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Example: Gaussian Mixture Model in NumPyro¶
This example illustrates how to construct an inference program based on the APGS sampler [1] for GMM. The details of GMM can be found in the sections 6.2 and F.1 of the reference. We will use the NumPyro (default) backend for this example.
References
Wu, Hao, et al. Amortized population Gibbs samplers with neural sufficient statistics. ICML 2020.
import argparse
from functools import partial
import coix
import flax.linen as nn
import jax
from jax import random
import jax.numpy as jnp
from matplotlib.patches import Ellipse
import matplotlib.pyplot as plt
import numpy as np
import numpyro
import numpyro.distributions as dist
from numpyro.ops.indexing import Vindex
import optax
import tensorflow as tf
First, let’s simulate a synthetic dataset of 2D Gaussian mixtures.
def simulate_clusters(num_instances=1, N=60, seed=0):
np.random.seed(seed)
tau = np.random.gamma(2, 0.5, (num_instances, 4, 2))
mu_base = np.random.normal(0, 1, (num_instances, 4, 2))
mu = mu_base / np.sqrt(0.1 * tau)
c = np.random.choice(np.arange(3), (num_instances, N))
mu_ = np.take_along_axis(mu, c[..., None], axis=1)
tau_ = np.take_along_axis(tau, c[..., None], axis=1)
eps = np.random.normal(0, 1, (num_instances, N, 2))
x = mu_ + eps / np.sqrt(tau_)
return x, c
def load_dataset(split, *, batch_size):
if split == "train":
num_data = 20000
num_points = 60
seed = 0
else:
num_data = batch_size
num_points = 100
seed = 1
data, label = simulate_clusters(num_data, num_points, seed=seed)
if split == "train":
ds = tf.data.Dataset.from_tensor_slices(data)
ds = ds.repeat()
ds = ds.shuffle(10 * batch_size, seed=0)
else:
ds = tf.data.Dataset.from_tensor_slices((data, label))
ds = ds.repeat()
ds = ds.batch(batch_size)
return ds.as_numpy_iterator()
Next, we define the neural proposals for the Gibbs kernels.
class GMMEncoderMeanTau(nn.Module):
@nn.compact
def __call__(self, x):
s = nn.Dense(2)(x)
t = nn.Dense(3)(x)
t = nn.softmax(t, -1)
s, t = jnp.expand_dims(s, -2), jnp.expand_dims(t, -1)
N = t.sum(-3)
x = (t * s).sum(-3)
x2 = (t * s**2).sum(-3)
mu0, nu0, alpha0, beta0 = (0, 0.1, 2, 2)
alpha = alpha0 + 0.5 * N
beta = (
beta0
+ 0.5 * (x2 - x**2 / N)
+ 0.5 * N * nu0 / (N + nu0) * (x / N - mu0) ** 2
)
mu = (mu0 * nu0 + x) / (nu0 + N)
nu = nu0 + N
return alpha, beta, mu, nu
class GMMEncoderC(nn.Module):
@nn.compact
def __call__(self, x):
x = nn.Dense(32)(x)
x = nn.tanh(x)
logits = nn.Dense(1)(x).squeeze(-1)
return logits + jnp.log(jnp.ones(3) / 3)
def broadcast_concatenate(*xs):
shape = jnp.broadcast_shapes(*[x.shape[:-1] for x in xs])
xs = [jnp.broadcast_to(x, shape + x.shape[-1:]) for x in xs]
return jnp.concatenate(xs, -1)
class GMMEncoder(nn.Module):
def setup(self):
self.encode_initial_mean_tau = GMMEncoderMeanTau()
self.encode_mean_tau = GMMEncoderMeanTau()
self.encode_c = GMMEncoderC()
def __call__(self, x): # N x D
# Heuristic procedure to setup initial parameters.
alpha, beta, mean, _ = self.encode_initial_mean_tau(x) # M x D
tau = alpha / beta # M x D
xmt = jax.vmap(broadcast_concatenate, (None, -2, -2), -2)(x, mean, tau)
logits = self.encode_c(xmt) # N x D
c = jnp.argmax(logits, -1) # N
xc = jnp.concatenate([x, jax.nn.one_hot(c, 3)], axis=-1)
return self.encode_mean_tau(xc)
Then, we define the target and kernels as in Section 6.2.
def gmm_target(inputs):
tau = numpyro.sample("tau", dist.Gamma(2, 2).expand([3, 2]).to_event())
mean = numpyro.sample(
"mean", dist.Normal(0, 1 / jnp.sqrt(tau * 0.1)).to_event()
)
with numpyro.plate("N", inputs.shape[-2], dim=-1):
c = numpyro.sample("c", dist.Categorical(probs=jnp.ones(4) / 4))
loc = Vindex(mean)[..., c, :]
scale = 1 / jnp.sqrt(Vindex(tau)[..., c, :])
x = numpyro.sample("x", dist.Normal(loc, scale).to_event(1), obs=inputs)
out = {"mean": mean, "tau": tau, "c": c, "x": x}
return (out,)
def gmm_kernel_mean_tau(network, inputs):
if not isinstance(inputs, dict):
inputs = {"x": inputs}
if "c" in inputs:
x = inputs["x"]
c = jax.nn.one_hot(inputs["c"], 3)
xc = broadcast_concatenate(x, c)
alpha, beta, mu, nu = network.encode_mean_tau(xc)
else:
alpha, beta, mu, nu = network.encode_initial_mean_tau(inputs["x"])
alpha, beta, mu, nu = jax.tree_util.tree_map(
lambda x: jnp.expand_dims(x, -3), (alpha, beta, mu, nu)
)
tau = numpyro.sample("tau", dist.Gamma(alpha, beta).to_event(2))
mean = numpyro.sample(
"mean", dist.Normal(mu, 1 / jnp.sqrt(tau * nu)).to_event(2)
)
out = {**inputs, **{"mean": mean, "tau": tau}}
return (out,)
def gmm_kernel_c(network, inputs):
x, mean, tau = inputs["x"], inputs["mean"], inputs["tau"]
xmt = jax.vmap(broadcast_concatenate, (None, -2, -2), -2)(x, mean, tau)
logits = network.encode_c(xmt)
with numpyro.plate("N", logits.shape[-2], dim=-1):
c = numpyro.sample("c", dist.Categorical(logits=logits))
out = {**inputs, **{"c": c}}
return (out,)
Finally, we create the gmm inference program, define the loss function, run the training loop, and plot the results.
def make_gmm(params, num_sweeps, num_particles):
network = coix.util.BindModule(GMMEncoder(), params)
# Add particle dimension and construct a program.
make_particle_plate = lambda: numpyro.plate("particle", num_particles, dim=-3)
target = make_particle_plate()(gmm_target)
kernel_mean_tau = make_particle_plate()(partial(gmm_kernel_mean_tau, network))
kernel_c = make_particle_plate()(partial(gmm_kernel_c, network))
kernels = [kernel_mean_tau, kernel_c]
program = coix.algo.apgs(target, kernels, num_sweeps=num_sweeps)
return program
def loss_fn(params, key, batch, num_sweeps, num_particles):
# Prepare data for the program.
shuffle_rng, rng_key = random.split(key)
batch = random.permutation(shuffle_rng, batch, axis=1)
# Run the program and get metrics.
program = make_gmm(params, num_sweeps, num_particles)
_, _, metrics = coix.traced_evaluate(program, seed=rng_key)(batch)
for metric_name in ["log_Z", "log_density", "loss"]:
metrics[metric_name] = metrics[metric_name] / batch.shape[0]
return metrics["loss"], metrics
def main(args):
lr = args.learning_rate
num_steps = args.num_steps
batch_size = args.batch_size
num_sweeps = args.num_sweeps
num_particles = args.num_particles
train_ds = load_dataset("train", batch_size=batch_size)
test_ds = load_dataset("test", batch_size=batch_size)
init_params = GMMEncoder().init(jax.random.PRNGKey(0), jnp.zeros((60, 2)))
gmm_params, _ = coix.util.train(
partial(loss_fn, num_sweeps=num_sweeps, num_particles=num_particles),
init_params,
optax.adam(lr),
num_steps,
train_ds,
)
program = make_gmm(gmm_params, num_sweeps, num_particles)
batch, label = next(test_ds)
out, _, _ = coix.traced_evaluate(program, seed=jax.random.PRNGKey(1))(batch)
out = out[0]
_, axes = plt.subplots(2, 3, figsize=(15, 10))
for i in range(6):
axes[i // 3][i % 3].scatter(
batch[i, :, 0],
batch[i, :, 1],
marker=".",
color=np.array(["c", "m", "y"])[label[i]],
)
for j, c in enumerate(["r", "g", "b"]):
ellipse = Ellipse(
xy=(out["mean"][0, i, 0, j, 0], out["mean"][0, i, 0, j, 1]),
width=4 / jnp.sqrt(out["tau"][0, i, 0, j, 0]),
height=4 / jnp.sqrt(out["tau"][0, i, 0, j, 1]),
fc=c,
alpha=0.3,
)
axes[i // 3][i % 3].add_patch(ellipse)
plt.show()
if __name__ == "__main__":
parser = argparse.ArgumentParser(description="Annealing example")
parser.add_argument("--batch-size", nargs="?", default=20, type=int)
parser.add_argument("--num-sweeps", nargs="?", default=5, type=int)
parser.add_argument("--num_particles", nargs="?", default=10, type=int)
parser.add_argument("--learning-rate", nargs="?", default=2.5e-4, type=float)
parser.add_argument("--num-steps", nargs="?", default=200000, type=int)
parser.add_argument(
"--device", default="gpu", type=str, help='use "cpu" or "gpu".'
)
args = parser.parse_args()
tf.config.experimental.set_visible_devices([], "GPU") # Disable GPU for TF.
numpyro.set_platform(args.device)
main(args)