DOC: Write stochastic variational inference to doc

doc/source/user_guide/advanced.rst

@@ -246,8 +246,90 @@ the next iteration with a new annealing value.
 Stochastic variational inference
 --------------------------------
 
+In stochastic variational inference :cite:`Hoffman:2013`, the idea is to use
+mini-batches of large datasets to compute noisy gradients and learn the VB
+distributions by using stochastic gradient ascent.  In order for it to be
+useful, the model must be such that it can be divided into "intermediate" and
+"global" variables.  The number of intermediate variables increases with the
+data but the number of global variables remains fixed.  The global variables are
+learnt in the stochastic optimization.
+
+By denoting the data as :math:`Y=[Y_1, \ldots, Y_N]`, the intermediate variables
+as :math:`Z=[Z_1, \ldots, Z_N]` and the global variables as :math:`\theta`, the
+model needs to have the following structure:
+
+.. math::
+
+   p(Y, Z, \theta) &= p(\theta) \prod^N_{n=1} p(Y_n|Z_n,\theta) p(Z_n|\theta)
+
+The algorithm consists of three steps which are iterated: 1) a random mini-batch
+of the data is selected, 2) the corresponding intermediate variables are updated
+by using normal VB update equations, and 3) the global variables are updated
+with (stochastic) gradient ascent as if there was as many replications of the
+mini-batch as needed to recover the original dataset size.
+
+The learning rate for the gradient ascent must satisfy:
+
+.. math::
+
+   \sum^\infty_{i=1} \alpha_i = \infty \qquad \text{and} \qquad
+   \sum^\infty_{i=1} \alpha^2 < \infty,
+
+where :math:`i` is the iteration number.  An example of a valid learning
+parameter is :math:`\alpha_i = (\delta + i)^{-\gamma}`, where :math:`\delta \geq
+0` is a delay and :math:`\gamma\in (0.5, 1]` is a forgetting rate.
+
+Stochastic variational inference is relatively easy to use in BayesPy.  The idea
+is that the user creates a model for the size of a mini-batch and specifies a
+multiplier for those plate axes that are replicated.  For the PCA example, the
+mini-batch model can be costructed as follows.  We decide to use ``X`` as an
+intermediate variable and the other variables are global.  The global variables
+``alpha``, ``C`` and ``tau`` are constructed identically as before.  The
+intermediate variable ``X`` is constructed as:
+
+>>> X = GaussianARD(0, 1,
+...                 shape=(D,),
+...                 plates=(1,5),
+...                 plates_multiplier=(1,20),
+...                 name='X')
+
+Note that the plates are ``(1,5)`` whereas they are ``(1,100)`` in the full
+model.  Thus, we need to provide a plates multiplier ``(1,20)`` to define how
+the plates are replicated to get the full dataset.  These multipliers do not
+need to be integers, in this case the latter plate axis is multiplied by
+:math:`100/5=20`.  The remaining variables are defined as before:
+
+>>> F = Dot(C, X)
+>>> Y = GaussianARD(F, tau, name='Y')
+
+Note that the plates of ``Y`` and ``F`` also correspond to the size of the
+mini-batch and they also deduce the plate multipliers from their parents, thus
+we do not need to specify the multiplier here explicitly (although it is ok to
+do so).
+
+Let us construct the inference engine for the new mini-batch model:
+
+>>> Q = VB(Y, C, X, alpha, tau)
+
+Use random initialization for ``C`` to break the symmetry in ``C`` and ``X``:
+
+>>> C.initialize_from_random()
+
+Then, stochastic variational inference algorithm could look as follows:
+
+>>> for n in range(200):
+...     subset = np.random.choice(100, 5)
+...     Y.observe(data[:,subset])
+...     Q.update(X)
+...     learning_rate = (n + 2.0) ** (-0.7)
+...     Q.gradient_step(C, alpha, tau, scale=learning_rate)
+
+The loop consists of three parts: 1) Draw a random mini-batch of the data (5
+samples from 100).  2) Update the intermediate variable ``X``.  3) Update global
+variables with gradient ascent using a proper learning rate.
+
 
 Black-box variational inference
 -------------------------------
 
-
+TODO