A new Gaussian process package, which facilitates user to integrate deep neural network into Gaussian process model (e.g. use neural network as mean function or kernel function to improve the power of their GP model). It use Zygote to compute derivatives w.r.t model parameters, and is naturally compatible with Flux.
- Building the GP mean function with a Flux's neural network
- Implement neural kernel network (arxiv 1806.04326), which makes it easy to build various composite kernels
This package is still under development, suggestions, bug report and pull request are welcome :), detailed documentation will come later...
Installing GPFlux requires run the following code in a Julia REPL:
] add GPFlux
Brief introduction to GP
Gaussian processe is a powerful algorithm in statistical machine learning and probabilistic modelling, it models the underlying distribution of a dataset by a prior belief ( which is a parametrized multivariate normal distribution ) and a Gaussian likelihood, learning is done by maximizing the log likelihood (MLE), which is tractable for Gaussian process. Gaussian process is widely used in surrogate function modelling, geostatitics, pattern recognition, etc.
Time series prediction
Gaussian process is determined by a mean function and a kernel function, they can be specified in GPFlux as follows
# mean function c = [0.0] # mean constant zero_mean = ConstantMean(c) # square exponential kernel ll = [0.0] # length scale in log scale lσ = [0.0] # scaling factor in log scale se_kernel = IsoGaussKernel(ll, lσ) # build Gauss process lnoise = [-2.0] # noise in log scale gp = GaussProcess(zero_mean, se_kernel, lnoise)
The parameters in the above
gp model are
lnoise, one can extract all parameters by:
ps = params(gp)
y, one can compute the negative log likelihood and it's gradient w.r.t all the parameters by:
negloglik(gp, X, y) # (X, y) is the dataset gradient(()->negloglik(gp, X, y), ps)
which are straight forward if you are familiar with Flux and Zygote.
One can also build composite kernel by using
AddCompositeKernel( Note: AD works for arbitrary composite kernels ).
se_kernel = IsoGaussKernel(ll, lσ) per_kernel = IsoPeriodKernel(lp, ll, lσ) se_mul_periodic_kernel = ProductCompositeKernel(se_kernel, per_kernel) se_add_periodic_kernel = AddCompositeKernel(se_kernel, per_kernel) params(se_mul_periodic_kernel) # provide parameters of se_mul_periodic_kernel params(se_add_periodic_kernel) # provide parameters of se_add_periodic_kernel
The most significant feature of GPFlux is that it allows to use Flux's neural network to build mean function and Neural Kernel Network (NKN) to build kernel function, the computation of negative log likelihood and it's gradient is same as above cases.
# build the mean function with neural network using Flux nn_mean = Chain(Dense(5, 10, relu), Dense(10, 1)) # build the kernel function with neural kernel network nkn = NeuralKernelNetwork(Primitive(se_kernel, per_kernel), Linear(2, 4), z->Product(z, step=4)) # build GP nn_gp = GaussProcess(nn_mean, nkn, lnoise) # compute negative log likelihood and gradient negloglik(nn_gp, X, y) gradient(()->negloglik(nn_gp, X, y), params(nn_gp))
Once we have negative log likelihood and gradients, we can either use Optim.jl or Flux's optimizers to do optimization.