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Example: Time Series Model - Bouncing MNIST in NumPyro
This example illustrates how to construct an inference program based on the APGS sampler [1] for BMNIST. The details of BMNIST can be found in the sections 6.4 and F.3 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
import matplotlib.animation as animation
from matplotlib.patches import Rectangle
import matplotlib.pyplot as plt
import numpyro
import numpyro.distributions as dist
import optax
import tensorflow as tf
import tensorflow_datasets as tfds
First, let’s load the moving mnist dataset.
def load_dataset(*, is_training, batch_size):
ds = tfds.load("moving_mnist:1.0.0", split="test")
ds = ds.repeat()
if is_training:
ds = ds.shuffle(10 * batch_size, seed=0)
map_fn = lambda x: x["image_sequence"][..., :10, :, :, 0] / 255
else:
map_fn = lambda x: x["image_sequence"][..., 0] / 255
ds = ds.batch(batch_size)
ds = ds.map(map_fn)
return iter(tfds.as_numpy(ds))
def get_digit_mean():
ds, ds_info = tfds.load("mnist:3.0.1", split="train", with_info=True)
ds = tfds.as_numpy(ds.batch(ds_info.splits["train"].num_examples))
digit_mean = next(iter(ds))["image"].squeeze(-1).mean(axis=0)
return digit_mean / 255
Next, we define the neural proposals for the Gibbs kernels and the neural decoder for the generative model.
def scale_and_translate(image, where, out_size):
translate = abs(image.shape[-1] - out_size) * (where[..., ::-1] + 1) / 2
return jax.image.scale_and_translate(
image,
(out_size, out_size),
(0, 1),
jnp.ones(2),
translate,
method="cubic",
antialias=False,
)
def crop_frames(frames, z_where, digit_size=28):
# frames: time.frame_size.frame_size
# z_where: (digits).time.2
# out: (digits).time.digit_size.digit_size
if frames.ndim == 2 and z_where.ndim == 1:
return scale_and_translate(frames, z_where, out_size=digit_size)
elif frames.ndim == 3 and z_where.ndim == 2:
in_axes = (0, 0)
elif frames.ndim == 3 and z_where.ndim == 3:
in_axes = (None, 0)
elif frames.ndim == z_where.ndim:
in_axes = (0, 0)
elif frames.ndim > z_where.ndim:
in_axes = (0, None)
else:
in_axes = (None, 0)
return jax.vmap(partial(crop_frames, digit_size=digit_size), in_axes)(
frames, z_where
)
def embed_digits(digits, z_where, frame_size=64):
# digits: (digits). .digit_size.digit_size
# z_where: (digits).(time).2
# out: (digits).(time).frame_size.frame_size
if digits.ndim == 2 and z_where.ndim == 1:
return scale_and_translate(digits, z_where, out_size=frame_size)
elif digits.ndim == 2 and z_where.ndim == 2:
in_axes = (None, 0)
elif digits.ndim >= z_where.ndim:
in_axes = (0, 0)
else:
in_axes = (None, 0)
return jax.vmap(partial(embed_digits, frame_size=frame_size), in_axes)(
digits, z_where
)
def conv2d(frames, digits):
# frames: (time).frame_size.frame_size
# digits: (digits). .digit_size.digit_size
# out: (digits).(time).conv_size .conv_size
if frames.ndim == 2 and digits.ndim == 2:
return jax.scipy.signal.convolve2d(frames, digits, mode="valid")
elif frames.ndim == digits.ndim:
in_axes = (0, 0)
elif frames.ndim > digits.ndim:
in_axes = (0, None)
else:
in_axes = (None, 0)
return jax.vmap(conv2d, in_axes=in_axes)(frames, digits)
class EncoderWhat(nn.Module):
@nn.compact
def __call__(self, digits):
x = digits.reshape(digits.shape[:-2] + (-1,))
x = nn.Dense(400)(x)
x = nn.relu(x)
x = nn.Dense(200)(x)
x = nn.relu(x)
x = x.sum(-2) # sum/mean across time
loc_raw = nn.Dense(10)(x)
scale_raw = 0.5 * nn.Dense(10)(x)
return loc_raw, jnp.exp(scale_raw)
class EncoderWhere(nn.Module):
@nn.compact
def __call__(self, frame_conv):
x = frame_conv.reshape(frame_conv.shape[:-2] + (-1,))
x = nn.softmax(x, -1)
x = nn.Dense(200)(x)
x = nn.relu(x)
x = nn.Dense(200)(x)
x = x.reshape(x.shape[:-1] + (2, 100))
x = nn.relu(x)
loc_raw = nn.Dense(2)(x[..., 0, :])
scale_raw = 0.5 * nn.Dense(2)(x[..., 1, :])
return nn.tanh(loc_raw), jnp.exp(scale_raw)
class DecoderWhat(nn.Module):
@nn.compact
def __call__(self, z_what):
x = nn.Dense(200)(z_what)
x = nn.relu(x)
x = nn.Dense(400)(x)
x = nn.relu(x)
x = nn.Dense(784)(x)
logits = x.reshape(x.shape[:-1] + (28, 28))
return nn.sigmoid(logits)
class BMNISTAutoEncoder(nn.Module):
digit_mean: jnp.ndarray
frame_size: int
def setup(self):
self.encode_what = EncoderWhat()
self.encode_where = EncoderWhere()
self.decode_what = DecoderWhat()
def __call__(self, frames):
# Heuristic procedure to setup initial parameters.
frames_conv = conv2d(frames, self.digit_mean)
z_where, _ = self.encode_where(frames_conv)
digits = crop_frames(frames, z_where, 28)
z_what, _ = self.encode_what(digits)
digit_recon = self.decode_what(z_what)
frames_recon = embed_digits(digit_recon, z_where, self.frame_size)
return frames_recon
Then, we define the target and kernels as in Section 6.4.
def bmnist_target(network, inputs, D=2, T=10):
z_what = numpyro.sample(
"z_what", dist.Normal(0, 1).expand([D, 10]).to_event()
)
digits = network.decode_what(z_what) # can cache this
z_where = []
# p = []
for d in range(D):
z_where_d = []
z_where_d_t = jnp.zeros(2)
for t in range(T):
scale = 1 if t == 0 else 0.1
z_where_d_t = numpyro.sample(
f"z_where_{d}_{t}", dist.Normal(z_where_d_t, scale).to_event(1)
)
z_where_d.append(z_where_d_t)
z_where_d = jnp.stack(z_where_d, -2)
z_where.append(z_where_d)
z_where = jnp.stack(z_where, -3)
p = embed_digits(digits, z_where, network.frame_size)
p = dist.util.clamp_probs(p.sum(-4)) # sum across digits
frames = numpyro.sample("frames", dist.Bernoulli(p).to_event(3), obs=inputs)
out = {
"frames": frames,
"frames_recon": p,
"z_what": z_what,
"digits": jax.lax.stop_gradient(digits),
**{f"z_where_{t}": z_where[..., t, :] for t in range(T)},
}
return (out,)
def kernel_where(network, inputs, D=2, t=0):
if not isinstance(inputs, dict):
inputs = {
"frames": inputs,
"digits": jnp.repeat(jnp.expand_dims(network.digit_mean, -3), D, -3),
}
frame = inputs["frames"][..., t, :, :]
z_where_t = []
for d in range(D):
digit = inputs["digits"][..., d, :, :]
x_conv = conv2d(frame, digit)
loc, scale = network.encode_where(x_conv)
z_where_d_t = numpyro.sample(
f"z_where_{d}_{t}", dist.Normal(loc, scale).to_event(1)
)
z_where_t.append(z_where_d_t)
frame_recon = embed_digits(digit, z_where_d_t, network.frame_size)
frame = frame - frame_recon
z_where_t = jnp.stack(z_where_t, -2)
out = {**inputs, **{f"z_where_{t}": z_where_t}}
return (out,)
def kernel_what(network, inputs, T=10):
z_where = jnp.stack([inputs[f"z_where_{t}"] for t in range(T)], -2)
digits = crop_frames(inputs["frames"], z_where, 28)
loc, scale = network.encode_what(digits)
z_what = numpyro.sample("z_what", dist.Normal(loc, scale).to_event(2))
out = {**inputs, **{"z_what": z_what}}
return (out,)
Finally, we create the bmnist inference program, define the loss function, run the training loop, and plot the results.
def make_bmnist(params, bmnist_net, T=10, num_sweeps=5, num_particles=10):
network = coix.util.BindModule(bmnist_net, params)
# Add particle dimension and construct a program.
vmap = lambda p: numpyro.plate("particle", num_particles, dim=-2)(p)
target = vmap(partial(bmnist_target, network, D=2, T=T))
kernels = []
for t in range(T):
kernels.append(vmap(partial(kernel_where, network, D=2, t=t)))
kernels.append(vmap(partial(kernel_what, network, T=T)))
program = coix.algo.apgs(target, kernels, num_sweeps=num_sweeps)
return program
def loss_fn(params, key, batch, bmnist_net, num_sweeps, num_particles):
# Prepare data for the program.
shuffle_rng, rng_key = random.split(key)
batch = random.permutation(shuffle_rng, batch, axis=1)
T = batch.shape[-3]
# Run the program and get metrics.
program = make_bmnist(params, bmnist_net, T, 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(is_training=True, batch_size=batch_size)
test_ds = load_dataset(is_training=False, batch_size=1)
digit_mean = get_digit_mean()
test_data = next(test_ds)
frame_size = test_data.shape[-1]
bmnist_net = BMNISTAutoEncoder(digit_mean=digit_mean, frame_size=frame_size)
init_params = bmnist_net.init(jax.random.PRNGKey(0), test_data[0])
bmnist_params, _ = coix.util.train(
partial(
loss_fn,
bmnist_net=bmnist_net,
num_sweeps=num_sweeps,
num_particles=num_particles,
),
init_params,
optax.adam(lr),
num_steps,
train_ds,
)
T_test = test_data.shape[-3]
program = make_bmnist(
bmnist_params, bmnist_net, T_test, num_sweeps, num_particles
)
out, _, _ = coix.traced_evaluate(program, seed=jax.random.PRNGKey(1))(
test_data
)
out = out[0]
prop_cycle = plt.rcParams["axes.prop_cycle"]
colors = prop_cycle.by_key()["color"]
fig, axes = plt.subplots(1, 2, figsize=(12, 6))
def animate(i):
axes[0].cla()
axes[0].imshow(test_data[0, i])
axes[1].cla()
axes[1].imshow(out["frames_recon"][0, 0, i])
for d in range(2):
where = 0.5 * (out[f"z_where_{i}"][0, 0, d] + 1) * (frame_size - 28) - 0.5
color = colors[d]
axes[0].add_patch(
Rectangle(where, 28, 28, edgecolor=color, lw=3, fill=False)
)
plt.rc("animation", html="jshtml")
plt.tight_layout()
ani = animation.FuncAnimation(fig, animate, frames=range(20), interval=300)
writer = animation.PillowWriter(fps=15)
ani.save("bmnist.gif", writer=writer)
plt.show()
if __name__ == "__main__":
parser = argparse.ArgumentParser(description="Annealing example")
parser.add_argument("--batch-size", nargs="?", default=5, 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=1e-4, type=float)
parser.add_argument("--num-steps", nargs="?", default=20000, 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)