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Use VIMCO to lower the variance of gradient[csiszar_function(log(Avg(u))]. (deprecated)

This function generalizes VIMCO [(Mnih and Rezende, 2016)][1] to Csiszar f-Divergences.

The VIMCO loss is:

vimco = f(log(Avg{u[i] : i=0,...,m-1}))
  logu[i] = log( p(x, h[i]) / q(h[i] | x) )
  h[i] iid~ q(H | x)

Interestingly, the VIMCO gradient is not the naive gradient of vimco. Rather, it is characterized by:

grad[vimco] - variance_reducing_term
  variance_reducing_term = Sum{ grad[log q(h[i] | x)] *
                                  (vimco - f(log Avg{h[j;i] : j=0,...,m-1}))
                               : i=0, ..., m-1 }
  h[j;i] = { u[j]                             j!=i
           { GeometricAverage{ u[k] : k!=i}   j==i

(We omitted stop_gradient for brevity. See implementation for more details.)

The Avg{h[j;i] : j} term is a kind of "swap-out average" where the i-th element has been replaced by the leave-i-out Geometric-average.

This implementation prefers numerical precision over efficiency, i.e., O(num_draws * num_batch_draws * prod(batch_shape) * prod(event_shape)). (The constant may be fairly large, perhaps around 12.)

f Python callable representing a Csiszar-function in log-space.
p_log_prob Python callable representing the natural-log of the probability under distribution p. (In variational inference p is the joint distribution.)
q tf.Distribution-like instance; must implement: sample(n, seed), and log_prob(x). (In variational inference q is the approximate posterior distribution.)
num_draws Integer scalar number of draws used to approximate the f-Divergence expectation.
num_batch_draws Integer scalar number of draws used to approximate the f-Divergence expectation.
seed PRNG seed for q.sample; see tfp.random.sanitize_seed for details.
name Python str name prefixed to Ops created by this function.

vimco The Csiszar f-Divergence generalized VIMCO objective.

ValueError if num_draws < 2.


[1]: Andriy Mnih and Danilo Rezende. Variational Inference for Monte Carlo objectives. In International Conference on Machine Learning, 2016.