Hamiltonian Monte Carlo with Dual Averaging

edward/__init__.py

@@ -11,7 +11,7 @@
 from edward.criticisms import evaluate, ppc, ppc_density_plot, \
     ppc_stat_hist_plot
 from edward.inferences import Inference, MonteCarlo, VariationalInference, \
-    HMC, MetropolisHastings, SGLD, SGHMC, \
+    HMC, HMCDA, MetropolisHastings, SGLD, SGHMC, \
     KLpq, KLqp, ReparameterizationKLqp, ReparameterizationKLKLqp, \
     ReparameterizationEntropyKLqp, ScoreKLqp, ScoreKLKLqp, ScoreEntropyKLqp, \
     GANInference, BiGANInference, WGANInference, ImplicitKLqp, MAP, Laplace, \
@@ -41,6 +41,7 @@
     'VariationalInference',
     'HMC',
     'MetropolisHastings',
+    'HMCDA',
     'SGLD',
     'SGHMC',
     'KLpq',

edward/inferences/__init__.py

@@ -9,6 +9,7 @@
 from edward.inferences.gan_inference import *
 from edward.inferences.gibbs import *
 from edward.inferences.hmc import *
+from edward.inferences.hmcda import *
 from edward.inferences.implicit_klqp import *
 from edward.inferences.inference import *
 from edward.inferences.klpq import *
@@ -44,6 +45,7 @@
     'MAP',
     'MetropolisHastings',
     'MonteCarlo',
+    'HMCDA',
     'SGLD',
     'SGHMC',
     'VariationalInference',

edward/inferences/hmcda.py

@@ -0,0 +1,329 @@
+from __future__ import absolute_import
+from __future__ import division
+from __future__ import print_function
+
+import six
+import tensorflow as tf
+
+from collections import OrderedDict
+from edward.inferences.monte_carlo import MonteCarlo
+from edward.models import RandomVariable
+from edward.util import copy, get_session
+
+try:
+  from edward.models import Normal, Uniform
+except Exception as e:
+  raise ImportError("{0}. Your TensorFlow version is not supported.".format(e))
+
+
+class HMCDA(MonteCarlo):
+  """Hamiltonian Monte Carlo with Dual Averaging
+  (Hoffman M. and Gelman A., 2014) - Algorithm 5
+
+  Notes
+  -----
+  In conditional inference, we infer :math:`z` in :math:`p(z, \\beta
+  \mid x)` while fixing inference over :math:`\\beta` using another
+  distribution :math:`q(\\beta)`.
+  ``HMC`` substitutes the model's log marginal density
+
+  .. math::
+
+    \log p(x, z) = \log \mathbb{E}_{q(\\beta)} [ p(x, z, \\beta) ]
+                \\approx \log p(x, z, \\beta^*)
+
+  leveraging a single Monte Carlo sample, where :math:`\\beta^* \sim
+  q(\\beta)`. This is unbiased (and therefore asymptotically exact as a
+  pseudo-marginal method) if :math:`q(\\beta) = p(\\beta \mid x)`.
+  """
+  def __init__(self, *args, **kwargs):
+    """
+    Examples
+    --------
+    >>> z = Normal(loc=0.0, scale=1.0)
+    >>> x = Normal(loc=tf.ones(10) * z, scale=1.0)
+    >>>
+    >>> qz = Empirical(tf.Variable(tf.zeros(500)))
+    >>> data = {x: np.array([0.0] * 10, dtype=np.float32)}
+    >>> inference = ed.HMCDA({z: qz}, data)
+    """
+    super(HMCDA, self).__init__(*args, **kwargs)
+
+  def initialize(self, n_adapt, delta=0.65, Lambda=0.15, *args, **kwargs):
+    """
+    Parameters
+    ----------
+    n_adapt : float
+      Number of samples with adaptation for epsilon
+    delta : float, optional
+      Target accept rate
+    Lambda : float, optional
+      Target leapfrog length
+    """
+    # store global scope for log joint calculations
+    self._scope = tf.get_default_graph().unique_name("inference") + '/'
+
+    # Find initial epsilon
+    step_size = self.find_good_eps()
+    sess = get_session()
+    init_op = tf.global_variables_initializer()
+    sess.run(init_op)
+    step_size = sess.run(step_size)
+
+    # Variables for Dual Averaging
+    self.epsilon = tf.Variable(step_size, trainable=False)
+    self.mu = tf.cast(tf.log(10.0 * step_size), tf.float32)
+    self.epsilon_B = tf.Variable(1.0, trainable=False, name="epsilon_bar")
+    self.H_B = tf.Variable(0.0, trainable=False, name="H_bar")
+
+    # Parameters for Dual Averaging
+    self.n_adapt = n_adapt
+    self.delta = tf.constant(delta, dtype=tf.float32)
+    self.Lambda = tf.constant(Lambda)
+
+    self.gamma = tf.constant(0.05)
+    self.t_0 = tf.constant(10)
+    self.kappa = tf.constant(0.75)
+
+    return super(HMCDA, self).initialize(*args, **kwargs)
+
+  def build_update(self):
+    """Simulate Hamiltonian dynamics using a numerical integrator.
+    Correct for the integrator's discretization error using an
+    acceptance ratio. The initial value of epsilon is heuristically chosen
+    with Algorithm 4.
+
+    Notes
+    -----
+    The updates assume each Empirical random variable is directly
+    parameterized by ``tf.Variable``s.
+    """
+    old_sample = {z: tf.gather(qz.params, tf.maximum(self.t - 1, 0))
+                  for z, qz in six.iteritems(self.latent_vars)}
+    old_sample = OrderedDict(old_sample)
+
+    # Sample momentum.
+    old_r_sample = OrderedDict()
+    for z, qz in six.iteritems(self.latent_vars):
+      event_shape = qz.event_shape
+      normal = Normal(loc=tf.zeros(event_shape), scale=tf.ones(event_shape))
+      old_r_sample[z] = normal.sample()
+
+    # Simulate Hamiltonian dynamics.
+    L_m = tf.maximum(1, tf.cast(tf.round(self.Lambda / self.epsilon), tf.int32))
+    new_sample, new_r_sample = leapfrog(old_sample, old_r_sample,
+                                        self.epsilon, self._log_joint, L_m)
+
+    # Calculate acceptance ratio.
+    ratio = kinetic_energy(old_r_sample)
+    ratio -= kinetic_energy(new_r_sample)
+    ratio += self._log_joint(new_sample)
+    ratio -= self._log_joint(old_sample)
+
+    # Accept or reject sample.
+    u = Uniform().sample()
+    alpha = tf.minimum(1.0, tf.exp(ratio))
+    accept = tf.log(u) < ratio
+
+    sample_values = tf.cond(accept, lambda: list(six.itervalues(new_sample)),
+                            lambda: list(six.itervalues(old_sample)))
+    if not isinstance(sample_values, list):
+      # ``tf.cond`` returns tf.Tensor if output is a list of size 1.
+      sample_values = [sample_values]
+
+    sample = {z: sample_value for z, sample_value in
+              zip(six.iterkeys(new_sample), sample_values)}
+
+    # Use Dual Averaging to adapt epsilon
+    should_adapt = self.t <= self.n_adapt
+    assign_ops = tf.cond(should_adapt,
+                         lambda: self._adapt_step_size(alpha),
+                         lambda: self._do_not_adapt_step_size(alpha))
+
+    # Update Empirical random variables.
+    for z, qz in six.iteritems(self.latent_vars):
+      variable = qz.get_variables()[0]
+      assign_ops.append(tf.scatter_update(variable, self.t, sample[z]))
+
+    # Increment n_accept (if accepted).
+    assign_ops.append(self.n_accept.assign_add(tf.where(accept, 1, 0)))
+    return tf.group(*assign_ops)
+
+  def _do_not_adapt_step_size(self, alpha):
+    # Do not adapt step size but assign last running averaged epsilon to epsilon
+    assign_ops = []
+    assign_ops.append(tf.assign(self.H_B, self.H_B).op)
+    assign_ops.append(tf.assign(self.epsilon_B, self.epsilon_B).op)
+    assign_ops.append(tf.assign(self.epsilon, self.epsilon_B).op)
+    return assign_ops
+
+  def _adapt_step_size(self, alpha):
+    # Adapt step size as described in Algorithm 5
+    assign_ops = []
+
+    factor_H = tf.cast(1 / (self.t + 1 + self.t_0), tf.float32)
+
+    H_B = (1 - factor_H) * self.H_B + factor_H * (self.delta - alpha)
+    epsilon = tf.exp(self.mu - tf.sqrt(tf.cast(self.t + 1, tf.float32)) /
+                     self.gamma * H_B)
+
+    t_powed = tf.pow(tf.cast(self.t + 1, tf.float32), -self.kappa)
+    epsilon_B = tf.exp(t_powed * tf.log(epsilon) +
+                       (1 - t_powed) * tf.log(self.epsilon_B))
+
+    # Return ops containing the updates
+    assign_ops.append(tf.assign(self.H_B, H_B).op)
+    assign_ops.append(tf.assign(self.epsilon, epsilon).op)
+    assign_ops.append(tf.assign(self.epsilon_B, epsilon_B).op)
+    return assign_ops
+
+  def find_good_eps(self):
+    # Heuristically find an inital espilon following Algorithm 4
+
+    # Sample momentum.
+    old_r = OrderedDict()
+
+    for z, qz in six.iteritems(self.latent_vars):
+      event_shape = qz.event_shape
+      normal = Normal(loc=tf.zeros(event_shape), scale=tf.ones(event_shape))
+      old_r[z] = normal.sample()
+
+    # Initialize espilon at 1.0
+    epsilon = tf.constant(1.0)
+
+    # Calculate log joint probability
+    old_z = {z: tf.gather(qz.params, 0)
+             for z, qz in six.iteritems(self.latent_vars)}
+    old_z = OrderedDict(old_z)
+
+    log_p_joint = -kinetic_energy(old_r)
+    log_p_joint += self._log_joint(old_z)
+
+    new_sample, new_r_sample = leapfrog(old_z, old_r,
+                                        epsilon, self._log_joint, 1)
+
+    log_p_joint_prime = -kinetic_energy(new_r_sample)
+    log_p_joint_prime += self._log_joint(new_sample)
+
+    log_p_joint_diff = log_p_joint_prime - log_p_joint
+
+    # See whether epsilon is too small or to big
+    condition = log_p_joint_diff >= tf.log(0.5)
+    a = 2.0 * tf.where(condition, 1.0, 0.0) - 1.0
+
+    # Save keys of (z, r) so that we can rebuild the Dict inside the while_loop
+    keys_r = list(six.iterkeys(old_r))
+    keys_z = list(six.iterkeys(old_z))
+
+    k = tf.constant(0)
+
+    def while_condition(k, _, log_p_joint_prime_loop, log_p_joint, *args):
+      accep_big_enough = tf.pow(tf.exp(
+          log_p_joint_prime_loop - log_p_joint), a) > tf.pow(2.0, -a)
+      to_many_iterations = k < 12
+      return tf.logical_and(to_many_iterations, accep_big_enough)
+
+    def body(k, epsilon_loop, _, log_p_joint, values_r, values_z):
+      new_epsilon_loop = tf.pow(2.0, a) * epsilon_loop
+
+      # Rebuild the Dicts inside the while_loop since we can only return lists
+      old_z_loop = OrderedDict()
+      old_r_loop = OrderedDict()
+      for i, key in enumerate(values_z):
+        old_z_loop[keys_z[i]] = values_z[i]
+        old_r_loop[keys_r[i]] = values_r[i]
+
+      new_z_loop, new_r_loop = leapfrog(old_z_loop, old_r_loop,
+                                        new_epsilon_loop, self._log_joint, 1)
+      new_log_p_joint_prime = -kinetic_energy(new_r_loop)
+      new_log_p_joint_prime += self._log_joint(new_z_loop)
+
+      return [k + 1, new_epsilon_loop, new_log_p_joint_prime,
+              log_p_joint, values_r, values_z]
+
+    _, new_epsilon, _, _, _, _ = tf.while_loop(
+        while_condition, body,
+        loop_vars=[k, epsilon, log_p_joint_prime, log_p_joint,
+                   list(six.itervalues(old_r.copy())),
+                   list(six.itervalues(old_z.copy()))])
+
+    return new_epsilon
+
+  def _log_joint(self, z_sample):
+    """Utility function to calculate model's log joint density,
+    log p(x, z), for inputs z (and fixed data x).
+
+    Parameters
+    ----------
+    z_sample : dict
+      Latent variable keys to samples.
+    """
+    scope = self._scope + tf.get_default_graph().unique_name("sample")
+    # Form dictionary in order to replace conditioning on prior or
+    # observed variable with conditioning on a specific value.
+    dict_swap = z_sample.copy()
+    for x, qx in six.iteritems(self.data):
+      if isinstance(x, RandomVariable):
+        if isinstance(qx, RandomVariable):
+          qx_copy = copy(qx, scope=scope)
+          dict_swap[x] = qx_copy.value()
+        else:
+          dict_swap[x] = qx
+
+    log_joint = 0.0
+    for z in six.iterkeys(self.latent_vars):
+      z_copy = copy(z, dict_swap, scope=scope)
+      log_joint += tf.reduce_sum(z_copy.log_prob(dict_swap[z]))
+
+    for x in six.iterkeys(self.data):
+      if isinstance(x, RandomVariable):
+        x_copy = copy(x, dict_swap, scope=scope)
+        log_joint += tf.reduce_sum(x_copy.log_prob(dict_swap[x]))
+
+    return log_joint
+
+
+def leapfrog(z_old, r_old, step_size, log_joint, n_steps):
+  # Use a stochastic while_loop since n_steps is a Tensor
+  z_new = z_old.copy()
+  r_new = r_old.copy()
+  keys_z_new = list(six.iterkeys(z_old.copy()))
+  first_grad_log_joint = tf.gradients(log_joint(z_new),
+                                      list(six.itervalues(z_new)))
+
+  k = tf.constant(0)
+
+  def while_condition(k, v_z_new, v_r_new, grad_log_joint):
+    return k < n_steps
+
+  def body(k, v_z_new, v_r_new, grad_log_joint):
+    z_new = OrderedDict()
+    for i, key in enumerate(v_z_new):
+      z, r = v_z_new[i], v_r_new[i]
+      z_new[keys_z_new[i]] = z  # Rebuild the Dict
+      v_r_new[i] = r
+      v_r_new[i] += 0.5 * step_size * tf.convert_to_tensor(grad_log_joint[i])
+      v_z_new[i] = z + step_size * v_r_new[i]
+
+    grad_log_joint = tf.gradients(log_joint(z_new),
+                                  list(six.itervalues(z_new)))
+    for i, key in enumerate(v_z_new):
+      v_r_new[i] += 0.5 * step_size * tf.convert_to_tensor(grad_log_joint[i])
+    return [k + 1, v_z_new, v_r_new, grad_log_joint]
+
+  _, v_z_new, v_r_new, _ = tf.while_loop(
+      while_condition, body,
+      loop_vars=[k, list(six.itervalues(z_new)), list(six.itervalues(r_new)),
+                 first_grad_log_joint])
+
+  # Rebuild the Dicts outside the while_loop since we can only pass lists
+  for i, key in enumerate(six.iterkeys(z_new)):
+    r_new[key] = v_r_new[i]
+    z_new[key] = v_z_new[i]
+
+  return z_new, r_new
+
+
+def kinetic_energy(momentum):
+  return tf.reduce_sum([0.5 * tf.reduce_sum(tf.square(r))
+                        for r in six.itervalues(momentum)])

tests/test-inferences/test_hmcda.py

@@ -0,0 +1,34 @@
+# -*- coding: utf-8 -*-
+
+from __future__ import absolute_import
+from __future__ import division
+from __future__ import print_function
+
+import edward as ed
+import numpy as np
+import tensorflow as tf
+
+from edward.models import Normal, Empirical
+
+
+class test_hmcda_class(tf.test.TestCase):
+
+  def test_normalnormal_run(self):
+    with self.test_session() as sess:
+      x_data = np.array([0.0] * 50, dtype=np.float32)
+
+      mu = Normal(loc=0.0, scale=1.0)
+      x = Normal(loc=tf.ones(50) * mu, scale=1.0)
+
+      qmu = Empirical(params=tf.Variable(tf.ones(5000)))
+
+      # analytic solution: N(loc=0.0, scale=\sqrt{1/51}=0.140)
+      inference = ed.HMCDA({mu: qmu}, data={x: x_data})
+      inference.run(n_adapt=1000)
+      self.assertAllClose(qmu.mean().eval(), 0, rtol=1e-2, atol=1e-2)
+      self.assertAllClose(qmu.stddev().eval(), np.sqrt(1 / 51),
+                          rtol=5e-2, atol=5e-2)
+
+if __name__ == '__main__':
+  ed.set_seed(42)
+  tf.test.main()