cs224d 作业 problem set1 (一) 主要是实现word2vector模型,SGD,CBOW,Softmax,算法

'''
Created on 2017年9月13日

@author: weizhen
'''
import numpy as np
def sigmoid(x):
    return 1 / (1 + np.exp(-x))

首先上来的是最简单的sigmoid激励函数，

至于为什么选他做激励函数，

1、因为这个函数能将定义域为(-inf,+inf)的值映射到(0,1)区间，便于计算(能够消除量纲的影响)

2、这个函数的变化曲线不是特变陡峭，每一点处都可导

3、这个函数的导数为y(1-y)，即用他原来的函数值就可以求出其导数的值，便于计算

 

如果感觉写得还行请大家抢个沙发啥的，小弟不胜感激。

 

 

 

误差的反向传递求grad_xi

 

 

 

令

  

　　

 

令

 

所以

 

 

 

 

 

 

令

 

 

则 

 

To Summary：

 

Tip3:上面公式中 的值是由输出向量w_c与每一轮的变量值x相乘得来的,使用交叉熵函数作为损失函数主要是 为了恒量输出向量与正确分布的相似程度

那么每次在误差的反向传播过程中，

如何将梯度下降方法应用到变量x和变量w_c上呢？

需要首先求出目标函数对于x和变量w_c的导数grad_xi和grad_wi

然后指定相应的步长d(可以对于x和变量w_c应用相同的步长)

然后使用梯度下降公式，来对x和w_c的值进行更新

x_i+1=x_i+d*grad_xi

w_i+1=w_i+d*gradwi

在将得到的两个新的值代入到 交叉熵函数里边CrossEntropy(x_i+1,w_i+1)<Predefine Error 则停止迭代

否则继续迭代。

 

上面所有的公式推导都是在求grad_xi,并且在这过程中需要使用到交叉熵关于  x和变量w_c隐函数的导数，也就是交叉熵对softmax函数求导，这时候可以进行相应的化简

这里边有两种情况

 当f_y=f_c时  $\frac{\partial Entropy}{\partial z_{y}}=z_{y}-1$ 即如果是正对的节点的误差传播，对后一节点的导数，可以用当前前向传播传播过程计算出来的值$z_{y}-1$来计算gradz

 当f_y不等于f_c时  $\frac{\partial Entropy}{\partial z_{y}}=z_{y}$ 即如果不是正对的节点的误差传播，对后一节点的导数，可以用当前前向传播传播过程计算出来的值$z_{y}$来计算gradz

 

如果实在不明白可以参考以下图片,第一张图片是交叉熵函数对x变量求偏导，第二张图片是交叉熵函数对w变量求偏导

 

下面这张图片是交叉熵损失函数对变量w求偏导，这里将w=(w_c,w_y)

 

接下来是sigmoid函数的导数 

'''
Created on Sep 11, 2017

@author: p0079482
'''
def sigmoid_graf(f):
    """
        计算Sigmoid的梯度
    """
    f = f * (1 - f)
    
    return f

上面这个就没啥可说的了，就是求神经元的导数

下面是softmax函数，能够将[3,1,3]类似这种的输出结果通过exp(3)/[exp(3)+exp(1)+exp(3)]这种形式映射到(0,1)的概率区间，

便于下一步我们进行交叉熵计算

下面的代码是数值求微分的方法，来检验上一步中计算的导数值和真实值的区别是否足够小，即通过公式算出来的导数是否精确

'''
Created on Sep 11, 2017

@author: p0079482
'''
import random
import numpy as np
def gradcheck_naive(f,x):
    """ 对一个函数f求梯度检验
        -f 输入x,然后输出loss和梯度的函数
        -x 就是输入
     """
     
    rndstate = random.getstate() 
      #获取当前随机数的环境状态，以便下次再使用
    random.setstate(rndstate)
    fx,grad=f(x)
    h=1e-4
     
     #遍历x的每一维
     #单个数组的迭代
     #nditer将输入数组视为只读对象。要修改数组元素，必须指定读写（ read-write）或只写（write-only）模式
     
    it = np.nditer(x,flags=['multi_index'],op_flags=['readwrite'])
    while not it.finished:
         ix=it.multi_index
         old_val=x[ix]
         x[ix]=old_val-h
         random.setstate(rndstate)
         (fxh1,_)=f(x)
         
         x[ix]=old_val+h
         random.setstate(rndstate)
         (fxh2,_)=f(x)
         
         numgrad = (fxh2-fxh1)/(2*h)
         x[ix] = old_val
         
         #对比梯度
         reldiff = abs(numgrad-grad[ix])/max(1,abs(numgrad),abs(grad[ix]))
         if reldiff>1e-5:
             print("Gradient check failed.")
             print("First gradient error found at index %s"%str(ix))
             print("Your gradient: %f \t Numerical gradient: %f"%(grad[ix],numgrad))
             return
         it.iternext()  #Step to next dimension
    print("Gradient check passed!")
      
    
         
         
         

下面的代码是计算一个一层全连接网络，一层softmax层的程序。并且计算了梯度的反向传播。

'''
Created on Sep 11, 2017

@author: p0079482
'''

"""写出只有一个隐藏层其激活函数为sigmoid的神经网络前向和后向传播代码"""
import numpy as np
from q1_sigmoid import sigmoid
from q1_softmax import softmax
from q2_sigmoid import sigmoid_graf
def forward_backward_prop(data, labels, params, verbose=False):
    """2 个隐层的神经网络的前向运算和反向传播"""
    
    if len(data.shape) >= 2:
        (N, _) = data.shape
    
    dimensions = [1] * 3
    dimensions[0]=N
    dimensions[1]=20
    dimensions[2]=labels.shape
    
    
    # ## 展开每一层神经网络的参数
    t = 0
    W1 = np.reshape(params[t:t + dimensions[0] * dimensions[1]], (dimensions[0], dimensions[1]))
    t += dimensions[0] * dimensions[1]
    b1 = np.reshape(params[t:t + dimensions[1]], (1, dimensions[1]))
    t += dimensions[1]
    W2 = np.reshape(params[t:t + dimensions[1] * dimensions[2]], (dimensions[1], dimensions[2]))
    t += dimensions[1] * dimensions[2]
    b2 = np.reshape(params[t:t + dimensions[2]], (1, dimensions[2]))
    
    # ##前向运算
    
    # 第一个隐层做内积
    a1 = sigmoid(data.dot(W1) + b1)
    # 第二个隐层做内积
    a2 = softmax(a1.dot(W2) + b2)
    
    cost = -np.sum(np.log(a2[labels == 1])) / N
    
    # ## 反向传播
    
    # Calculate analytic gradient for the cross entropy function
    grad_a2 = (a2 - labels) / N
    
    # Backpropagate through the second latent layer
    gradW2 = np.dot(a1.T, grad_a2)
    gradb2 = np.sum(grad_a2, axis=0, keepdims=True)
    
    # Backpropagate through the first latent layer
    grad_a1 = np.dot(grad_a2, W2.T) * sigmoid_graf(a1)
    
    gradW1 = np.dot(data.T, grad_a1)
    gradb1 = np.sum(grad_a1, axis=0, keepdims=True)
    
    if verbose:  # Verbose mode for logging information
        print("W1 shape:{}".format(str(W1.shape)))
        print("W1 gradient shape: {}".format(str(gradW1.shape)))
        print("b1 shape: {}".format(str(b1.shape)))
        print("b1 gradient shape: {}".format(str(gradb1.shape)))
    
    # ## 梯度拼起来
    grad = np.concatenate((gradW1.flatten(), gradb1.flatten(), gradW2.flatten(), gradb2.flatten()))
    return cost, grad

    

 

下面的代码是随机梯度下降，SGD真的是和普通的梯度下降算法相比特别特别地块和有效来着

'''
Created on Sep 12, 2017

@author: p0079482

实现随机梯度下降
随机梯度下降每1000轮，就保存一下现在训练得到的参数
'''
SAVE_PARAMS_EVERY = 1000

import glob
import os.path as op
import pickle  as pickle
import sys
import random

def load_saved_params():
    """
        载入之前的参数以免从头开始训练
    """
    st = 0
    for f in glob.glob("saved_params_*.npy"):
        iter = int(op.splitext(op.basename(f))[0].split("_")[2])
        if(iter > st):
            st = iter
    
    if st > 0:
        with open("saved_params_%d.npy" % st, "rb") as f:
            params = pickle.load(f)
            state = pickle.load(f)
        return st, params, state
    else:
        return st, None, None
    
def save_params(iter, params):
    with open("saved_params_%d.npy" % iter, "wb") as f:
        pickle.dump(params, f)
        pickle.dump(random.getstate(), f)

def sgd(f, x0, step, iterations, postprocessing=None, useSaved=False, PRINT_EVERY=10, ANNEAL_EVERY=20000):
    """随机梯度下降
    输入:
    f:需要最优化的函数
    x0:SGD的初始值
    step:SGD的步长
    iterations：总的迭代次数
    postprocessing:参数后处理(比如word2vec里需要对词向量做归一化处理)
    PRINT_EVERY:指明多少次迭代以后输出一下状态
  输出:
    x:SGD完成后的输出参数
    """
    if useSaved:
        start_iter, oldx, state = load_saved_params()
        if start_iter > 0:
            x0 = oldx;
            step *= 0.5
        if state:
            random.setstate(state)
    else:
        start_iter = 0
    
    x = x0
    
    if not postprocessing:
        postprocessing = lambda x:x
    
    expcost = None
    
    for iter in range(start_iter + 1, iterations + 1):
        cost, grad = f(x)
        x = x - step * grad
        x = postprocessing(x)
        
        if iter % PRINT_EVERY == 0:
            print("iter#{},cost={}".format(iter, cost))
            sys.stdout.flush()
        
        if iter % SAVE_PARAMS_EVERY == 0 and useSaved:
            save_params(iter, x)
        
        if iter % ANNEAL_EVERY == 0:
            step *= 0.5
    return x

    

 

下面的代码是softmax函数的误差反向传播，

skip-gram模型的实现

'''
Created on Sep 11, 2017

@author: p0079482
'''
"""实现word2vec模型，并使用随机梯度下降方法SGD训练属于自己的词向量
      首先写一个辅助函数对矩阵中的每一行进行归一化，
      同样在这个文件中完成对softmax,
      负采样损失函数
      以及梯度计算函数的实现
      然后完成面向skip-gram的梯度损失函数
"""
import numpy as np
import random

from q1_softmax import softmax
from q2_gradcheck import gradcheck_naive
from q2_sigmoid import sigmoid_graf
from q1_sigmoid import sigmoid

def normalizeRows(x):
    """
       行归一化函数
    """
    N=x.shape[0]
    x/=np.sqrt(np.sum(x**2,axis=1)).reshape((N,1))+1e-30
    
    return x

def test_normalize_rows():
    print("Testing normalizeRows")
    x = normalizeRows(np.array([[3.0,4.0],[1,2]]))
    """结果应该是[[0.6,0.8],[0.4472136,0.89442719]]"""
    print(x)
    assert(np.amax(np.fabs(x-np.array([[0.6,0.8],[0.4472136,0.89442719]])))<= 1e-6)
    print(" ")

def softmaxCostAndGradient(predicted,target,outputVectors,dataset):
    """word2vec的Softmax损失函数"""
    
    """
       输入:
       predicted：预测词向量的numpy数组
       target：目标词的下标
       outputVectors：所有token的"output"向量(行形式)
       dataset：用来做负例采样的，这里其实没用着
    
        输出:
        cost:输出的互熵损失
        gradPred:the gradient with respect to the predicted word vector
        grad:    the gradient with respect to all the other word vectors
    """
    probabilities = softmax(predicted.dot(outputVectors.T))
    cost = -np.log(probabilities[target])
    delta = probabilities
    delta[target] -= 1
    N = delta.shape[0]
    D = predicted.shape[0]
    grad = delta.reshape((N,1))*predicted.reshape((1,D))
    gradPred = (delta.reshape((1,N)).dot(outputVectors)).flatten()
    
    return cost,gradPred,grad

def negSamplingCostAndGradient(predicted,target,outputVectors,dataset,K=10):
    """
        Word2vec模型负例采样后的损失函数和梯度
    """
    grad = np.zeros(outputVectors.shape)
    gradPred = np.zeros(predicted.shape)
    
    indices = [target]
    for k in range(K):
        newidx = dataset.sampleTokenIdx()
        while newidx == target:
            newidx = dataset.sampleTokenIdx()
        indices+=[newidx]
    
    labels = np.array([1]+[-1 for k in range(K)])
    vecs = outputVectors[indices,:]
    
    t=sigmoid(vecs.dot(predicted)*labels)
    cost = -np.sum(np.log(t))
    
    delta = labels*(t-1)
    gradPred = delta.reshape((1,K+1)).dot(vecs).flatten()
    gradtemp = delta.reshape((K+1,1)).dot(predicted.reshape((1,predicted.shape[0])))
    for k in range(K+1):
        grad[indices[k]]+=gradtemp[k,:]
    
    t = sigmoid(predicted.dot(outputVectors[target,:]))
    cost = -np.log(t)
    delta = t - 1
    
    gradPred+=delta*outputVectors[target,:]
    grad[target,:] += delta*predicted
    
    for k in range(K):
        idx = dataset.sampleTokenIdx()
        
        t = sigmoid(-predicted.dot(outputVectors[idx,:]))
        cost += -np.log(t)
        delta = 1-t
        
        gradPred += delta*outputVectors[idx,:]
        grad[idx,:]+=delta*predicted
    
    return cost,gradPred,grad

def skipgram(currentWord,C,contextWords,tokens,inputVectors,outputVectors,
             dataset,word2vecCostAndGradient=softmaxCostAndGradient):
    """
        Skip-gram model in word2vec
        
        skip-gram模型的实现
        输入:
        currentWord:当前中心词所对应的串
        C:上下文大小(词窗大小)
        contextWords:最多2*C个词
        tokens:对应词向量中词下标的字典
        inputVectors:"input" word vectors (as rows) for all tokens
        outputVectors: "output" word vectors (as rows) for all tokens
        word2vecCostAndGradient: the cost and gradient function for a prediction vector
                                 given the target word vectors,
                                 could be one of the two cost functions you implemented
                                 above
                                 
        输出:
        cost:skip-gram模型算得的损失值
        grad:词向量对应的梯度
    """
    currentI=tokens[currentWord]
    predicted = inputVectors[currentI,:]
    
    cost=0.0
    gradIn = np.zeros(inputVectors.shape)
    gradOut = np.zeros(outputVectors.shape)
    for cwd in contextWords:
        idx = tokens[cwd]
        cc,gp,gg = word2vecCostAndGradient(predicted,idx,outputVectors,dataset)
        cost+=cc
        gradOut += gg
        gradIn[currentI,:]+=gp
    return cost,gradIn,gradOut

def word2vec_sgd_wrapper(word2vecModel,tokens,wordVectors,dataset,C,word2vecCostAndGradient=softmaxCostAndGradient):
    batchsize = 50
    cost =0.0
    grad = np.zeros(wordVectors.shape)
    N=wordVectors.shape[0]
    splitIndex=int(N/2)
    inputVectors = wordVectors[:splitIndex,:]
    outputVectors = wordVectors[splitIndex:,:]
    for i in range(batchsize):
        C1 = random.randint(1,C)
        centerword,context=dataset.getRandomContext(C1)
        
        if word2vecModel == skipgram:
            denom =1
        else:
            denom =1
        
        c,gin,gout = word2vecModel(centerword,C1,context,tokens,inputVectors,outputVectors,dataset,word2vecCostAndGradient)
        cost+=c/batchsize/denom
        grad[:splitIndex,:]+=gin/batchsize/denom
        grad[splitIndex:,:]+=gout/batchsize/denom
     
    return cost, grad

def test_word2vec():
    """
         Interface to the dataset for negative sampling
    """
 
    dataset = type('dummy',(),{})()
    def dummySampleTokenIdx():
        return random.randint(0,4)
     
    def getRandomContext(C):
        tokens = ["a","b","c","d","e"]
        return tokens[random.randint(0,4)],[tokens[random.randint(0,4)] for i in range(2*C)]
     
    dataset.sampleTokenIdx = dummySampleTokenIdx
    dataset.getRandomContext = getRandomContext
     
    random.seed(31415)
    np.random.seed(9265)
    dummy_vectors = normalizeRows(np.random.randn(10,3))
    dummy_tokens = dict([("a",0),("b",1),("c",2),("d",3),("e",4)])
    print("===== Gradient check for skip-gram ======")
    #gradcheck_naive(lambda vec:word2vec_sgd_wrapper(skipgram,dummy_tokens,vec,dataset,5),dummy_vectors)
    gradcheck_naive(lambda vec:word2vec_sgd_wrapper(skipgram,dummy_tokens,vec,dataset,5,negSamplingCostAndGradient),dummy_vectors)
    print("\n===== Gradient check for CBOW")
    #gradcheck_naive(lambda vec:word2vec_sgd_wrapper(cbow,dummy_tokens,vec,dataset,5),dummy_vectors)
    #gradcheck_naive(lambda vec:word2vec_sgd_wrapper(cbow,dummy_tokens,vec,dataset,5,negSamplingCostAndGradient),dummy_vectors)
     
    print("\n===== Results ===")
    print(skipgram("c",3,["a","b","e","d","b","c"],dummy_tokens,dummy_vectors[:5,:],dummy_vectors[5:,:],dataset))
    #print(skipgram("c",1,["a","b"],dummy_tokens,dummy_vectors[:5,:],dummy_vectors[5:,:],dataset,negSamplingCostAndGradient))
    #print(cbow("a",2,["a","b","c","a"],dummy_tokens,dummy_vectors[:5,:],dummy_vectors[5:,:],dataset))
    #print(cbow("a",2,["a","b","a","c"],dummy_tokens,dummy_vectors[:5,:],dummy_vectors[5:,:],dataset,negSamplingCostAndGradient))
     
if __name__=="__main__":
    test_normalize_rows()
    test_word2vec()
    
        

 

 

下面的模型是word2vector模型的训练，真正的纯手写版的，没有利用任何包。

(看来我还是默默地再天朝搬砖吧，感觉一旦进了斯坦福都毕业不了)

'''
Created on 2017年9月17日

@author: weizhen
'''
import random
import numpy as np
from q3_word2vec import skipgram,word2vec_sgd_wrapper,negSamplingCostAndGradient
from q3_sgd import sgd,load_saved_params
from data_utils import StanfordSentiment
# Reset the random seed to make sure that everyone gets the same results
dataset = StanfordSentiment()
tokens = dataset.tokens()
nWords = len(tokens)

#going to train 10-dimensional vectors for this assignment
dimVectors = 10

#Context size
C=5
# Train word vectors (this could take a while!)

# Reset the random seed to make sure that everyone gets the same results
random.seed(31415)
np.random.seed(9265)
wordVectors = np.concatenate(((np.random.rand(nWords, dimVectors) - .5) / dimVectors, 
                              np.zeros((nWords, dimVectors))), axis=0)
wordVectors0 = sgd(lambda vec: word2vec_sgd_wrapper(skipgram, tokens, vec, dataset, C, negSamplingCostAndGradient), 
                   wordVectors, 0.3, 40000, None, True, PRINT_EVERY=10)
# sanity check: cost at convergence should be around or below 10

# sum the input and output word vectors

#st, wordVectors0, state2 = load_saved_params();
wordVectors = (wordVectors0[:int(nWords),:] + wordVectors0[int(nWords):,:])

print("\n=== For autograder ===")
checkWords = ["the", "a", "an", "movie", "ordinary", "but", "and"]
checkIdx = [tokens[word] for word in checkWords]
checkVecs = wordVectors[checkIdx, :]
print(checkVecs)

输出的log如下所示

iter#10,cost=20.43394753392233
iter#20,cost=21.958872793932695
iter#30,cost=21.19652773138706
iter#40,cost=20.434104390006
iter#50,cost=22.568897721749693
iter#60,cost=23.7887821643947
iter#70,cost=23.788746846678066
iter#80,cost=20.43372965152089
iter#90,cost=21.04397549635104
iter#100,cost=24.093923168600938
iter#110,cost=23.941407071171348
iter#120,cost=22.263861158139957
iter#130,cost=21.34917296135529
iter#140,cost=23.789000157131245
iter#150,cost=20.58633854560291
iter#160,cost=21.04411292794054
iter#170,cost=21.348785582867343
iter#180,cost=20.128956212458444
iter#190,cost=17.68927651078614
iter#200,cost=24.551281877202594
iter#210,cost=19.365943158625402
iter#220,cost=23.48358084824119
iter#230,cost=20.739473371395963
iter#240,cost=23.331734839583465
iter#250,cost=21.349294746785546
iter#260,cost=22.569291250356102
iter#270,cost=22.111001535622552
iter#280,cost=23.483817878664986
iter#290,cost=22.41604937462105
iter#300,cost=20.43404997423582
iter#310,cost=20.738462084119163
iter#320,cost=21.65431510819316
iter#330,cost=20.433424632590597
iter#340,cost=20.586194398205283
iter#350,cost=21.043497293197134
iter#360,cost=20.434210753312154
iter#370,cost=22.1115615477318
iter#380,cost=19.823479540802577
iter#390,cost=22.26238128164097
iter#400,cost=17.3851857415029
iter#410,cost=23.02602502114436
iter#420,cost=20.280452941633577
iter#430,cost=19.365398822916518
iter#440,cost=21.19539162111903
iter#450,cost=18.756499819557227
iter#460,cost=18.298244927755217
iter#470,cost=21.042527034737663
iter#480,cost=21.043436272476
iter#490,cost=23.78649975282614
iter#500,cost=21.65376695903663
iter#510,cost=23.02365296377378
iter#520,cost=20.585225334304308
iter#530,cost=21.805357129741573
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纯手写版的，真的是山外有山，人外有人呀。

既然tensorflow一行代码就行实现，那么为啥还要，自己写呢？

不为啥，就为了给自己装逼用。哈哈哈