.. _tutcomputinggrads:
=======================
Derivatives in PyTensor
=======================
Computing Gradients
===================
Now let's use PyTensor for a slightly more sophisticated task: create a
function which computes the derivative of some expression ``y`` with
respect to its parameter ``x``. To do this we will use the macro `at.grad`.
For instance, we can compute the gradient of :math:`x^2` with respect to
:math:`x`. Note that: :math:`d(x^2)/dx = 2 \cdot x`.
Here is the code to compute this gradient:
.. If you modify this code, also change :
.. tests/test_tutorial.py:T_examples.test_examples_4
>>> import numpy
>>> import pytensor
>>> import pytensor.tensor as at
>>> from pytensor import pp
>>> x = at.dscalar('x')
>>> y = x ** 2
>>> gy = at.grad(y, x)
>>> pp(gy) # print out the gradient prior to optimization
'((fill((x ** TensorConstant{2}), TensorConstant{1.0}) * TensorConstant{2}) * (x ** (TensorConstant{2} - TensorConstant{1})))'
>>> f = pytensor.function([x], gy)
>>> f(4)
array(8.0)
>>> numpy.allclose(f(94.2), 188.4)
True
In this example, we can see from ``pp(gy)`` that we are computing
the correct symbolic gradient.
``fill((x**2), 1.0)`` means to make a matrix of the same shape as
``x**2`` and fill it with ``1.0``.
.. note::
PyTensor's rewrites simplify the symbolic gradient expression. You can see
this by digging inside the internal properties of the compiled function.
.. testcode::
pp(f.maker.fgraph.outputs[0])
'(2.0 * x)'
After rewriting, there is only one `Apply` node left in the graph.
We can also compute the gradient of complex expressions such as the
logistic function defined above. It turns out that the derivative of the
logistic is: :math:`ds(x)/dx = s(x) \cdot (1 - s(x))`.
.. figure:: dlogistic.png
A plot of the gradient of the logistic function, with :math:`x` on the x-axis
and :math:`ds(x)/dx` on the :math:`y`-axis.
.. If you modify this code, also change :
.. tests/test_tutorial.py:T_examples.test_examples_5
>>> x = at.dmatrix('x')
>>> s = at.sum(1 / (1 + at.exp(-x)))
>>> gs = at.grad(s, x)
>>> dlogistic = pytensor.function([x], gs)
>>> dlogistic([[0, 1], [-1, -2]])
array([[ 0.25 , 0.19661193],
[ 0.19661193, 0.10499359]])
In general, for any **scalar** expression ``s``, ``at.grad(s, w)`` provides
the PyTensor expression for computing :math:`\frac{\partial s}{\partial w}`. In
this way PyTensor can be used for doing **efficient** symbolic differentiation
(as the expression returned by `at.grad` will be optimized during compilation), even for
function with many inputs. (see `automatic differentiation `_ for a description
of symbolic differentiation).
.. note::
The second argument of `at.grad` can be a list, in which case the
output is also a list. The order in both lists is important: element
``i`` of the output list is the gradient of the first argument of
`at.grad` with respect to the ``i``-th element of the list given as second argument.
The first argument of `at.grad` has to be a scalar (a tensor
of size 1). For more information on the semantics of the arguments of
`at.grad` and details about the implementation, see
:ref:`this` section of the library.
Additional information on the inner workings of differentiation may also be
found in the more advanced tutorial :ref:`Extending PyTensor`.
Computing the Jacobian
======================
In PyTensor's parlance, the term **Jacobian** designates the tensor comprising the
first partial derivatives of the output of a function with respect to its inputs.
(This is a generalization of to the so-called Jacobian matrix in Mathematics.)
PyTensor implements the :func:`pytensor.gradient.jacobian` macro that does all
that is needed to compute the Jacobian. The following text explains how
to do it manually.
In order to manually compute the Jacobian of some function ``y`` with
respect to some parameter ``x`` we need to use `scan`. What we
do is to loop over the entries in ``y`` and compute the gradient of
``y[i]`` with respect to ``x``.
.. note::
`scan` is a generic op in PyTensor that allows writing in a symbolic
manner all kinds of recurrent equations. While creating
symbolic loops (and optimizing them for performance) is a hard task,
effort is being done for improving the performance of `scan`. We
shall return to :ref:`scan` later in this tutorial.
>>> import pytensor
>>> import pytensor.tensor as at
>>> x = at.dvector('x')
>>> y = x ** 2
>>> J, updates = pytensor.scan(lambda i, y, x : at.grad(y[i], x), sequences=at.arange(y.shape[0]), non_sequences=[y, x])
>>> f = pytensor.function([x], J, updates=updates)
>>> f([4, 4])
array([[ 8., 0.],
[ 0., 8.]])
What we do in this code is to generate a sequence of integers from ``0`` to
``y.shape[0]`` using `at.arange`. Then we loop through this sequence, and
at each step, we compute the gradient of element ``y[i]`` with respect to
``x``. `scan` automatically concatenates all these rows, generating a
matrix which corresponds to the Jacobian.
.. note::
There are some pitfalls to be aware of regarding `at.grad`. One of them is that you
cannot re-write the above expression of the Jacobian as
``pytensor.scan(lambda y_i,x: at.grad(y_i,x), sequences=y, non_sequences=x)``,
even though from the documentation of scan this
seems possible. The reason is that ``y_i`` will not be a function of
``x`` anymore, while ``y[i]`` still is.
Computing the Hessian
=====================
In PyTensor, the term **Hessian** has the usual mathematical meaning: It is the
matrix comprising the second order partial derivative of a function with scalar
output and vector input. PyTensor implements :func:`pytensor.gradient.hessian` macro that does all
that is needed to compute the Hessian. The following text explains how
to do it manually.
You can compute the Hessian manually similarly to the Jacobian. The only
difference is that now, instead of computing the Jacobian of some expression
``y``, we compute the Jacobian of ``at.grad(cost,x)``, where ``cost`` is some
scalar.
>>> x = at.dvector('x')
>>> y = x ** 2
>>> cost = y.sum()
>>> gy = at.grad(cost, x)
>>> H, updates = pytensor.scan(lambda i, gy,x : at.grad(gy[i], x), sequences=at.arange(gy.shape[0]), non_sequences=[gy, x])
>>> f = pytensor.function([x], H, updates=updates)
>>> f([4, 4])
array([[ 2., 0.],
[ 0., 2.]])
Jacobian times a Vector
=======================
Sometimes we can express the algorithm in terms of Jacobians times vectors,
or vectors times Jacobians. Compared to evaluating the Jacobian and then
doing the product, there are methods that compute the desired results while
avoiding actual evaluation of the Jacobian. This can bring about significant
performance gains. A description of one such algorithm can be found here:
- Barak A. Pearlmutter, "Fast Exact Multiplication by the Hessian", Neural Computation, 1994
While in principle we would want PyTensor to identify these patterns automatically for us,
in practice, implementing such optimizations in a generic manner is extremely
difficult. Therefore, we provide special functions dedicated to these tasks.
R-operator
----------
The **R operator** is built to evaluate the product between a Jacobian and a
vector, namely :math:`\frac{\partial f(x)}{\partial x} v`. The formulation
can be extended even for :math:`x` being a matrix, or a tensor in general, case in
which also the Jacobian becomes a tensor and the product becomes some kind
of tensor product. Because in practice we end up needing to compute such
expressions in terms of weight matrices, PyTensor supports this more generic
form of the operation. In order to evaluate the R-operation of
expression ``y``, with respect to ``x``, multiplying the Jacobian with ``V``
you need to do something similar to this:
>>> W = at.dmatrix('W')
>>> V = at.dmatrix('V')
>>> x = at.dvector('x')
>>> y = at.dot(x, W)
>>> JV = pytensor.gradient.Rop(y, W, V)
>>> f = pytensor.function([W, V, x], JV)
>>> f([[1, 1], [1, 1]], [[2, 2], [2, 2]], [0,1])
array([ 2., 2.])
:ref:`List ` of Op that implement Rop.
L-operator
----------
In similitude to the R-operator, the **L-operator** would compute a row vector times
the Jacobian. The mathematical formula would be :math:`v \frac{\partial
f(x)}{\partial x}`. The L-operator is also supported for generic tensors
(not only for vectors). Similarly, it can be implemented as follows:
>>> W = at.dmatrix('W')
>>> v = at.dvector('v')
>>> x = at.dvector('x')
>>> y = at.dot(x, W)
>>> VJ = pytensor.gradient.Lop(y, W, v)
>>> f = pytensor.function([v,x], VJ)
>>> f([2, 2], [0, 1])
array([[ 0., 0.],
[ 2., 2.]])
.. note::
``v``, the point of evaluation, differs between the L-operator and the R-operator.
For the L-operator, the point of evaluation needs to have the same shape
as the output, whereas for the R-operator this point should
have the same shape as the input parameter. Furthermore, the results of these two
operations differ. The result of the L-operator is of the same shape
as the input parameter, while the result of the R-operator has a shape similar
to that of the output.
:ref:`List of op with r op support `.
Hessian times a Vector
======================
If you need to compute the Hessian times a vector, you can make use of the
above-defined operators to do it more efficiently than actually computing
the exact Hessian and then performing the product. Due to the symmetry of the
Hessian matrix, you have two options that will
give you the same result, though these options might exhibit differing performances.
Hence, we suggest profiling the methods before using either one of the two:
>>> x = at.dvector('x')
>>> v = at.dvector('v')
>>> y = at.sum(x ** 2)
>>> gy = at.grad(y, x)
>>> vH = at.grad(at.sum(gy * v), x)
>>> f = pytensor.function([x, v], vH)
>>> f([4, 4], [2, 2])
array([ 4., 4.])
or, making use of the R-operator:
>>> x = at.dvector('x')
>>> v = at.dvector('v')
>>> y = at.sum(x ** 2)
>>> gy = at.grad(y, x)
>>> Hv = pytensor.gradient.Rop(gy, x, v)
>>> f = pytensor.function([x, v], Hv)
>>> f([4, 4], [2, 2])
array([ 4., 4.])
Final Pointers
==============
- The `grad` function works symbolically: it receives and returns PyTensor variables.
- `grad` can be compared to a macro since it can be applied repeatedly.
- Scalar costs only can be directly handled by `grad`. Arrays are handled through repeated applications.
- Built-in functions allow to compute efficiently vector times Jacobian and vector times Hessian.
- Work is in progress on the optimizations required to compute efficiently the full
Jacobian and the Hessian matrix as well as the Jacobian times vector.