Theano快速突击(二)

下面是logic函数：

原文描述：

Logistic Function=================Here's another straightforward example, though a bit more elaboratethan adding two numbers together. Let's say that you want to computethe logistic curve, which is given by:.. math::   s(x) = \frac{1}{1 + e^{-x}}You want to compute the function :ref:`elementwise<libdoc_tensor_elementwise>` on matrices of doubles, which means thatyou want to apply this function to each individual element of thematrix.Well, what you do is this:.. If you modify this code, also change :.. theano/tests/test_tutorial.py:T_examples.test_examples_1>>> import theano>>> import theano.tensor as T>>> x = T.dmatrix('x')>>> s = 1 / (1 + T.exp(-x))>>> logistic = theano.function([x], s)>>> logistic([[0, 1], [-1, -2]])array([[ 0.5       ,  0.73105858],       [ 0.26894142,  0.11920292]])The reason logistic is performed elementwise is because all of itsoperations---division, addition, exponentiation, and division---arethemselves elementwise operations.

一个doubles矩阵上元素依次(elementwise)计算，意味着你想把这个函数应用到矩阵的每个独立的元素上。
logistic依次执行的原因是因为所有它的运算-除法，加法，指数运算 - 是它们自己的元素依次运算。

It is also the case that:.. math::    s(x) = \frac{1}{1 + e^{-x}} = \frac{1 + \tanh(x/2)}{2}We can verify that this alternate form produces the same values:.. If you modify this code, also change :.. theano/tests/test_tutorial.py:T_examples.test_examples_2>>> s2 = (1 + T.tanh(x / 2)) / 2>>> logistic2 = theano.function([x], s2)>>> logistic2([[0, 1], [-1, -2]])array([[ 0.5       ,  0.73105858],       [ 0.26894142,  0.11920292]])Computing More than one Thing at the Same Time==============================================Theano supports functions with multiple outputs. For example, we cancompute the :ref:`elementwise <libdoc_tensor_elementwise>` difference, absolute difference, andsquared difference between two matrices *a* and *b* at the same time:.. If you modify this code, also change :.. theano/tests/test_tutorial.py:T_examples.test_examples_3>>> a, b = T.dmatrices('a', 'b')>>> diff = a - b>>> abs_diff = abs(diff)>>> diff_squared = diff**2>>> f = theano.function([a, b], [diff, abs_diff, diff_squared])>>> f([[1, 1], [1, 1]], [[0, 1], [2, 3]])[array([[ 1.,  0.],       [-1., -2.]]), array([[ 1.,  0.],       [ 1.,  2.]]), array([[ 1.,  0.],       [ 1.,  4.]])]Setting a Default Value for an Argument=======================================.. theano/tests/test_tutorial.py:T_examples.test_examples_6>>> from theano import In>>> from theano import function>>> x, y = T.dscalars('x', 'y')>>> z = x + y>>> f = function([x, In(y, value=1)], z)>>> f(33)array(34.0)>>> f(33, 2)array(35.0)This makes use of the :ref:`In <function_inputs>` class which allowsyou to specify properties of your function's parameters with greater detail. Here wegive a default value of 1 for *y* by creating a ``In`` instance withits ``value`` field set to 1.Inputs with default values must follow inputs without defaultvalues (like Python's functions).  There can be multiple inputs with default values. These parameters canbe set positionally or by name, as in standard Python:.. If you modify this code, also change :.. theano/tests/test_tutorial.py:T_examples.test_examples_7>>> x, y, w = T.dscalars('x', 'y', 'w')>>> z = (x + y) * w>>> f = function([x, In(y, value=1), In(w, value=2, name='w_by_name')], z)>>> f(33)array(68.0)>>> f(33, 2)array(70.0)>>> f(33, 0, 1)array(33.0)>>> f(33, w_by_name=1)array(34.0)>>> f(33, w_by_name=1, y=0)array(33.0).. note::   ``In`` does not know the name of the local variables *y* and *w*   that are passed as arguments.  The symbolic variable objects have name   attributes (set by ``dscalars`` in the example above) and *these* are the   names of the keyword parameters in the functions that we build.  This is   the mechanism at work in ``In(y, value=1)``.  In the case of ``In(w,   value=2, name='w_by_name')``. We override the symbolic variable's name   attribute with a name to be used for this function.You may like to see :ref:`Function<usingfunction>` in the library for more detail... _functionstateexample:Using Shared Variables======================It is also possible to make a function with an internal state. Forexample, let's say we want to make an accumulator: at the beginning,the state is initialized to zero. Then, on each function call, the stateis incremented by the function's argument.First let's define the *accumulator* function. It adds its argument to theinternal state, and returns the old state value... If you modify this code, also change :.. theano/tests/test_tutorial.py:T_examples.test_examples_8>>> from theano import shared>>> state = shared(0)>>> inc = T.iscalar('inc')>>> accumulator = function([inc], state, updates=[(state, state+inc)]).. theano/tests/test_tutorial.py:T_examples.test_examples_8>>> print(state.get_value())0>>> accumulator(1)array(0)>>> print(state.get_value())1>>> accumulator(300)array(1)>>> print(state.get_value())301It is possible to reset the state. Just use the ``.set_value()`` method:>>> state.set_value(-1)>>> accumulator(3)array(-1)>>> print(state.get_value())2As we mentioned above, you can define more than one function to use the sameshared variable.  These functions can all update the value... If you modify this code, also change :.. theano/tests/test_tutorial.py:T_examples.test_examples_8>>> decrementor = function([inc], state, updates=[(state, state-inc)])>>> decrementor(2)array(2)>>> print(state.get_value())0.. theano/tests/test_tutorial.py:T_examples.test_examples_8>>> fn_of_state = state * 2 + inc>>> # The type of foo must match the shared variable we are replacing>>> # with the ``givens``>>> foo = T.scalar(dtype=state.dtype)>>> skip_shared = function([inc, foo], fn_of_state, givens=[(state, foo)])>>> skip_shared(1, 3)  # we're using 3 for the state, not state.valuearray(7)>>> print(state.get_value())  # old state still there, but we didn't use it0Copying functions=================Theano functions can be copied, which can be useful for creating similarfunctions but with different shared variables or updates. This is done usingthe :func:`copy()<theano.compile.function_module.Function.copy>` method of ``function`` objects. The optimized graph of the original function is copied,so compilation only needs to be performed once.Let's start from the accumulator defined above:>>> import theano>>> import theano.tensor as T>>> state = theano.shared(0)>>> inc = T.iscalar('inc')>>> accumulator = theano.function([inc], state, updates=[(state, state+inc)])We can use it to increment the state as usual:>>> accumulator(10)array(0)>>> print(state.get_value())10We can use ``copy()`` to create a similar accumulator but with its own internal stateusing the ``swap`` parameter, which is a dictionary of shared variables to exchange:>>> new_state = theano.shared(0)>>> new_accumulator = accumulator.copy(swap={state:new_state})>>> new_accumulator(100)[array(0)]>>> print(new_state.get_value())100The state of the first function is left untouched:>>> print(state.get_value())10We now create a copy with updates removed using the ``delete_updates``parameter, which is set to ``False`` by default:>>> null_accumulator = accumulator.copy(delete_updates=True)As expected, the shared state is no longer updated:>>> null_accumulator(9000)[array(10)]>>> print(state.get_value())10.. _using_random_numbers:Using Random Numbers====================Brief Example-------------Here's a brief example.  The setup code is:.. If you modify this code, also change :.. theano/tests/test_tutorial.py:T_examples.test_examples_9.. testcode::    from theano.tensor.shared_randomstreams import RandomStreams    from theano import function    srng = RandomStreams(seed=234)    rv_u = srng.uniform((2,2))    rv_n = srng.normal((2,2))    f = function([], rv_u)    g = function([], rv_n, no_default_updates=True)    #Not updating rv_n.rng    nearly_zeros = function([], rv_u + rv_u - 2 * rv_u)Here, 'rv_u' represents a random stream of 2x2 matrices of draws from a uniformdistribution.  Likewise,  'rv_n' represents a random stream of 2x2 matrices ofdraws from a normal distribution.  The distributions that are implemented aredefined in :class:`RandomStreams` and, at a lower level,in :ref:`raw_random<libdoc_tensor_raw_random>`. They only work on CPU.See `Other Implementations`_ for GPU version.Now let's use these objects.  If we call f(), we get random uniform numbers.The internal state of the random number generator is automatically updated,so we get different random numbers every time.>>> f_val0 = f()>>> f_val1 = f()  #different numbers from f_val0When we add the extra argument ``no_default_updates=True`` to``function`` (as in *g*), then the random number generator state isnot affected by calling the returned function.  So, for example, calling*g* multiple times will return the same numbers.>>> g_val0 = g()  # different numbers from f_val0 and f_val1>>> g_val1 = g()  # same numbers as g_val0!An important remark is that a random variable is drawn at most once during anysingle function execution.  So the *nearly_zeros* function is guaranteed toreturn approximately 0 (except for rounding error) even though the *rv_u*random variable appears three times in the output expression.>>> nearly_zeros = function([], rv_u + rv_u - 2 * rv_u)Seeding Streams---------------Random variables can be seeded individually or collectively.You can seed just one random variable by seeding or assigning to the``.rng`` attribute, using ``.rng.set_value()``.>>> rng_val = rv_u.rng.get_value(borrow=True)   # Get the rng for rv_u>>> rng_val.seed(89234)                         # seeds the generator>>> rv_u.rng.set_value(rng_val, borrow=True)    # Assign back seeded rngYou can also seed *all* of the random variables allocated by a :class:`RandomStreams`object by that object's ``seed`` method.  This seed will be used to seed atemporary random number generator, that will in turn generate seeds for eachof the random variables.>>> srng.seed(902340)  # seeds rv_u and rv_n with different seeds eachSharing Streams Between Functions---------------------------------As usual for shared variables, the random number generators used for randomvariables are common between functions.  So our *nearly_zeros* function willupdate the state of the generators used in function *f* above.For example:>>> state_after_v0 = rv_u.rng.get_value().get_state()>>> nearly_zeros()       # this affects rv_u's generatorarray([[ 0.,  0.],       [ 0.,  0.]])>>> v1 = f()>>> rng = rv_u.rng.get_value(borrow=True)>>> rng.set_state(state_after_v0)>>> rv_u.rng.set_value(rng, borrow=True)>>> v2 = f()             # v2 != v1>>> v3 = f()             # v3 == v1Copying Random State Between Theano Graphs------------------------------------------An example of how "random states" can be transferred from one theano functionto another is shown below.>>> from __future__ import print_function>>> import theano>>> import numpy>>> import theano.tensor as T>>> from theano.sandbox.rng_mrg import MRG_RandomStreams>>> from theano.tensor.shared_randomstreams import RandomStreams>>> class Graph():...     def __init__(self, seed=123):...         self.rng = RandomStreams(seed)...         self.y = self.rng.uniform(size=(1,))>>> g1 = Graph(seed=123)>>> f1 = theano.function([], g1.y)>>> g2 = Graph(seed=987)>>> f2 = theano.function([], g2.y)>>> # By default, the two functions are out of sync.>>> f1()array([ 0.72803009])>>> f2()array([ 0.55056769])>>> def copy_random_state(g1, g2):...     if isinstance(g1.rng, MRG_RandomStreams):...         g2.rng.rstate = g1.rng.rstate...     for (su1, su2) in zip(g1.rng.state_updates, g2.rng.state_updates):...         su2[0].set_value(su1[0].get_value())>>> # We now copy the state of the theano random number generators.>>> copy_random_state(g1, g2)>>> f1()array([ 0.59044123])>>> f2()array([ 0.59044123])Other Random Distributions--------------------------There are :ref:`other distributions implemented <libdoc_tensor_raw_random>`... _example_other_random:Other Implementations---------------------There are 2 other implementations based on :ref:`MRG31k3p<libdoc_rng_mrg>` and :class:`CURAND <theano.sandbox.cuda.rng_curand>`.The RandomStream only work on the CPU, MRG31k3pwork on the CPU and GPU. CURAND only work on the GPU... note::    To use you the MRG version easily, you can just change the import to:        .. code-block:: python            from theano.sandbox.rng_mrg import MRG_RandomStreams as RandomStreams.. _logistic_regression:A Real Example: Logistic Regression===================================The preceding elements are featured in this more realistic example.It will be used repeatedly... testcode::    import numpy    import theano    import theano.tensor as T    rng = numpy.random    N = 400                                   # training sample size    feats = 784                               # number of input variables    # generate a dataset: D = (input_values, target_class)    D = (rng.randn(N, feats), rng.randint(size=N, low=0, high=2))    training_steps = 10000    # Declare Theano symbolic variables    x = T.dmatrix("x")    y = T.dvector("y")    # initialize the weight vector w randomly    #    # this and the following bias variable b    # are shared so they keep their values    # between training iterations (updates)    w = theano.shared(rng.randn(feats), name="w")    # initialize the bias term    b = theano.shared(0., name="b")    print("Initial model:")    print(w.get_value())    print(b.get_value())    # Construct Theano expression graph    p_1 = 1 / (1 + T.exp(-T.dot(x, w) - b))   # Probability that target = 1    prediction = p_1 > 0.5                    # The prediction thresholded    xent = -y * T.log(p_1) - (1-y) * T.log(1-p_1) # Cross-entropy loss function    cost = xent.mean() + 0.01 * (w ** 2).sum()# The cost to minimize    gw, gb = T.grad(cost, [w, b])             # Compute the gradient of the cost                                              # w.r.t weight vector w and                                              # bias term b                                              # (we shall return to this in a                                              # following section of this tutorial)    # Compile    train = theano.function(              inputs=[x,y],              outputs=[prediction, xent],              updates=((w, w - 0.1 * gw), (b, b - 0.1 * gb)))    predict = theano.function(inputs=[x], outputs=prediction)    # Train    for i in range(training_steps):        pred, err = train(D[0], D[1])    print("Final model:")    print(w.get_value())    print(b.get_value())    print("target values for D:")    print(D[1])    print("prediction on D:")    print(predict(D[0])).. testoutput::   :hide:   :options: +ELLIPSIS   Initial model:   ...   0.0   Final model:   ...   target values for D:   ...   prediction on D:   ...

翻译参考：http://geek.csdn.net/news/detail/131362