python - MLP神经网络: calculating the gradient (matrices)

Question

在 n 层神经网络中计算梯度的良好实现是什么？

权重层:

1. 第一层权重:    (n_inputs+1, n_units_layer)-matrix
2. 隐藏层权重:(n_units_layer+1, n_units_layer)-matrix
3. 最后一层权重:    (n_units_layer+1, n_outputs)-matrix

注释:

• 如果只有一个隐藏层，我们将仅使用两个(权重)层来表示网络:
  输入 --first_layer-> network_unit --second_layer-> 输出
• 对于具有多个隐藏层的 n 层网络，我们需要实现第 (2) 步骤。

---

有点模糊的伪代码:

    weight_layers = [ layer1, layer2 ]             # a list of layers as described above
    input_values  = [ [0,0], [0,0], [1,0], [0,1] ] # our test set (corresponds to XOR)
    target_output = [ 0, 0, 1, 1 ]                 # what we want to train our net to output
    output_layers = []                             # output for the corresponding layers

    for layer in weight_layers:
        output <-- calculate the output     # calculate the output from the current layer
        output_layers <-- output            # store the output from each layer
    
    n_samples = input_values.shape[0]
    n_outputs = target_output.shape[1]
    
    error = ( output-target_output )/( n_samples*n_outputs )

    """ calculate the gradient here """

---

最终实现

The final implementation is available at github .

Answer 1

使用 Python 和 numpy 这很容易。

您有两个选择:

1. 您可以并行计算 num_instances 个实例的所有内容，或者
2. 您可以计算一个实例的梯度(这实际上是 1 的一种特例。)。

我现在将给出一些如何实现选项 1 的提示。我建议您创建一个名为 Layer 的新类。它应该有两个功能:

forward:
    inputs:
    X: shape = [num_instances, num_inputs]
        inputs
    W: shape = [num_outputs, num_inputs]
        weights
    b: shape = [num_outputs]
        biases
    g: function
        activation function
    outputs:
    Y: shape = [num_instances, num_outputs]
        outputs


backprop:
    inputs:
    dE/dY: shape = [num_instances, num_outputs]
        backpropagated gradient
    W: shape = [num_outputs, num_inputs]
        weights
    b: shape = [num_outputs]
        biases
    gd: function
        calculates the derivative of g(A) = Y
        based on Y, i.e. gd(Y) = g'(A)
    Y: shape = [num_instances, num_outputs]
        outputs
    X: shape = [num_instances, num_inputs]
        inputs
    outputs:
    dE/dX: shape = [num_instances, num_inputs]
        will be backpropagated (dE/dY of lower layer)
    dE/dW: shape = [num_outputs, num_inputs]
        accumulated derivative with respect to weights
    dE/db: shape = [num_outputs]
        accumulated derivative with respect to biases

The implementation is simple:

def forward(X, W, b):
    A = X.dot(W.T) + b # will be broadcasted
    Y = g(A)
    return Y

def backprop(dEdY, W, b, gd, Y, X):
    Deltas = gd(Y) * dEdY # element-wise multiplication
    dEdX = Deltas.dot(W)
    dEdW = Deltas.T.dot(X)
    dEdb = Deltas.sum(axis=0)
    return dEdX, dEdW, dEdb

第一层的

X 是从数据集中获取的，然后将每个 Y 作为下一层的 X 传递前向传球。

输出层的 dE/dY 计算(对于 softmax 激活函数和交叉熵误差函数，或者对于线性激活函数和误差平方和)为 Y-T，其中 Y 是网络的输出 (shape = [num_instances, num_outputs])，T (shape = [num_instances, num_outputs]) 是所需的输出。然后你可以反向传播，即每层的dE/dX是前一层的dE/dY。

现在您可以使用每层的dE/dW和dE/db来更新W和b .

以下是 C++ 的示例:OpenANN .

顺便说一句。您可以比较实例方式和批量方式前向传播的速度:

In [1]: import timeit

In [2]: setup = """import numpy
   ...: W = numpy.random.rand(10, 5000)
   ...: X = numpy.random.rand(1000, 5000)"""

In [3]: timeit.timeit('[W.dot(x) for x in X]', setup=setup, number=10)
Out[3]: 0.5420958995819092

In [4]: timeit.timeit('X.dot(W.T)', setup=setup, number=10)
Out[4]: 0.22001314163208008