Caffe, we can use this knowledge to be faster than cuDNN. As the implementation is unknown, we cannot easily define a heuristic for choosing between the cuDNN algorithms. However, the meta-optimizer can just try all applicable algorithms and see which one is the fastest. I found it to suffice to just try two algorithms per convolution:

• For the forward pass, try the “correct” algorithm and the gradient wrt. weights (both are valid convolutions)
• For the gradient wrt. weights, try the “correct” algorithm and the forward pass
• For the gradient wrt. inputs, try the “correct” algorithm and the forward pass (with additional zero padding to make it a full convolution)

I call this the “swapping trick” because it often leads to the first two algorithms being swapped.

Implementation

To understand why Theano was a perfect fit to add automatic algorithm selection, we will need to explain a bit of its inner workings.

First, Theano is not a neural network library, but a mathematical expression compiler. In contrast to, say, Caffe, its basic components are not neural network layers, but mathematical operations. Implementing a neural network is done by composing the expression for the forward pass (which will probably include matrix multiplications, vector additions, elementwise nonlinearities and possibly batched convolution and pooling), using this to build an expression for the training cost, and then letting Theano transform it into expressions for the gradients wrt. the parameters to be learned. Finally, the expressions are compiled into functions that evaluate them for specific settings of the free variables (such as a mini-batch of training data).

But right before an expression is compiled, it is optimized, and this is where all the magic happens. The expression is represented as a graph of Apply nodes (operations) and Variable nodes (the inputs and outputs of an operation), and Theano comes with a bunch of graph optimizers that modify the graph to produce the same result either more efficiently or more numerically stable.
One particular graph optimizer moves convolution operations from the CPU to the GPU by replacing the respective Apply node and adding the necessary transfer operations around it. A whole set of graph optimizers then replaces the legacy GPU convolution operation with one of the more efficient implementations available in Theano. These optimizers have relative priorities and can be enabled and disabled by the user.

The new meta-optimizer is just another graph optimizer with a twist: When it encounters a convolution operation, it applies each of the set of available graph optimizers (plus the cuDNN “swapping trick” optimizer) in sequence, each time compiling and executing the subgraph performing the convolution, and chooses the one resulting in the best performance. (Finally, this explains why it’s called meta-optimization.)
As the basic components in Theano are the mathematical operations, there is no extra work needed to be able to choose different implementations for the three convolutions per layer: All Theano sees when optimizing and compiling an expression is a graph containing several anonymous convolution operations, so it will naturally optimize each of them separately.

Practical gains

Let us now put the meta-optimizer to test using the benchmark script mentioned in the cherry-picking section:

$ THEANO_FLAGS=metaopt.verbose=1 SKIP=legacy,gemm,fft,convnet,dnn python pylearn2_benchmark.py i128x36x12,k64x6x3,b256
Using gpu device 0: GeForce GTX 780 Ti

CONFIG: input = 128 x 36 x 12 * ker = 128 x 64 x 6 x 3 ( bs = 256 , stride = 1 )
ConvMetaOptimizer meta-optimizing GpuConv{valid, (1, 1), None, (3, 6), True, (128, 12, 36), (3, 6)}(GpuFromHost.0, GpuFromHost.0) (5 choices):
* local_conv_fft_full: not applicable
* local_conv_fft_valid: 0.012958 sec
* local_conv_dnn: 0.021169 sec
* local_conv_gemm: 0.03973 sec
* local_conv_dnn_alternative: 0.044379 sec
= local_conv_fft_valid
(experimental) meta-optimizer                      ==> fprop         ==>      12
ConvMetaOptimizer meta-optimizing GpuConv{full, (1, 1), None, (3, 6), True, (64, 10, 31), (3, 6)}(GpuFromHost.0, GpuFromHost.0) (5 choices):
* local_conv_fft_full: 0.019099 sec
* local_conv_fft_valid: not applicable
* local_conv_dnn: 0.032979 sec
* local_conv_gemm: 0.028478 sec
* local_conv_dnn_alternative: 0.015099 sec
= local_conv_dnn_alternative
(experimental) meta-optimizer                      ==> bprop inputs  ==>      15
ConvMetaOptimizer meta-optimizing GpuConv{valid, (1, 1), None, (10, 31), False, (256, 12, 36), (10, 31)}(GpuFromHost.0, GpuFromHost.0) (5 choices):
* local_conv_fft_full: not applicable
* local_conv_fft_valid: 0.011441 sec
* local_conv_dnn: 0.030338 sec
* local_conv_gemm: 0.025984 sec
* local_conv_dnn_alternative: 0.031552 sec
= local_conv_fft_valid
(experimental) meta-optimizer                      ==> bprop weights ==>      12

In verbose mode, the meta-optimizer reports which implementations are tested, how each of them performs and which one is finally chosen. For the configuration at hands, it turns out that the FFT-based implementation is fastest for the forward pass and the gradient wrt. weights, and cuDNN is fastest for the gradient wrt. inputs – but only when using the “wrong” algorithm for it (namely, cuDNN’s forward pass algorithm with zero padding, tried according to the swapping trick). In all three instances, the optimal algorithm is about twice as fast as just choosing cuDNN, which would have been Theano’s current default behavior.

When training a full network, the impact will generally be smaller, because the convolution operations only constitute a part of the expressions evaluated (but often the most costly part). The improvement also heavily depends on the input and kernel shapes – for a wide range of configurations, just using cuDNN for all convolutions is nearly optimal. Still, a colleague of Sander reported a threefold performance improvement for a network trained for a Kaggle competition, with the meta-optimizer combining FFT, Caffe, and cuDNN with and without the swapping trick.

To get an estimate on how much Theano could help for your use case, just run the benchmark script for the configurations occurring in a forward pass through your network. If you already use Theano, just set THEANO_FLAGS=optimizer_including=conv_meta to rest assured you will always make the most out of the time (and electricity!) you spend on training your networks.

Future

While the basic machinery is in place and works fine, there are a lot of conceivable improvements:

• The meta-optimizer should cache its results on disk to speed up repeated compilations of the same graph.
• Right now, the meta-optimizer uses all available convolution operations in Theano; it should be possible to control this.
• As cuda-convnet is not included in Theano, but an external project (Pylearn2), it is not included in the meta-optimizer. However, it is possible to register additional optimizers at runtime via theano.sandbox.cuda.opt.conv_metaopt.register(). It would be nice to write such a pluggable optimizer for cuda-convnet.
• Similarly, it would be nice to have a wrapper for cuda-convnet2 (in a separate repository) along with an optimizer to be registered with the meta-optimizer.
• Currently, meta-optimization can only be used for non-strided valid or full convolutions, because this is what the legacy implementation is limited to. Changing this would require some refactoring, but lead to cleaner code and slightly improved performance.
• Finally, it could be worthwhile to repeat the same for the pooling operation of CNNs: Port additional implementations to Theano, benchmark them and add a meta-optimizer.

Watch Issue #2072 on github for any progress on this, or even better, step in and implement one of these features if you can use it! Both that issue and theano-dev are well-suited to ask for hints about implementing any of these TODOs – we’d be glad to have you on board.