# InfoGAN: using the variational bound on mutual information (twice)

Many people have recommended me the infoGAN paper, but I hadn't taken the time to read it until recently. It is actually quite cool:

#### Summary of this note

• I show how the original GAN algorithm can be derived using exactly the same variational lower-bound that the authors use in this paper (see also this blog post by Yingzhen)
• However, GANs use the bound in the wrong direction and end up minimising a lower bound which is not a good thing to do
• InfoGANs can be expressed purely in terms of mutual information, and applying the variational bound twice: once in the correct direction, once in the wrong direction
• I believe that the unstable behaviour of GANs is partially explained by using the bound in the incorrect way

## Mini-review

The InfoGAN idea is pretty simple. The paper presents an extension to the GAN objective. A new term encourages high mutual information between generated samples and a small subset of latent variables cc. The hope is that by forcing high information content, we cram the most interesting aspects of the representation into cc.

If we were successful, cc ends up representing the most salient and most meaningful sources of variation in the data, while the rest of the noise variables zz will account for additional, meaningless sources of variation and can essentially be dismissed as uncompressible noise.

In order to maximise the mutual information, the authors make use of a variational lower bound. This, conveniently, results in a recognition model, similar to the one we see in variational autoencoders. The recognition model infers latent representation cc from data.

The paper is pretty cool, the results are convincing. I found the notation and derivation a bit confusing, so here is my mini-review:

• I think the introduction, I don't think it's fair to say "To the best of our knowledge, the only other unsupervised method that learns disentangled representations is hossRBM". There are loads of other methods that attempt this.
• I believe Lemma 5.1 is basically a trivial application of the theorem of total expectation, and I really don't see the need to provide a proof for that (maybe reviewers asked for a proof).

ref paper eqn 5

## My view on InfoGANs

I think there is an interesting connection that the authors did not mention (frankly, it probably would have overcomplicated the presentation). The connection is that original GAN objective itself can be derived from mutual information, and in fact, the discriminator D can be thought of as a variational auxillary variable, exactly the same role as the recognition model q(c|x)q(c|x) in the InfoGAN paper.

The connection relies on the interpretation of Jensen-Shannon divergence as mutual information (see e.g. Yingzen's blog postGANs, mutual information, and possibly algorithm selection?). Here is my graphical model view on InfoGANs that may put things in a slightly different light:

Let's consider the joint distribution of a bunch of varibles:

Now, the main problem is with this derivation is that we were supposed to minimise ℓGANℓGAN, so we really would like an upper bound instead of a lower bound. But the variational method only provides a lower bound. Therefore,

GANs minimise a lower bound, which I believe accounts for some of their unstable behaviour

### InfoGANs use the bound twice

Recall that the idealised InfoGAN objective is the weighted difference of two mutual information terms.

To arrive at the algorithm the authors used, one uses the bound on both mutual information terms.

• When you apply the bound on the first term, you get a lower bound, and you introduce an auxillary distribution that ends up being called the discriminator. This application of the bound is wrong because it bounds the loss function from the wrong side.
• When you apply the bound on the second term, you end up upper bounding the loss function, because of the negative sign. This is a good thing. The combination of a lower bound and an upper bound means that you don't even know which direction you're bounding or approximating the loss function from anymore, it's neither an upper or a lower bound.

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