Continuous Normalizing Flows with GalacticOptim.jl

Now, we study a single layer neural network that can estimate the density p_x of a variable of interest x by re-parameterizing a base variable z with known density p_z through the Neural Network model passed to the layer.

Copy-Pasteable Code

Before getting to the explanation, here's some code to start with. We will follow a full explanation of the definition and training process:

using DiffEqFlux, DifferentialEquations, GalacticOptim, Distributions

nn = Chain(
    Dense(1, 3, tanh),
    Dense(3, 1, tanh),
) |> f32
tspan = (0.0f0, 10.0f0)
ffjord_mdl = FFJORD(nn, tspan, Tsit5())

data_dist = Normal(6.0f0, 0.7f0)
train_data = rand(data_dist, 1, 100)

function loss(θ)
    logpx, λ₁, λ₂ = ffjord_mdl(train_data, θ)
    -mean(logpx)
end

adtype = GalacticOptim.AutoZygote()
res1 = DiffEqFlux.sciml_train(loss, ffjord_mdl.p, ADAM(0.1), adtype; maxiters=100)
res2 = DiffEqFlux.sciml_train(loss, res1.u, LBFGS(), adtype; allow_f_increases=false)


using Distances

actual_pdf = pdf.(data_dist, train_data)
learned_pdf = exp.(ffjord_mdl(train_data, res2.u)[1])
train_dis = totalvariation(learned_pdf, actual_pdf) / size(train_data, 2)

We can use DiffEqFlux.jl to define, train and output the densities computed by CNF layers. In the same way as a neural ODE, the layer takes a neural network that defines its derivative function (see [1] for a reference). A possible way to define a CNF layer, would be:

using DiffEqFlux, DifferentialEquations, GalacticOptim, Distributions

nn = Chain(
    Dense(1, 3, tanh),
    Dense(3, 1, tanh),
) |> f32
tspan = (0.0f0, 10.0f0)
ffjord_mdl = FFJORD(nn, tspan, Tsit5())

where we also pass as an input the desired timespan for which the differential equation that defines log p_x and z(t) will be solved.

Training a CNF layer

First, let's get an array from a normal distribution as the training data

data_dist = Normal(6.0f0, 0.7f0)
train_data = rand(data_dist, 1, 100)

Now we define a loss function that we wish to minimize

function loss(θ)
    logpx, λ₁, λ₂ = ffjord_mdl(train_data, θ)
    -mean(logpx)
end

In this example, we wish to choose the parameters of the network such that the likelihood of the re-parameterized variable is maximized. Other loss functions may be used depending on the application. Furthermore, the CNF layer gives the log of the density of the variable x, as one may guess from the code above.

We then train the neural network to learn the distribution of x.

Here we showcase starting the optimization with ADAM to more quickly find a minimum, and then honing in on the minimum by using LBFGS.

adtype = GalacticOptim.AutoZygote()
res1 = DiffEqFlux.sciml_train(loss, ffjord_mdl.p, ADAM(0.1), adtype; maxiters=100)

# output
* Status: success

* Candidate solution
   u: [-1.88e+00, 2.44e+00, 2.01e-01,  ...]
   Minimum:   1.240627e+00

* Found with
   Algorithm:     ADAM
   Initial Point: [9.33e-01, 1.13e+00, 2.92e-01,  ...]

We then complete the training using a different optimizer starting from where ADAM stopped.

res2 = DiffEqFlux.sciml_train(loss, res1.u, LBFGS(), adtype; allow_f_increases=false)

# output
* Status: success

* Candidate solution
   u: [-1.06e+00, 2.24e+00, 8.77e-01,  ...]
   Minimum:   1.157672e+00

* Found with
   Algorithm:     L-BFGS
   Initial Point: [-1.88e+00, 2.44e+00, 2.01e-01,  ...]

* Convergence measures
   |x - x'|               = 0.00e+00 ≰ 0.0e+00
   |x - x'|/|x'|          = 0.00e+00 ≰ 0.0e+00
   |f(x) - f(x')|         = 0.00e+00 ≤ 0.0e+00
   |f(x) - f(x')|/|f(x')| = 0.00e+00 ≤ 0.0e+00
   |g(x)|                 = 4.09e-03 ≰ 1.0e-08

* Work counters
   Seconds run:   514  (vs limit Inf)
   Iterations:    44
   f(x) calls:    244
   ∇f(x) calls:   244

Evaluation

For evaluating the result, we can use totalvariation function from Distances.jl. First, we compute densities using actual distribution and FFJORD model. then we use a distance function.

using Distances

actual_pdf = pdf.(data_dist, train_data)
learned_pdf = exp.(ffjord_mdl(train_data, res2.u)[1])
train_dis = totalvariation(learned_pdf, actual_pdf) / size(train_data, 2)

References:

[1] W. Grathwohl, R. T. Q. Chen, J. Bettencourt, I. Sutskever, D. Duvenaud. FFJORD: Free-Form Continuous Dynamic For Scalable Reversible Generative Models. arXiv preprint at arXiv1810.01367, 2018.