GSM-VI: Gaussian score-based variational inference algorithms

This repository provides a Python implementation of two score-based (black-box) variational inference algorithms:

1. GSM-VI, which is described in the NeurIPS 2023 paper https://arxiv.org/abs/2307.07849.
2. Batch and match (BaM), which is described in the ICML 2024 paper https://arxiv.org/pdf/2402.14758.

We describe each of these in the following two sections.

Variational Inference (VI) with Gaussian Score Matching (GSM) (NeurIPS 2023)

GSM-VI fits a multivariate Gasussian distribution with dense covaraince matrix to the target distribution by score matching. It only requires access to the score function i.e. the gradient of the target log-probability distribution and implements analytic updates for the variational parameters (mean and covariance matrix).

Installation:

The code is available on PyPI

pip install gsmvi

Usage

The simplest version of the algorithm is written in numpy. The following is the minimal code to use GSM to fit the parameters x of a model given its log_prob and log_prob_grad functions. See example/example_gsm_numpy.py for a full example.

dimensions = D
def log_prob(x):
  # return log_prbability at sample x
  ...

def log_prob_grad(x):
  # return the score fuction i.e. the gradient of log_prbability at sample x
  ...

from gsmvi.gsm_numpy import GSM
gsm = GSM(D=D, lp=log_prob, lp_g=log_prob_grad)
random_seed = 99
number_of_iterations = 500
mean_fit, cov_fit = gsm.fit(key=random_seed, niter=number_of_iterations)

A more efficient version of the algorithm is implemented in Jax where it can benefit from jit compilation. The basic signature stays the same. See example/example_gsm.py for a full example.

dimensions = D
model =  setup_model(D=D)     # Ths example sets up a numpyro model which has log_prob attribute implemented
lp = jit(lambda x: jnp.sum(model.log_prob(x)))
lp_g = jit(grad(lp, argnums=0))

from gsmvi.gsm import GSM
gsm = GSM(D=D, lp=lp, lp_g=lp_g)
mean_fit, cov_fit = gsm.fit(key=random.PRNGKey(99), niter=500)

Other utilities:

• For comparison, we also provide implementation of ADVI algorithm (https://arxiv.org/abs/1603.00788), another common approach to fit a multivariate Gaussian variational distribution which maximizes ELBO.
• We provide LBFGS initilization for the variational distribution which can be used with GSM and ADVI.
• We also provide a Monitor class to monitor the KL divergence over iterations as the algorithms progress.

Code Dependencies

The vanilla code is written in python3 and does not have any dependencies.

Optional dependencies

These will not be installed with the package and should be installed by user depending on the use-case.

The Jax version of the code requires jax and jaxlib.
The target distributions in example files other than example_gsm_numpy.py are implemented in numpyro.
ADVI algorithm uses optax for maximizing ELBO.
LBFGS initialization for initializing variational distributions uses scipy.

Starting point

We provide simple examples in examples/ folder to fit a target multivariate Gaussian distribution with GSM and ADVI.

cd examples
python3 example_gsm_numpy.py  # vanilla example in numpy, no dependencies
python3 example_gsm.py        # jax version, requires jax and numpyro
python3 example_advi.py       # jax version, requires jax, numpyro and optax

An example on how to use the Monitor class and LBFGS initialization is in examples/example_initializers.py

cd examples
python3 example_initializers.py   # jax version, requires jax, numpyro, optax and scipy

Batch and match: black-box variational inference with a score-based divergence (ICML 2024)

Batch and match (BaM) also fits a full covariance multivariate Gaussian and recovers (a version of) GSM as a special case. In the BaM algorithm, a score-based divergence is minimized. The code is set up similarly to the GSM code. Currently, it is not yet available on PyPI.

To install, run

pip install -e .

The example usage code is in `examples/example_bam.py':

cd examples
python3 example_bam.py        # jax version, requires jax and numpyro

Note that this installation approach also includes the GSM and ADVI code examples above.