Using the AggregatedGradientIMNN
================================

With a generative model built in ``jax`` for example we can calculate
the derivatives of some simulations with which we can use the chain rule
to calculate the derivatives of the network outputs with respect to the
physical model parameters necessary to fit an IMNN. Furthermore, if the
simulations are too numerous or too large to fit into memory or could be
accelerated over several devices, we can aggregate the gradients too.
This is an expensive operation and should only be used if memory over a
single device is really an issue. Note that with a fixed data set for
training an IMNN it is important to have a validation set for early
stopping. This is because, with a limited training set there *will* be
accidental correlations which look like they are due to parameters and
the IMNN *will* extract these features. Using a validation set for early
stopping makes sure that once all the features in the validation set
have been extracted then no extra information can be incorrectly
processed.

For this example we are going to summaries the unknown mean,
:math:`\mu`, and variance, :math:`\Sigma`, of :math:`n_{\bf d}=10` data
points of two 1D random Gaussian field,
:math:`{\bf d}=\{d_i\sim\mathcal{N}(\mu,\Sigma)|i\in[1, n_{\bf d}]\}`.
This is an interesting problem since we know the likelihood
analytically, but it is non-Gaussian

.. math:: \mathcal{L}({\bf d}|\mu,\Sigma) = \prod_i^{n_{\bf d}}\frac{1}{\sqrt{2\pi|\Sigma|}}\exp\left[-\frac{1}{2}\frac{(d_i-\mu)^2}{\Sigma}\right]

As well as knowing the likelihood for this problem, we also know what
sufficient statistics describe the mean and variance of the data - they
are the mean and the variance

.. math:: \frac{1}{n_{\bf d}}\sum_i^{n_{\bf d}}d_i = \mu\textrm{  and  }\frac{1}{n_{\bf d}-1}\sum_i^{n_{\bf d}}(d_i-\mu)^2=\Sigma

What makes this an interesting problem for the IMNN is the fact that the
sufficient statistic for the variance is non-linear, i.e. it is a sum of
the square of the data, and so linear methods like MOPED would be lossy
in terms of information.

We can calculate the Fisher information by taking the negative second
derivative of the likelihood taking the expectation by inserting the
relations for the sufficient statistics, i.e. and examining at the
fiducial parameter values

.. math:: {\bf F}_{\alpha\beta} = -\left.\left(\begin{array}{cc}\displaystyle-\frac{n_{\bf d}}{\Sigma}&0\\0&\displaystyle-\frac{n_{\bf d}}{2\Sigma^2}\end{array}\right)\right|_{\Sigma=\Sigma^{\textrm{fid}}}.

Choosing fiducial parameter values of :math:`\mu^\textrm{fid}=0` and
:math:`\Sigma^\textrm{fid}=1` we find that the determinant of the Fisher
information matrix is :math:`|{\bf F}_{\alpha\beta}|=50`.

.. code:: ipython3

    from imnn import AggregatedGradientIMNN

    from imnn.utils import value_and_jacfwd

    import jax
    import jax.numpy as np
    from jax.experimental import stax, optimizers

We’re going to use 1000 summary vectors, with a length of two, at a time
to make an estimate of the covariance of network outputs and the
derivative of the mean of the network outputs with respect to the two
model parameters.

.. code:: ipython3

    n_s = 1000
    n_d = n_s

    n_params = 2
    n_summaries = n_params

    input_shape = (10,)

The simulator is simply

.. code:: ipython3

    def simulator(key, θ):
        return θ[0] + jax.random.normal(key, shape=input_shape) * np.sqrt(θ[1])

Our fiducial parameter values are :math:`\mu^\textrm{fid}=0` and
:math:`\Sigma^\textrm{fid}=1`.

.. code:: ipython3

    θ_fid = np.array([0., 1.])

We can generate the simulations using:

.. code:: ipython3

    get_sims_and_ders = value_and_jacfwd(simulator, argnums=1)

For initialising the neural network a random number generator and we’ll
grab another for generating the data:

.. code:: ipython3

    rng = jax.random.PRNGKey(1)
    rng, model_key, data_key = jax.random.split(rng, num=3)

We’ll make the keys for each of the simulations for fitting and
validation

.. code:: ipython3

    data_keys = np.array(jax.random.split(rng, num=2 * n_s))

.. code:: ipython3

    fiducial, derivative = jax.vmap(get_sims_and_ders)(
        data_keys[:n_s], np.repeat(np.expand_dims(θ_fid, 0), n_s, axis=0))
    validation_fiducial, validation_derivative = jax.vmap(get_sims_and_ders)(
        data_keys[n_s:], np.repeat(np.expand_dims(θ_fid, 0), n_s, axis=0))

We’re going to use ``jax``\ ’s stax module to build a simple network
with three hidden layers each with 128 neurons and which are activated
by leaky relu before outputting the two summaries. The optimiser will be
a ``jax`` Adam optimiser with a step size of 0.001.

.. code:: ipython3

    model = stax.serial(
        stax.Dense(128),
        stax.LeakyRelu,
        stax.Dense(128),
        stax.LeakyRelu,
        stax.Dense(128),
        stax.LeakyRelu,
        stax.Dense(n_summaries))
    optimiser = optimizers.adam(step_size=1e-3)

We will use the CPU as the host memory and use the GPUs for calculating
the summaries.

.. code:: ipython3

    host = jax.devices("cpu")[0]
    devices = jax.devices("gpu")

Now lets say that we know that we can process 100 simulations at a time
per device before running out of memory, we therefore can set

.. code:: ipython3

    n_per_device = 100

.. code:: ipython3

    imnn = AggregatedGradientIMNN(
        n_s=n_s, n_d=n_d, n_params=n_params, n_summaries=n_summaries,
        input_shape=input_shape, θ_fid=θ_fid, model=model,
        optimiser=optimiser, key_or_state=model_key, host=host,
        devices=devices, n_per_device=n_per_device,
        fiducial=fiducial, derivative=derivative,
        validation_fiducial=validation_fiducial,
        validation_derivative=validation_derivative,
        prefetch=None, cache=True)

To set the scale of the regularisation we use a coupling strength
:math:`\lambda` whose value should mean that the determinant of the
difference between the covariance of network outputs and the identity
matrix is larger than the expected initial value of the determinant of
the Fisher information matrix from the network. How close to the
identity matrix the covariance should be is set by :math:`\epsilon`.
These parameters should not be very important, but they will help with
convergence time.

.. code:: ipython3

    λ = 10.
    ϵ = 0.1

Fitting can then be done simply by calling:

.. code:: ipython3

    imnn.fit(λ, ϵ, patience=10, max_iterations=1000, print_rate=1)


Here we have included a ``print_rate`` for a progress bar, but leaving
this out will massively reduce fitting time (at the expense of not
knowing how many iterations have been run). The IMNN will be fit for a
maximum of ``max_iterations = 1000`` iterations, but with early stopping
which can turn on after ``min_iterations = 100`` iterations and after
``patience = 10`` iterations where the maximum determinant of the Fisher
information matrix has not increased. ``imnn.w`` is set to the values of
the network parameters which obtained the highest value of the
determinant of the Fisher information matrix, but the values at the
final iteration can be set using ``best = False``.

To continue training one can simply rerun fit

.. code:: python

   imnn.fit(λ, ϵ, patience=10, max_iterations=1000, print_rate=1)

although we will not run it in this example.

To visualise the fitting history we can plot the results:

.. code:: ipython3

    imnn.plot(expected_detF=50);



.. image:: output_27_0.png
