Neural Networks and Deep Learning 习题1

Sigmoid neurons simulating perceptrons, part I

Suppose we take all the weights and biases in a network of perceptrons, and multiply them by a positive constant, c>0c>0. Show that the behaviour of the network doesn't change. 对于perceptron rule :

两边同时乘上c，等式不变

Sigmoid neurons simulating perceptrons, part II

Suppose we have the same setup as the last problem - a network of perceptrons. Suppose also that the overall input to the network of perceptrons has been chosen. We won't need the actual input value, we just need the input to have been fixed. Suppose the weights and biases are such that w⋅x+b≠0 for the input x to any particular perceptron in the network. Now replace all the perceptrons in the network by sigmoid neurons, and multiply the weights and biases by a positive constant c>0. Show that in the limit as c→∞the behaviour of this network of sigmoid neurons is exactly the same as the network of perceptrons. How can this fail when w⋅x+b=0 for one of the perceptrons?

对于sigmoid 函数来讲，(wx + b)同时乘上c，不影响wx + b > 0 或 wx + b < 0的结果，所以对于σ(wx + b) > 0.5 或 σ(wx + b) < 0.5 的断定没有影响。可是当wx + b = 0时，σ(wx + b) = 0.5，判断不告终果的类别，所以不能进行二元分类。

Four output neural exercise

There is a way of determining the bitwise representation of a digit by adding an extra layer to the three-layer network above. The extra layer converts the output from the previous la!yer into a binary representation, as illustrated in the figure below. Find a set of weights and biases for the new output layer. Assume that the first 3 layers of neurons are such that the correct output in the third layer (i.e., the old output layer) has activation at least 0.99, and incorrect outputs have activation less than 0.01.

咱们先列出0~9的二进制数字：

• 0 ：0000

• 1 ：0001

• 2 ：0010

• 3 ：0011

• 4 ：0100

• 5 ：0101

• 6 ：0110

• 7 ：0111

• 8 ：1000

• 9 ：1001

  output layer从上至下依次表示2^0, 2^1, 2^2, 2^3，那么1,3,5,7,9会使2^0的output neural输出1,而2,3,6,7会使2^1的output neural输出1，依次类推。 那么对于第一个output neural，咱们能够令向量w=-1,1,-1,1,-1,1,-1,1,-1,1，b = 0则对于这个neural来讲，wx + b = -1x0 + x1 - x2 + x3 - x4 + x5 - x6 + x7 - x8 + x9 > 0 当且仅当输入中x1,x3,x5,x7,x9的值的和大于其他的和（正常状况下只有一项为0.99，其他都是0.01）。好比x5=0.99,其他是0.01，那么wx + b = 0.98 > 0 ,因此σ（wx + b）> 0.5，2^0位输出为1。 同理，获得4个ouput neural的w：

1. -1, 1,-1, 1,-1, 1,-1, 1,-1, 1

2. -1,-1, 1, 1,-1,-1, 1, 1,-1,-1

3. -1,-1,-1,-1, 1, 1, 1, 1,-1,-1

4. -1,-1,-1,-1,-1,-1,-1,-1, 1, 1

   注意前面给出的条件是：正确的结果必定大于0.99，错误的必定小于0.01，这样就避免了好比数字0为0.8最大，其他的是0.7比0.8小，可是其余的和加起来大于0.8，输出错误的状况。

proof that gradient descent is the optimal strategy for minimizing a cost function

Prove the assertion of the last paragraph. Hint: If you're not already familiar with the Cauchy-Schwarz inequality, you may find it helpful to familiarize yourself with it.

看了网上的一个答案：

geometric interpretation of what gradient descent is doing in the one-dimensional case

I explained gradient descent when C is a function of two variables, and when it's a function of more than two variables. What happens when C is a function of just one variable? Can you provide a geometric interpretation of what gradient descent is doing in the one-dimensional case?

如图，Δy = Δx * y`，容易看出y`的方向是y降低最快的方向。例如y=x^2，则y`=2x。迭代过程图像以下：

compared stochastic gradient descent with incremental gradient descent

An extreme version of gradient descent is to use a mini-batch size of just 1. That is, given a training input, x, we update our weights and biases according to the rules wk→w′k=wk−η∂Cx/∂wk and bl→b′l=bl−η∂Cx/∂bl. Then we choose another training input, and update the weights and biases again. And so on, repeatedly. This procedure is known as online, on-line, or incremental learning. In online learning, a neural network learns from just one training input at a time (just as human beings do). Name one advantage and one disadvantage of online learning, compared to stochastic gradient descent with a mini-batch size of, say, 20.

在Andrew Ng的machine learning课上，第二课就讲了这个问题。不过对方法的名字有点疑惑。Andrew Ng课上，online learning方法叫stochastic gradient descent (also incremental gradient descent)，而按参数循环迭代的方法叫batch gradient descent。可是本文stochastic/incremental指两个不一样的方法。。。。 抛开名字不说，在线的方法相对于离线方法来讲，不用必须遍历整个训练集，每循环一个样本，就迭代一次，所以一般在线方法比离线方法更快的收敛到最小值。可是，在先方法可能永远到不了绝对的最小值，会一直在最小值附近徘徊。

activations vector in component form

Write out Equation a′=σ(wa+b) in component form, and verify that it gives the same result as the rule 1/(1+exp⁡(−∑jwjxj−b)) for computing the output of a sigmoid neuron.

creating a network with just two layers

Try creating a network with just two layers - an input and an output layer, no hidden layer - with 784 and 10 neurons, respectively. Train the network using stochastic gradient descent. What classification accuracy can you achieve?

In [8]: net = Network.Network([784, 10])
In [9]: net.SGD(training_data, 10, 10, 3, test_data=test_data)  
Epoch 0: 1903 / 10000  
Epoch 1: 1903 / 10000    
Epoch 2: 1903 / 10000    
Epoch 3: 1903 / 10000
Epoch 4: 1903 / 10000
Epoch 5: 1903 / 10000
Epoch 6: 1903 / 10000
Epoch 7: 1903 / 10000
Epoch 8: 1903 / 10000
Epoch 9: 1903 / 10000

准确率真是至关地低啊。