# Language Games With Neural Networks

#### Code

##### Noah Trenaman

The limits of my language mean the limits of my world.
— Wittgenstein

In 1948, Claude Shannon published A Mathematical Theory of Communication, which introduces some math for thinking about how to set up a communication system. To communicate a message correctly from person A to person B, we don't want to lose any of the important "information" in the message.

In the real world, the systems we use to communicate information tend to be called protocols or languages. For example, TCP is a protocol for passing information through the Internet, and natural languages like English are spoken so that we can pass information along to other people.1 Communication systems can be designed, or they can emerge spotaneously and evolve over time — like most spoken languages. This observation begs the question: "Could we design environments which encourage novel communication systems to emerge and evolve?"

A helpful tool from information theory is the Shannon-Weaver model, which universally describes the elements in a communication system:

1. An information source
the message we want to communicate
2. A transmitter
a process for representing that message
3. A channel
how the message is represented as it travels from the source to the destination
4. A receiver
a process for reconstructing the message
5. A destination
the message that was received
Original Representation Transformation Modified Representation

## Using Neural Networks To Generate Simple Writing Systems

As universal function approximators, neural networks can learn how to transform information from one form to another. An autoencoder is probably the closest match between a neural network and the Shannon-Weaver model. The input and the output of an autoencoder are the same. The interesting part of an autoencoder is what is called the bottleneck, or latent space, which lines up with the idea of a communication channel. The latent space is a transformed (and often compressed) representation of some input, which the autoencoder learns to reconstruct as accurately as possible on the other end.

I've been interested in exploring how these systems could generate patterns that are interpretable. Particularly visual patterns generated by neural networks that operate similarly to writing systems. This could have a whole lot of interesting use-cases such as generating highly readable fonts, turning source code into pictures, or transforming entire sentences or paragraphs into abstract images using a language model such as BERT. Joel Simon did some really interesting experiments with generating writing systems using neural networks in 2018. His project inspired my initial curiosity.

One straightforward idea would be to generate an alphabet of a particular size. For an alphabet with 10 characters, one can train an autoencoder to produce 10 unique visual patterns. The inputs and outputs are simply the numbers 1-10 (as one-hot vectors), and the latent space is an image. The loss function should encourage the autoencoder to reconstruct the original message by accurately predicting what the input was, based on the image. We can do this with a simple cross-entropy loss which encourages the predicted message to match the input message.

If we add some constraints to the latent space such as rotation or blur, we'll get patterns that are more coherent and distinct. An information-theoretic interpretation is that adding noise forces the model to learn error-correcting codes. This is a simple way to try making the alphabet more humanlike and interesting. If you tilt your head or squint your eyes, you can still read the letters in a word, because our written languages are robust enough to a survive bit of blur and rotation.

The resulting system contains an encoder, which takes a message and turns it into an image, as well as a decoder, which takes an image and predicts its message. Based on the design of Joel Simon's experiment, Ryan Murdock first called these systems Cooperative Communication Networks.

The result of training an autoencoder this way may look like this:

This animation interpolates between the different learned symbols. We get a very distinct visual pattern for each character in the artificial alphabet.

These generative systems grow in complexity in proportion to the complexity of the environment that they are generated in. We can set additional constraints on the communication channel to encourage more interesting patterns to emerge. We could also define different kinds of messages beyond simple alphabets to represent visually. It seems that there's a huge space of constraints and objectives one could experiment with, with many interesting patterns to be generated!

### Attempting To Learn A Math Notation System

One kind of message I enjoyed experimenting with was arithmetic expressions. I was curious if an autoencoder would, by turning syntax trees into images, learn a kind of mathematical notation system. Just like with a simple 10-character alphabet, we can train an autoencoder to learn representations of the message by defining a reconstruction loss function. The loss function for reconstructing syntax trees is a little trickier. A syntax tree predicted by a graph autoencoder (GraphVAE) is based on an adjacency matrix and a set of node feature matrices (in this case there was one for node type and one for node order.) So, the message is a collection of one-hot matrices that are predicted all at once by the graph decoder.

This model predicts both the adjacency matrix and node feature matrix representing a syntax tree. The two prediction loss scores are summed together to give the full reconstruction loss (with an additional trick to admit permutation-invariance of graphs3).

These patterns are visually distinct, but much less interpretable compared to the simple alphabet. The interpolations are also very shaky.

## Inspiration from GANs: Introducing A Spy

Maybe one of the most interesting constraints on an environment is to have another agent that is actively trying to learn the same language, and trying to deceive the decoder network. We'll find that this is very similar to a GAN. Although in this system there are two generators, and the images are not constrained to any dataset. In this system, there is an honest generator that is designing our language, and a spy generator that is trying to discover the language. The true generator and decoder will (ideally) collaborate to outsmart the spy.

Our system has two generators and one decoder. One is the "true" generator $G_t$ which wants to generate symbols that the decoder $D$ can interpret. Their relationship is cooperative. At the same time, there is a "spy" generator $G_s$ which attempts to imitate $G_t$ and thus trick $D$. Their relationship is adversarial.

The spy has no access to the true generator, but it can learn to imitate $G_t$ via backpropagation of its loss function — $G_s$ is trained to trick $D$ by making symbols that are indistinguishable from $G_t$. Thus, both generators attempt to transform a message $M$ into a representation that $D$ can understand:

$z_t = G_t(M)$

$z_s = G_s(M)$

And if both generators succeed, $D$ will be able to reconstruct the original message from the generated representations:

$D(z_t) = D(z_s) = M$

### Catching The Spy

So far, there's nothing in this system that would be able to catch the insidious spy. We need to train $G_t$ and $D$ to collaborate in outsmarting $G_s$. We can train $D$ itself to tell the difference between $z_t$ and $z_s$, but couldn't we help the true generator more directly?

We can borrow the idea of adversarial training from GANs but update $G_t$ and $D$ based on how much $D$ is being deceived. I think the closest example to the loss function is a Wasserstein GAN. A WGAN has a decoder network (called a "critic") that predicts the realness of an image. In this case, rather than predicting a scalar of "realness," we can encourage the decoder to be more certain with real images $z_t$, and more uncertain when exposed to fake images $z_s$.

### Cooperative And Adversarial Loss

Like in a GAN, we have two loss functions described by competing networks.

Our shorthand for the loss of a prediction $M'$ of a message $M$ will be $\mathcal{L}(M', M)$.

#### Cooperative loss (with $G_s$ frozen)

$\mathcal{L}_c = \mathcal{L}(D(G_t(M)), M) - \mathcal{L}(D(G_s(M)), M)$

Read as: try to reconstruct messages from the true generator as accurately as possible, and try to reconstruct messages from the spy generator as inaccurately as possible (with maximum uncertainty).

#### Adversarial loss (with $D$ frozen)

$\mathcal{L}_a = \mathcal{L}(D(G_s(M)), M)$

Read as: try to trick the decoder into thinking that the spy generator's representations are actually from the true generator.

Cross-entropy loss can be unstable and produce very large gradients when the cooperative networks are doing well in outsmarting the spy generator, to the point where the gradients for outsmarting the spy are greater than the cooperative objective itself. A bespoke fix is to make the adversarial loss linear rather than logistic. You can do this by switching out the cross-entropy-based loss function $\mathcal{L}$ with $L_1$ (mean absolute error) loss:

$\mathcal{L}_c = \mathcal{L}(D(G_t(M)), M) - L_1(D(G_s(M)), M)$

Here are some examples of spies attempting to copy the true generator. The top row is the original generator, and the bottom row is the spy (the middle row is a visualization of some image augmentations also used during training3):

A GAN uses a training dataset as its source of ground truth, but we're using the symbols generated from $G_t$. This makes our "dataset" differentiable.

### Example Results

Cooperative (left) and adversarial (right) generators trained to represent syntax trees.

Generators trained to represent unique symbols (the alphabet task.)

In both cases, the adversarial generator collapsed.

I haven't done an in-depth study on the hyperparameters of both generators, but have found good results by giving the spy half as many parameters as the true generator (you can do this by cutting the number of feature channels in half.) This might encourage the true generator to make use of its extra parameters to outsmart the adversarial generator and could be a reasonable way to encourage more complex and diverse visual patterns.

1. But information theory is only one tool for thinking about natural language — "ideas" and "meaning" have no valid mathematical description. See semiotics.
2. Calculating the loss for GraphVAE is a little more complicated than this because a prediction $M'$ could be isomorphic to $M$ and still be completely valid. We need to do graph matching before calculating the loss. If you're interested, see section 3.3 of GraphVAE for more details.