Autobatching log-densities example¶

This notebook demonstrates a simple Bayesian inference example where autobatching makes user code easier to write, easier to read, and less likely to include bugs.

Inspired by a notebook by @davmre.

import functools
import itertools
import re
import sys
import time

from matplotlib.pyplot import *

import jax

from jax import lax
import jax.numpy as jnp
import jax.scipy as jsp
from jax import random

import numpy as np
import scipy as sp

Generate a fake binary classification dataset¶

np.random.seed(10009)

num_features = 10
num_points = 100

true_beta = np.random.randn(num_features).astype(jnp.float32)
all_x = np.random.randn(num_points, num_features).astype(jnp.float32)
y = (np.random.rand(num_points) < sp.special.expit(all_x.dot(true_beta))).astype(jnp.int32)

array([0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0,
       1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0,
       1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0,
       0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1,
       1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0], dtype=int32)

Write the log-joint function for the model¶

We’ll write a non-batched version, a manually batched version, and an autobatched version.

Non-batched¶

def log_joint(beta):
    result = 0.
    # Note that no `axis` parameter is provided to `jnp.sum`.
    result = result + jnp.sum(jsp.stats.norm.logpdf(beta, loc=0., scale=1.))
    result = result + jnp.sum(-jnp.log(1 + jnp.exp(-(2*y-1) * jnp.dot(all_x, beta))))
    return result

log_joint(np.random.randn(num_features))

WARNING:absl:No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)

DeviceArray(-213.23558, dtype=float32)

# This doesn't work, because we didn't write `log_prob()` to handle batching.
try:
  batch_size = 10
  batched_test_beta = np.random.randn(batch_size, num_features)

  log_joint(np.random.randn(batch_size, num_features))
except ValueError as e:
  print("Caught expected exception " + str(e))

Caught expected exception Incompatible shapes for broadcasting: ((100, 10), (1, 100))

Manually batched¶

def batched_log_joint(beta):
    result = 0.
    # Here (and below) `sum` needs an `axis` parameter. At best, forgetting to set axis
    # or setting it incorrectly yields an error; at worst, it silently changes the
    # semantics of the model.
    result = result + jnp.sum(jsp.stats.norm.logpdf(beta, loc=0., scale=1.),
                           axis=-1)
    # Note the multiple transposes. Getting this right is not rocket science,
    # but it's also not totally mindless. (I didn't get it right on the first
    # try.)
    result = result + jnp.sum(-jnp.log(1 + jnp.exp(-(2*y-1) * jnp.dot(all_x, beta.T).T)),
                           axis=-1)
    return result

batch_size = 10
batched_test_beta = np.random.randn(batch_size, num_features)

batched_log_joint(batched_test_beta)

DeviceArray([-147.84033, -207.02205, -109.26076, -243.80832, -163.02911,
             -143.8485 , -160.28772, -113.7717 , -126.60543, -190.81989],            dtype=float32)

Autobatched with vmap¶

It just works.

vmap_batched_log_joint = jax.vmap(log_joint)
vmap_batched_log_joint(batched_test_beta)

DeviceArray([-147.84033, -207.02205, -109.26076, -243.80832, -163.02911,
             -143.8485 , -160.28772, -113.7717 , -126.60543, -190.81989],            dtype=float32)

Self-contained variational inference example¶

A little code is copied from above.

Set up the (batched) log-joint function¶

@jax.jit
def log_joint(beta):
    result = 0.
    # Note that no `axis` parameter is provided to `jnp.sum`.
    result = result + jnp.sum(jsp.stats.norm.logpdf(beta, loc=0., scale=10.))
    result = result + jnp.sum(-jnp.log(1 + jnp.exp(-(2*y-1) * jnp.dot(all_x, beta))))
    return result

batched_log_joint = jax.jit(jax.vmap(log_joint))

Define the ELBO and its gradient¶

def elbo(beta_loc, beta_log_scale, epsilon):
    beta_sample = beta_loc + jnp.exp(beta_log_scale) * epsilon
    return jnp.mean(batched_log_joint(beta_sample), 0) + jnp.sum(beta_log_scale - 0.5 * np.log(2*np.pi))
 
elbo = jax.jit(elbo)
elbo_val_and_grad = jax.jit(jax.value_and_grad(elbo, argnums=(0, 1)))

Optimize the ELBO using SGD¶

def normal_sample(key, shape):
    """Convenience function for quasi-stateful RNG."""
    new_key, sub_key = random.split(key)
    return new_key, random.normal(sub_key, shape)

normal_sample = jax.jit(normal_sample, static_argnums=(1,))

key = random.PRNGKey(10003)

beta_loc = jnp.zeros(num_features, jnp.float32)
beta_log_scale = jnp.zeros(num_features, jnp.float32)

step_size = 0.01
batch_size = 128
epsilon_shape = (batch_size, num_features)
for i in range(1000):
    key, epsilon = normal_sample(key, epsilon_shape)
    elbo_val, (beta_loc_grad, beta_log_scale_grad) = elbo_val_and_grad(
        beta_loc, beta_log_scale, epsilon)
    beta_loc += step_size * beta_loc_grad
    beta_log_scale += step_size * beta_log_scale_grad
    if i % 10 == 0:
        print('{}\t{}'.format(i, elbo_val))

-180.8538818359375
-113.06045532226562
-102.73727416992188
-99.787353515625
-98.90898895263672
-98.29745483398438
-98.18630981445312
-97.57972717285156
-97.28600311279297
-97.46996307373047
-97.4771728515625
-97.58065795898438
-97.4943618774414
-97.50271606445312
-96.86395263671875
-97.44197845458984
-97.06941223144531
-96.84028625488281
-97.21337890625
-97.56503295898438

-97.26397705078125
-97.11979675292969
-97.39595031738281
-97.16831970214844
-97.11841583251953
-97.24345397949219
-97.2978744506836
-96.69285583496094
-96.96438598632812
-97.30055236816406
-96.63592529296875
-97.03518676757812
-97.52909851074219
-97.28813171386719
-97.0732192993164
-97.15620422363281
-97.25881958007812
-97.19514465332031
-97.13092041015625
-97.11727905273438

-96.93873596191406
-97.26676940917969
-97.35324096679688
-97.21007537841797
-97.28434753417969
-97.1630859375
-97.2612533569336
-97.21343994140625
-97.23997497558594
-97.14913177490234
-97.23528289794922
-96.9342041015625
-97.21209716796875
-96.82575988769531
-97.01286315917969
-96.94176483154297
-97.16520690917969
-97.29165649414062
-97.42939758300781
-97.24371337890625

-97.15219116210938
-97.49845123291016
-96.99069213867188
-96.88956451416016
-96.89968872070312
-97.13793182373047
-97.43705749511719
-96.99236297607422
-97.15623474121094
-97.18690490722656
-97.11161041259766
-97.78105163574219
-97.23226928710938
-97.16207885742188
-96.99581909179688
-96.66722106933594
-97.16795349121094
-97.51435089111328
-97.28900909423828
-96.9122543334961

-97.1709976196289
-97.29047393798828
-97.16242980957031
-97.1910629272461
-97.56382751464844
-97.00194549560547
-96.86555480957031
-96.76338195800781
-96.83660888671875
-97.12178039550781
-97.09554290771484
-97.0682373046875
-97.11949157714844
-96.8792953491211
-97.45625305175781
-96.69280242919922
-97.29376220703125
-97.33531188964844
-97.34962463378906
-97.09674835205078

Display the results¶

Coverage isn’t quite as good as we might like, but it’s not bad, and nobody said variational inference was exact.

figure(figsize=(7, 7))
plot(true_beta, beta_loc, '.', label='Approximated Posterior Means')
plot(true_beta, beta_loc + 2*jnp.exp(beta_log_scale), 'r.', label='Approximated Posterior $2\sigma$ Error Bars')
plot(true_beta, beta_loc - 2*jnp.exp(beta_log_scale), 'r.')
plot_scale = 3
plot([-plot_scale, plot_scale], [-plot_scale, plot_scale], 'k')
xlabel('True beta')
ylabel('Estimated beta')
legend(loc='best')

<matplotlib.legend.Legend at 0x7f869c7b6be0>