A pathway to more efficient generative models | Will Grathwohl | 2018 Summer Intern Open House

already yeah so I'm going to talk to you about some of the work I've been doing here with Sonic and alia on sort of an Avenue that we think might be a promising way to make invertible generative models potentially more efficient and more expressive and so as I mentioned when you're defining or designing an invertible generative model you have two main constraints when doing this you need the model to be invertible and you also need the model you need the to be able to compute the log determinant of the Jacobian of the model and those are two pretty big constraints and because of that we typically restrict ourselves to simpler architectures where those things are easier to compute and because of that we are using these simple transformations we must compose a whole bunch of them together to build an expressive model and as you as Sayaka mentioned is the case with the gloww model and if we could potentially use more expressive transformations at every step of our flow then there might be a step for us to build invertible models that are more expressive use less parameters and potentially use less computations so in some of the work we've been doing we took kind of a different look at these types of models and if you look at them in a different way something the math changes in a way that provides an avenue for maybe improving the models so you can think of

a flow based model as parameterizing a discrete-time dynamics process where you start at time 0 at the data and then to get your next time step you just apply every step of the flow and then you encode your data by moving along all the way from time 0 to the finish time and then you can compute the likelihood of your data simply by the likelihood of the final sample under the prior plus the sum of the log determinants along the way and if we kind of take the limit of time to infinity then this basically looks like a continuous time dynamics process where instead of having individual flow models parameterize the derivative at every single time step we can throw all of that into one model that's going to take in the current data point and time as well and then parameter i's the gradient exactly and then what this gives us is the exact parameter a ssin of an ordinary differential equation initial value problem and the coolest thing here is that if we look at the log probability from the change of variables formula now in the continuous case we have a difference a different term here instead of the sum of the log determinants we actually have the integral of the divergence of the function that defines the gradient and just looking at those two things next to each other the form is very similar except now we have this integral of the divergence instead of the sum of the log determinants and that's a very interesting dichotomy there because the determinant has a lot of different properties than the divergence and specifically in general if we have an arbitrary function that goes from RN to RN then computing the log determinant of the Jacobian is going to work in n cubed time once we have the Jacobian which is also challenging to compute and there's really no efficient way to get a to estimate this and there is no like efficient unbiased estimator but with the divergence we can actually produce an efficient unbiased estimator just using automatic differentiation and that estimator basically just works by sampling like a Gaussian probe vector and then pre and post multiplying the Jacobian Phi that vector and then in expectation that's going to give us the divergence of the vector field and then we can similarly use this in the integral form of the log-likelihood to get an unbiased estimate for the log-likelihood which is not something we can do in the discrete-time sits situation and here's just a little three line kind of tensorflow implementation of how you would implement this estimator and the cool thing here is that using this now we have a way to parameterize these invertible generative models using an arbitrary neural network and we also have a way to efficiently train them just using back propagation or standard automatic differentiation tools which are you know very common today so there are some problems here because we have alleviated a source of complexity but we have added another one because now instead of a very simple like known computation process where we just apply the you know n flows that we've defined we must now integrate actually this OD e and that is potentially challenging and it's kind of been the main area of work that we've been doing trying to make these models actually tractable and training them around the same scale of time that we could before and an even more challenging we have to now back propagate through the solution of an OD and get the gradients of the outputs with respect to its inputs and the parameters that define the flow function thankfully there was some recent work from the University of Toronto that presented a method that to do this and we've been building upon that method in this work and the basic way that works is you can actually get the gradients of everything you need just by solving an Augmented system of ordinary differential equations and while this is kind of out of left field from the deep learning kind of a type of work that we do there's actually been quite a number of decades of history on numerical methods for solving au des so there's decades of work that we can draw upon to a to help us out with that part of the

problem and so here's a somewhat of an example of some of the results I've gotten so here is the continuous normalizing flow on the left here and on the right we have the glow model and so in general I have not quite beaten the results of the glue model yet and I think that is mainly because the models currently take too long to train so I haven't been able to make them as large as I would like to but I have noticed that I have been able to get competitive results with real MVP and in some datasets I have beaten the glue model and so I think it's a very promising Avenue and I just think there's a couple kind of little issues that we need to solve before we can really get these models to deliver the goods and I guess just

here's a little kind of visualization of one of these models in the working and basically we've started from the prior distribution there and that was the continuous normalizing flow model actually integrating samples from the prior distribution to match the target distribution and so that was just in a simple two dimensional problem but this also works quite well on some

higher dimensional data so here is M this and as you can see over here these are actually going to be the digits that are being warped by this is the gradient field here that is being applied to them and then that is being integrated and actually warping them into what should look like Gaussian noise and once this finishes it will go backwards and it's pretty fun to look at yeah and the cool thing about this model is that this is just like one neural network that kind of has like an auto encoder type architecture and that neural network just takes in these the images at every time step that they've been warped and - the time itself and it just parameterised the gradient and all we need to do to apply this transformation is integrated and yes that's most of what I've been working on here and if you anyone is interested feel free to reach out to me there's my email and you have any questions oh that's typically just to got a standard isotropic distribution yeah I mean yeah that's the standard stuff I mean I've tried some other like more structured prior distributions and that like I did some earlier work here where I was working on using these models for like semi-supervised learning so you could put like a mixture of gaussians on some of the you know some of the vectors and that has an interesting effect of modeling you know class conditional probabilities pretty well but for all this work it was just standard Gaussian distributions just log like we herded the data yeah like that was the if you go well if we could go back a couple slides the bits per dimension is typically what people do which is sort of like you know log probability would be typically for you know written in Nats and then you just compute that from that's two bits and then average it over the dimensions that's the standard metric people use in the density estimation space