I found out about this paper from Ferenc's blog post here, which is a nice introduction.
The Variational Rényi Lower Bound
In the talk we covered the foundations of variational inference, and the derivation of the ELBO, the variational lower bound to the normalising factor in Bayesian inference. Maximising the ELBO is equivalent to minimising the KL divergence between the variational distribution and the true data-generating distribution.
Here are some nice notes on VI
The Renyi bound is a more general definition of the divergence between two distributions, and has the KL as a special case. Interestingly, it has a free parameter which, when negative, yields an upper bound instead of a lower one. So in principle one can sandwich a solution between upper and lower bounds.
In order to try this out we modified one of the examples from the autograd package to do variational inference in 1d with a mixture of gaussians and a renyi divergence. It doesn't behave well for negative alpha (upper bound) so feel free to hack around and try to make that case work.
# Implements black-box variational inference, where the variational
# distribution is a mixture of Gaussians.
#
# This trick was written up by Alex Graves in this note:
# [<http://arxiv.org/abs/1607.05690>](<http://arxiv.org/abs/1607.05690>)
from __future__ import absolute_import
from __future__ import print_function
import matplotlib.pyplot as plt
import autograd.numpy as np
import autograd.numpy.random as npr
import autograd.scipy.stats.norm as norm
from autograd.scipy.misc import logsumexp
from autograd import grad
#from optimizers import adam
from optimizers import sgd as adam
def diag_gaussian_log_density(x, mu, log_std):
return np.sum(norm.logpdf(x, mu, np.exp(log_std)), axis=-1)
def unpack_gaussian_params(params):
# Variational dist is a diagonal Gaussian.
D = np.shape(params)[0] / 2
mean, log_std = params[:D], params[D:]
return mean, log_std
def variational_log_density_gaussian(params, x):
mean, log_std = unpack_gaussian_params(params)
return diag_gaussian_log_density(x, mean, log_std)
def sample_diag_gaussian(params, num_samples, rs):
mean, log_std = unpack_gaussian_params(params)
D = np.shape(mean)[0]
return rs.randn(num_samples, D) * np.exp(log_std) + mean
"""
def variational_lower_bound(params, t, logprob, sampler, log_density,
num_samples, rs):
"Provides a stochastic estimate of the variational lower bound,\\
for any variational family and model density."
samples = sampler(params, num_samples, rs)
log_qs = log_density(params, samples)
log_ps = logprob(samples, t)
log_ps = np.reshape(log_ps, (num_samples, -1))
log_qs = np.reshape(log_qs, (num_samples, -1))
return np.mean(log_ps - log_qs)
def renyi_bound(params, t, logprob, sampler, log_density,
num_samples, rs,alpha=-0.5):
samples = sampler(params, num_samples, rs)
log_qs = log_density(params, samples)
log_ps = logprob(samples, t)
log_ps = np.reshape(log_ps, (num_samples, -1))
log_qs = np.reshape(log_qs, (num_samples, -1))
qs = np.exp(log_qs)
ps = np.exp(log_ps)
c=1.-alpha
return (1./c)*np.log(np.mean(np.power(ps/qs,c)))
"""
def init_gaussian_var_params(D, mean_mean=-1, log_std_mean=-5,
scale=0.1, rs=npr.RandomState(0)):
init_mean = mean_mean * np.ones(D) + rs.randn(D) * scale
init_log_std = log_std_mean * np.ones(D) + rs.randn(D) * scale
return np.concatenate([init_mean, init_log_std])
def log_normalize(x):
return x - logsumexp(x)
def build_mog_bbsvi(logprob, num_samples, k=10, rs=npr.RandomState(0)):
init_component_var_params = init_gaussian_var_params
component_log_density = variational_log_density_gaussian
component_sample = sample_diag_gaussian
def unpack_mixture_params(mixture_params):
log_weights = log_normalize(mixture_params[:k])
var_params = np.reshape(mixture_params[k:], (k, -1))
return log_weights, var_params
def init_var_params(D, rs=npr.RandomState(0), **kwargs):
log_weights = np.ones(k)
component_weights = [init_component_var_params(D, rs=rs, **kwargs) for i in range(k)]
return np.concatenate([log_weights] + component_weights)
def sample(var_mixture_params, num_samples, rs):
"""Sample locations aren't a continuous function of parameters
due to multinomial sampling."""
log_weights, var_params = unpack_mixture_params(var_mixture_params)
samples = np.concatenate([component_sample(params_k, num_samples, rs)[:, np.newaxis, :]
for params_k in var_params], axis=1)
ixs = np.random.choice(k, size=num_samples, p=np.exp(log_weights))
return np.array([samples[i, ix, :] for i, ix in enumerate(ixs)])
def mixture_log_density(var_mixture_params, x):
"""Returns a weighted average over component densities."""
log_weights, var_params = unpack_mixture_params(var_mixture_params)
component_log_densities = np.vstack([component_log_density(params_k, x)
for params_k in var_params]).T
return logsumexp(component_log_densities + log_weights, axis=1, keepdims=False)
def mixture_elbo(var_mixture_params, t):
# We need to only sample the continuous component parameters,
# and integrate over the discrete component choice
def mixture_lower_bound(params,alpha=100):
"""Provides a stochastic estimate of the variational lower bound."""
samples = component_sample(params, num_samples, rs)
log_qs = mixture_log_density(var_mixture_params, samples)
log_ps = logprob(samples, t)
log_ps = np.reshape(log_ps, (num_samples, -1))
log_qs = np.reshape(log_qs, (num_samples, -1))
#return np.mean(log_ps - log_qs)
factor = 1. #if (alpha < 0.) else 1.
argument = np.exp(log_ps - log_qs)
c=1.-alpha
qs = np.exp(log_qs)
ps = np.exp(log_ps)
return factor*(1./c)*np.log(np.mean(np.power(argument,c)))
log_weights, var_params = unpack_mixture_params(var_mixture_params)
component_elbos = np.vstack(
[mixture_lower_bound(params_k) for params_k in var_params])
return np.sum(component_elbos + log_weights)
return init_var_params, mixture_elbo, mixture_log_density, sample
if __name__ == '__main__':
# Specify an inference problem by its unnormalized log-density.
D = 1
def log_density(x, t):
dens = np.exp(-1*np.square(x[:,0]))*(np.square(np.sin(x[:,0]))+0.3)#+np.abs(np.cos(x[:,0]))
return np.log(dens)
init_var_params, elbo, variational_log_density, variational_sampler = \\
build_mog_bbsvi(log_density, num_samples=400, k=10)
def objective(params, t):
return -elbo(params, t)
def plot_lines(ax, func, xlimits=[-2, 2], ylimits=[0, 1],
numticks=101, cmap=None):
x = np.linspace(*xlimits, num=numticks).reshape((numticks,1))
f = func(x)
plt.plot(x,f)
ax.set_yticks([])
ax.set_xticks([])
ax.set_xlim(-4,4)
#ax.set_ylim(0,2)
# Set up plotting code
fig = plt.figure(figsize=(8,8), facecolor='white')
ax = fig.add_subplot(111, frameon=False)
plt.ion()
plt.show(block=False)
num_plotting_samples = 51
def callback(params, t, g):
print("Iteration {} lower bound {}".format(t, -objective(params, t)))
plt.cla()
target_distribution = lambda x: np.exp(log_density(x, t))
var_distribution = lambda x: np.exp(variational_log_density(params, x))
plot_lines(ax, target_distribution)
plot_lines(ax, var_distribution)
ax.set_autoscale_on(False)
rs = npr.RandomState(0)
samples = variational_sampler(params, num_plotting_samples, rs)
plt.plot(samples[:, 0],np.ones(num_plotting_samples), 'x')
plt.draw()
plt.pause(1.0/30.0)
print("Optimizing variational parameters...")
variational_params = adam(grad(objective), init_var_params(D,log_std_mean=-2), step_size=0.1,
num_iters=1000, callback=callback)