Relative Entropy Gradient Sampler for Unnormalized Distributions

This repository is the official implementation of [Relative Entropy Gradient Sampler for Unnormalized Distributions].

Introduction

The Relative Entropy Gradient Sampler (REGS) is proposed for sampling from unnormalized densties by the techniques including Wasserstein gradient flows, numerical ODEs, density-ratio estimation and deep neural networks. REGS achieves a fantastic numerical performance on 2D mixtrues of Gaussian distributions with a large number of modes, a small variance, and a large distance between any two modes.

Dependencies

• Python 3.7.7
• PyTorch 1.4.0
• torchvision 0.5.0

Experimental Results

Toy Examples

• These toy examples will yield Figure 2 in the main paper for the mixtures of 8 Gaussians with equal or unqual weights.

• To run REGS for the target distribution being a mixtrue of 2 Gaussians on GPU devices, use:

python mixGauss2_2d.py train

• To run REGS for the target distribution being a mixtrue of 8 Gaussians on GPU devices, use:

python mixGauss8_2d.py train

Usage

Useful arguments

max_epoch 		= 50000 # the max epoches
dataSize 		= 2000  # the size of particles
gpuDevice   	= True  # if GPU is used
lrd         	= 5e-4  # 'learning rate for DNN, default=0.0005'
batchSize		= 1000  # 'batch size'
data_target 	= 'gm2_2d'
eta				= 5e-4  # 'learning rate for particle update'
nlD          	= 128   # 'width of hidden layers'
nm_gauss		= 2     # 'number of components of mixture gaussian'

radius			= 4.0    # 'radius of mixed Gaussian u(x), the distance of mean from origin'
target_std		= np.sqrt(0.03)   # 'std of mixed Gaussian u(x), default is 0.4'

weights 		= 'equal'  # or 'unequal', or np.ones(nm_gauss)
base_mean		= 0.0 	# 'mean of base normal
base_std		= 3.0   # 'std of base normal'
init_std 		= 3.0   # 'std of initial normal
init_mean 		= 0.0   # 'mean of initial normal

• Run on CPU devices:

python mixGauss2_2d.py train --gpuDevice=False

• Change the radius of target distribution:

python mixGauss2_2d.py train --radius=3.0

• Set unequal weights of target distribution (default is equal wights):

python mixGauss2_2d.py train --weights='unequal'

• or set specified weights of target distribution:

python mixGauss2_2d.py train --weights=[1,1]

• Evaluate the numerical performance after 50000 iterations via E[H(x)] including mean, variance and cos(alpha*X + 0.5)

python mixGauss2_2d.py evaluate --Gz_nepoch=50000

More details can be found in the codes file.

Competitor: SVGD, MALA, ULA

Toy Examples

To run SVGD for the target distribution being a mixtrue of 2 or 8 Gaussians, use:

python gm2_compare.py train --algorithm='SVGD'


python gm8_compare.py train --algorithm='SVGD'

• Set unequal weights of target distribution (default is equal wights):

   python gm2_compare.py train --algorithm='SVGD' --weights='unequal'
  

• or set specified weights of target distribution:

   python gm2_compare.py train --algorithm='SVGD' --weights=[1,1]
  

To run MALA for the target distribution being a mixtrue of 2 or 8 Gaussians, use:

python gm2_compare.py train --algorithm='MALA'


python gm8_compare.py train --algorithm='MALA'

• Set unequal weights of target distribution (default is equal wights):

   python gm2_compare.py train --algorithm='MALA' --weights='unequal'
  

• or set specified weights of target distribution:

   python gm2_compare.py train --algorithm='MALA' --weights=[1,1]
  

To run ULA for the target distribution being a mixtrue of 2 or 8 Gaussians, use:

python gm2_compare.py train --algorithm='ULA'


python gm8_compare.py train --algorithm='ULA'

• Set unequal weights of target distribution (default is equal wights):

   python gm2_compare.py train --algorithm='ULA' --weights='unequal'
  

• or set specified weights of target distribution:

   python gm2_compare.py train --algorithm='ULA' --weights=[1,1]
  

To run ULA or MALA with multiple chains, use:

python gm2_compare.py train --algorithm='ULA' --num_chains=10

python gm8_compare.py train --algorithm='MALA' --num_chains=10