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)

y

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([-147.84032, -207.02205, -109.26076, -243.80833, -163.02908,
             -143.84848, -160.28772, -113.7717 , -126.60544, -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([-147.84032, -207.02205, -109.26076, -243.80833, -163.02908,
             -143.84848, -160.28772, -113.7717 , -126.60544, -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))

0	-180.85389709472656
10	-113.06045532226562
20	-102.73727416992188
30	-99.787353515625
40	-98.90898895263672
50	-98.29744720458984
60	-98.18630981445312
70	-97.57972717285156
80	-97.28599548339844
90	-97.46996307373047
100	-97.4771728515625
110	-97.58067321777344
120	-97.49435424804688
130	-97.50271606445312
140	-96.86395263671875
150	-97.44197082519531
160	-97.06940460205078
170	-96.84028625488281
180	-97.21337890625

190	-97.56502532958984
200	-97.26398468017578
210	-97.11979675292969
220	-97.39595031738281
230	-97.16831970214844
240	-97.118408203125
250	-97.24345397949219
260	-97.2978744506836
270	-96.69286346435547
280	-96.96438598632812
290	-97.30055236816406
300	-96.63592529296875
310	-97.0351791381836
320	-97.52909851074219
330	-97.28812408447266
340	-97.0732192993164
350	-97.15620422363281
360	-97.25881958007812
370	-97.19514465332031

380	-97.13092041015625
390	-97.11727905273438
400	-96.93872833251953
410	-97.26675415039062
420	-97.35324096679688
430	-97.21007537841797
440	-97.28434753417969
450	-97.1630859375
460	-97.2612533569336
470	-97.21343994140625
480	-97.23997497558594
490	-97.14913177490234
500	-97.23527526855469
510	-96.9342041015625
520	-97.21209716796875
530	-96.82577514648438
540	-97.01285552978516
550	-96.94176483154297
560	-97.16520690917969

570	-97.29165649414062
580	-97.42939758300781
590	-97.24371337890625
600	-97.15219116210938
610	-97.49845123291016
620	-96.9906997680664
630	-96.88956451416016
640	-96.89968872070312
650	-97.137939453125
660	-97.43707275390625
670	-96.99235534667969
680	-97.15623474121094
690	-97.18690490722656
700	-97.11161041259766
710	-97.78105163574219
720	-97.23226165771484
730	-97.16206359863281
740	-96.99583435058594
750	-96.66722106933594

760	-97.16796112060547
770	-97.51434326171875
780	-97.28901672363281
790	-96.9122543334961
800	-97.1709976196289
810	-97.29047393798828
820	-97.16242980957031
830	-97.1910629272461
840	-97.5638198852539
850	-97.00194549560547
860	-96.86555480957031
870	-96.76338195800781
880	-96.83660888671875
890	-97.12179565429688
900	-97.09554290771484
910	-97.0682373046875
920	-97.1194839477539
930	-96.8792953491211
940	-97.45625305175781

950	-96.69280242919922
960	-97.29376983642578
970	-97.33531188964844
980	-97.34962463378906
990	-97.09674072265625

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 0x7f1b603fbc10>