真正掌握一种算法，最实际的方法，完全手写出来。

LSTM（Long Short Tem Memory）特殊递归神经网络，神经元保存历史记忆，解决自然语言处理统计方法只能考虑最近n个词语而忽略更久前词语的问题。用途：word representation（embedding）(词语向量)、sequence to sequence learning（输入句子预测句子）、机器翻译、语音识别等。

100多行原始python代码实现基于LSTM二进制加法器。https://iamtrask.github.io/2015/11/15/anyone-can-code-lstm/ ，翻译http://blog.csdn.net/zzukun/article/details/49968129 ：

import copy, numpy as np
np.random.seed(0)

最开始引入numpy库，矩阵操作。

def sigmoid(x):
    output = 1/(1+np.exp(-x))
    return output

声明sigmoid激活函数，神经网络基础内容，常用激活函数sigmoid、tan、relu等，sigmoid取值范围[0, 1]，tan取值范围[-1,1]，x是向量，返回output是向量。

def sigmoid_output_to_derivative(output):
    return output*(1-output)

声明sigmoid求导函数。
加法器思路：二进制加法是二进制位相加，记录满二进一进位，训练时随机c=a+b样本，输入a、b输出c是整个lstm预测过程，训练由a、b二进制向c各种转换矩阵和权重，神经网络。

int2binary = {}

声明词典，由整型数字转成二进制，存起来不用随时计算，提前存好读取更快。

binary_dim = 8

largest_number = pow(2,binary_dim)
声明二进制数字维度，8，二进制能表达最大整数2^8=256，largest_number。

binary = np.unpackbits(
                       np.array([range(largest_number)],dtype=np.uint8).T,axis=1)
for i in range(largest_number):
    int2binary[i] = binary[i]

预先把整数到二进制转换词典存起来。

alpha = 0.1
input_dim = 2
hidden_dim = 16
output_dim = 1

设置参数，alpha是学习速度，input_dim是输入层向量维度，输入a、b两个数，是2，hidden_dim是隐藏层向量维度，隐藏层神经元个数，output_dim是输出层向量维度，输出一个c，是1维。从输入层到隐藏层权重矩阵是216维，从隐藏层到输出层权重矩阵是161维，隐藏层到隐藏层权重矩阵是16*16维：

synapse_0 = 2*np.random.random((input_dim,hidden_dim)) - 1
synapse_1 = 2*np.random.random((hidden_dim,output_dim)) - 1
synapse_h = 2*np.random.random((hidden_dim,hidden_dim)) - 1

2x-1，np.random.random生成从0到1之间随机浮点数，2x-1使其取值范围在[-1, 1]。

synapse_0_update = np.zeros_like(synapse_0)
synapse_1_update = np.zeros_like(synapse_1)
synapse_h_update = np.zeros_like(synapse_h)

声明三个矩阵更新，Delta。

for j in range(10000):

进行10000次迭代。

a_int = np.random.randint(largest_number/2)
a = int2binary[a_int]
b_int = np.random.randint(largest_number/2)
b = int2binary[b_int]
c_int = a_int + b_int
c = int2binary[c_int]

随机生成样本，包含二进制a、b、c，c=a+b，a_int、b_int、c_int分别是a、b、c对应整数格式。

d = np.zeros_like(c)

d存模型对c预测值。

overallError = 0

全局误差，观察模型效果。
layer_2_deltas = list()
存储第二层(输出层)残差，输出层残差计算公式推导公式http://deeplearning.stanford.edu/wiki/index.php/%E5%8F%8D%E5%90%91%E4%BC%A0%E5%AF%BC%E7%AE%97%E6%B3%95 。

layer_1_values = list()
layer_1_values.append(np.zeros(hidden_dim))

存储第一层(隐藏层)输出值，赋0值作为上一个时间值。

for position in range(binary_dim):

遍历二进制每一位。

X = np.array([[a[binary_dim - position - 1],b[binary_dim - position - 1]]])
y = np.array([[c[binary_dim - position - 1]]]).T

X和y分别是样本输入和输出二进制值第position位，X对于每个样本有两个值，分别是a和b对应第position位。把样本拆成每个二进制位用于训练，二进制加法存在进位标记正好适合利用LSTM长短期记忆训练，每个样本8个二进制位是一个时间序列。

layer_1 = sigmoid(np.dot(X,synapse_0) + np.dot(layer_1_values[-1],synapse_h))

公式Ct = sigma(W0·Xt + Wh·Ct-1)

layer_2 = sigmoid(np.dot(layer_1,synapse_1))

这里使用的公式是C2 = sigma(W1·C1)，

layer_2_error = y - layer_2

计算预测值和真实值误差。

layer_2_deltas.append((layer_2_error)*sigmoid_output_to_derivative(layer_2))

反向传导，计算delta，添加到数组layer_2_deltas

overallError += np.abs(layer_2_error[0])

计算累加总误差，用于展示和观察。

d[binary_dim - position - 1] = np.round(layer_2[0][0])

存储预测position位输出值。

layer_1_values.append(copy.deepcopy(layer_1))

存储中间过程生成隐藏层值。

future_layer_1_delta = np.zeros(hidden_dim)

存储下一个时间周期隐藏层历史记忆值，先赋一个空值。

for position in range(binary_dim):

遍历二进制每一位。

X = np.array([[a[position],b[position]]])

取出X值，从大位开始更新，反向传导按时序逆着一级一级更新。

layer_1 = layer_1_values[-position-1]

取出位对应隐藏层输出。

prev_layer_1 = layer_1_values[-position-2]

取出位对应隐藏层上一时序输出。

layer_2_delta = layer_2_deltas[-position-1]

取出位对应输出层delta。

layer_1_delta = (future_layer_1_delta.dot(synapse_h.T) + layer_2_delta.dot(synapse_1.T)) * sigmoid_output_to_derivative(layer_1)

神经网络反向传导公式，加上隐藏层?值。

synapse_1_update += np.atleast_2d(layer_1).T.dot(layer_2_delta)

累加权重矩阵更新，对权重(权重矩阵)偏导等于本层输出与下一层delta点乘。

synapse_h_update += np.atleast_2d(prev_layer_1).T.dot(layer_1_delta)

前一时序隐藏层权重矩阵更新，前一时序隐藏层输出与本时序delta点乘。

synapse_0_update += X.T.dot(layer_1_delta)

输入层权重矩阵更新。

future_layer_1_delta = layer_1_delta

记录本时序隐藏层delta。

synapse_0 += synapse_0_update * alpha
synapse_1 += synapse_1_update * alpha
synapse_h += synapse_h_update * alpha

权重矩阵更新。

synapse_0_update *= 0
synapse_1_update *= 0
synapse_h_update *= 0

更新变量归零。

if(j % 1000 == 0):
        print "Error:" + str(overallError)
        print "Pred:" + str(d)
        print "True:" + str(c)
        out = 0
        for index,x in enumerate(reversed(d)):
            out += x*pow(2,index)
        print str(a_int) + " + " + str(b_int) + " = " + str(out)
        print "------------"

每训练1000个样本输出总误差信息，运行时看收敛过程。
LSTM最简单实现，没有考虑偏置变量，只有两个神经元。

完整LSTM python实现。完全参照论文great intro paper实现,代码来源https://github.com/nicodjimenez/lstm ，作者解释http://nicodjimenez.github.io/2014/08/08/lstm.html ，具体过程参考http://colah.github.io/posts/2015-08-Understanding-LSTMs/ 图。

import random
import numpy as np
import math

def sigmoid(x):
    return 1. / (1 + np.exp(-x))

声明sigmoid函数。

def rand_arr(a, b, *args):
    np.random.seed(0)
    return np.random.rand(*args) * (b - a) + a

生成随机矩阵，取值范围[a,b)，shape用args指定。

class LstmParam:
    def __init__(self, mem_cell_ct, x_dim):
        self.mem_cell_ct = mem_cell_ct
        self.x_dim = x_dim
        concat_len = x_dim + mem_cell_ct
        # weight matrices
        self.wg = rand_arr(-0.1, 0.1, mem_cell_ct, concat_len)
        self.wi = rand_arr(-0.1, 0.1, mem_cell_ct, concat_len)
        self.wf = rand_arr(-0.1, 0.1, mem_cell_ct, concat_len)
        self.wo = rand_arr(-0.1, 0.1, mem_cell_ct, concat_len)
        # bias terms
        self.bg = rand_arr(-0.1, 0.1, mem_cell_ct)
        self.bi = rand_arr(-0.1, 0.1, mem_cell_ct)
        self.bf = rand_arr(-0.1, 0.1, mem_cell_ct)
        self.bo = rand_arr(-0.1, 0.1, mem_cell_ct)
        # diffs (derivative of loss function w.r.t. all parameters)
        self.wg_diff = np.zeros((mem_cell_ct, concat_len))
        self.wi_diff = np.zeros((mem_cell_ct, concat_len))
        self.wf_diff = np.zeros((mem_cell_ct, concat_len))
        self.wo_diff = np.zeros((mem_cell_ct, concat_len))
        self.bg_diff = np.zeros(mem_cell_ct)
        self.bi_diff = np.zeros(mem_cell_ct)
        self.bf_diff = np.zeros(mem_cell_ct)
        self.bo_diff = np.zeros(mem_cell_ct)

LstmParam类传递参数，mem_cell_ct是lstm神经元数目，x_dim是输入数据维度，concat_len是mem_cell_ct与x_dim长度和，wg是输入节点权重矩阵，wi是输入门权重矩阵，wf是忘记门权重矩阵，wo是输出门权重矩阵，bg、bi、bf、bo分别是输入节点、输入门、忘记门、输出门偏置，wg_diff、wi_diff、wf_diff、wo_diff分别是输入节点、输入门、忘记门、输出门权重损失，bg_diff、bi_diff、bf_diff、bo_diff分别是输入节点、输入门、忘记门、输出门偏置损失，初始化按照矩阵维度初始化，损失矩阵归零。

    def apply_diff(self, lr = 1):
        self.wg -= lr * self.wg_diff
        self.wi -= lr * self.wi_diff
        self.wf -= lr * self.wf_diff
        self.wo -= lr * self.wo_diff
        self.bg -= lr * self.bg_diff
        self.bi -= lr * self.bi_diff
        self.bf -= lr * self.bf_diff
        self.bo -= lr * self.bo_diff
        # reset diffs to zero
        self.wg_diff = np.zeros_like(self.wg)
        self.wi_diff = np.zeros_like(self.wi)
        self.wf_diff = np.zeros_like(self.wf)
        self.wo_diff = np.zeros_like(self.wo)
        self.bg_diff = np.zeros_like(self.bg)
        self.bi_diff = np.zeros_like(self.bi)
        self.bf_diff = np.zeros_like(self.bf)
        self.bo_diff = np.zeros_like(self.bo)

定义权重更新过程，先减损失，再把损失矩阵归零。

class LstmState:
    def __init__(self, mem_cell_ct, x_dim):
        self.g = np.zeros(mem_cell_ct)
        self.i = np.zeros(mem_cell_ct)
        self.f = np.zeros(mem_cell_ct)
        self.o = np.zeros(mem_cell_ct)
        self.s = np.zeros(mem_cell_ct)
        self.h = np.zeros(mem_cell_ct)
        self.bottom_diff_h = np.zeros_like(self.h)
        self.bottom_diff_s = np.zeros_like(self.s)
        self.bottom_diff_x = np.zeros(x_dim)

LstmState存储LSTM神经元状态，包括g、i、f、o、s、h，s是内部状态矩阵(记忆)，h是隐藏层神经元输出矩阵。

class LstmNode:
    def __init__(self, lstm_param, lstm_state):
        # store reference to parameters and to activations
        self.state = lstm_state
        self.param = lstm_param
        # non-recurrent input to node
        self.x = None
        # non-recurrent input concatenated with recurrent input
        self.xc = None

LstmNode对应样本输入，x是输入样本x，xc是用hstack把x和递归输入节点拼接矩阵（hstack是横拼矩阵，vstack是纵拼矩阵）。

    def bottom_data_is(self, x, s_prev = None, h_prev = None):
        # if this is the first lstm node in the network
        if s_prev == None: s_prev = np.zeros_like(self.state.s)
        if h_prev == None: h_prev = np.zeros_like(self.state.h)
        # save data for use in backprop
        self.s_prev = s_prev
        self.h_prev = h_prev

        # concatenate x(t) and h(t-1)
        xc = np.hstack((x,  h_prev))
        self.state.g = np.tanh(np.dot(self.param.wg, xc) + self.param.bg)
        self.state.i = sigmoid(np.dot(self.param.wi, xc) + self.param.bi)
        self.state.f = sigmoid(np.dot(self.param.wf, xc) + self.param.bf)
        self.state.o = sigmoid(np.dot(self.param.wo, xc) + self.param.bo)
        self.state.s = self.state.g * self.state.i + s_prev * self.state.f
        self.state.h = self.state.s * self.state.o
        self.x = x
        self.xc = xc

bottom和top是两个方向，输入样本从底部输入，反向传导从顶部向底部传导，bottom_data_is是输入样本过程，把x和先前输入拼接成矩阵，用公式wx+b分别计算g、i、f、o值，激活函数tanh和sigmoid。
每个时序神经网络有四个神经网络层(激活函数)，最左边忘记门，直接生效到记忆C，第二个输入门，依赖输入样本数据，按照一定“比例”影响记忆C，“比例”通过第三个层(tanh)实现，取值范围是[-1,1]可以正向影响也可以负向影响，最后一个输出门，每一时序产生输出既依赖输入样本x和上一时序输出，还依赖记忆C，设计模仿生物神经元记忆功能。

    def top_diff_is(self, top_diff_h, top_diff_s):
        # notice that top_diff_s is carried along the constant error carousel
        ds = self.state.o * top_diff_h + top_diff_s
        do = self.state.s * top_diff_h
        di = self.state.g * ds
        dg = self.state.i * ds
        df = self.s_prev * ds

        # diffs w.r.t. vector inside sigma / tanh function
        di_input = (1. - self.state.i) * self.state.i * di
        df_input = (1. - self.state.f) * self.state.f * df
        do_input = (1. - self.state.o) * self.state.o * do
        dg_input = (1. - self.state.g ** 2) * dg

        # diffs w.r.t. inputs
        self.param.wi_diff += np.outer(di_input, self.xc)
        self.param.wf_diff += np.outer(df_input, self.xc)
        self.param.wo_diff += np.outer(do_input, self.xc)
        self.param.wg_diff += np.outer(dg_input, self.xc)
        self.param.bi_diff += di_input
        self.param.bf_diff += df_input
        self.param.bo_diff += do_input
        self.param.bg_diff += dg_input

        # compute bottom diff
        dxc = np.zeros_like(self.xc)
        dxc += np.dot(self.param.wi.T, di_input)
        dxc += np.dot(self.param.wf.T, df_input)
        dxc += np.dot(self.param.wo.T, do_input)
        dxc += np.dot(self.param.wg.T, dg_input)

        # save bottom diffs
        self.state.bottom_diff_s = ds * self.state.f
        self.state.bottom_diff_x = dxc[:self.param.x_dim]
        self.state.bottom_diff_h = dxc[self.param.x_dim:]

反向传导，整个训练过程核心。假设在t时刻lstm输出预测值h(t)，实际输出值是y(t)，之间差别是损失，假设损失函数为l(t) = f(h(t), y(t)) = ||h(t) - y(t)||^2，欧式距离，整体损失函数是L(t) = ∑l(t)，t从1到T，T表示整个事件序列最大长度。最终目标是用梯度下降法让L(t)最小化，找到一个最优权重w使得L(t)最小，当w发生微小变化L(t)不再变化，达到局部最优，即L对w偏导梯度为0。
dL/dw表示当w发生单位变化L变化多少，dh(t)/dw表示当w发生单位变化h(t)变化多少，dL/dh(t)表示当h(t)发生单位变化时L变化多少，(dL/dh(t)) * (dh(t)/dw)表示第t时序第i个记忆单元w发生单位变化L变化多少，把所有由1到M的i和所有由1到T的t累加是整体dL/dw。

第i个记忆单元，h(t)发生单位变化，整个从1到T时序所有局部损失l的累加和，是dL/dh(t)，h(t)只影响从t到T时序局部损失l。

假设L(t)表示从t到T损失和，L(t) = ∑l(s)。

h(t)对w导数。

L(t) = l(t) + L(t+1)，dL(t)/dh(t) = dl(t)/dh(t) + dL(t+1)/dh(t)，用下一时序导数得出当前时序导数，规律推导，计算T时刻导数往前推，在T时刻，dL(T)/dh(T) = dl(T)/dh(T)。

class LstmNetwork():
    def __init__(self, lstm_param):
        self.lstm_param = lstm_param
        self.lstm_node_list = []
        # input sequence
        self.x_list = []

    def y_list_is(self, y_list, loss_layer):
        """
        Updates diffs by setting target sequence
        with corresponding loss layer.
        Will *NOT* update parameters.  To update parameters,
        call self.lstm_param.apply_diff()
        """
        assert len(y_list) == len(self.x_list)
        idx = len(self.x_list) - 1
        # first node only gets diffs from label ...
        loss = loss_layer.loss(self.lstm_node_list[idx].state.h, y_list[idx])
        diff_h = loss_layer.bottom_diff(self.lstm_node_list[idx].state.h, y_list[idx])
        # here s is not affecting loss due to h(t+1), hence we set equal to zero
        diff_s = np.zeros(self.lstm_param.mem_cell_ct)
        self.lstm_node_list[idx].top_diff_is(diff_h, diff_s)
        idx -= 1

        ### ... following nodes also get diffs from next nodes, hence we add diffs to diff_h
        ### we also propagate error along constant error carousel using diff_s
        while idx >= 0:
            loss += loss_layer.loss(self.lstm_node_list[idx].state.h, y_list[idx])
            diff_h = loss_layer.bottom_diff(self.lstm_node_list[idx].state.h, y_list[idx])
            diff_h += self.lstm_node_list[idx + 1].state.bottom_diff_h
            diff_s = self.lstm_node_list[idx + 1].state.bottom_diff_s
            self.lstm_node_list[idx].top_diff_is(diff_h, diff_s)
            idx -= 1

        return loss

diff_h(预测结果误差发生单位变化损失L多少，dL(t)/dh(t)数值计算)，由idx从T往前遍历到1，计算loss_layer.bottom_diff和下一个时序bottom_diff_h和作为diff_h(第一次遍历即T不加bottom_diff_h)。
loss_layer.bottom_diff：

    def bottom_diff(self, pred, label):
        diff = np.zeros_like(pred)
        diff[0] = 2 * (pred[0] - label)
        return diff

l(t) = f(h(t), y(t)) = ||h(t) - y(t)||^2导数l'(t) = 2 * (h(t) - y(t))
。当s(t)发生变化，L(t)变化来源s(t)影响h(t)和h(t+1)，影响L(t)。
h(t+1)不会影响l(t)。
左边式子(dL(t)/dh(t)) * (dh(t)/ds(t))，由t+1到t来逐级反推dL(t)/ds(t)。
神经元self.state.h = self.state.s self.state.o，h(t) = s(t) o(t)，dh(t)/ds(t) = o(t)，dL(t)/dh(t)是top_diff_h。

top_diff_is，Bottom means input to the layer, top means output of the layer. Caffe also uses this terminology. bottom表示神经网络层输入，top表示神经网络层输出，和caffe概念一致。
def top_diff_is(self, top_diff_h, top_diff_s):
top_diff_h表示当前t时序dL(t)/dh(t), top_diff_s表示t+1时序记忆单元dL(t)/ds(t)。

        ds = self.state.o * top_diff_h + top_diff_s
        do = self.state.s * top_diff_h
        di = self.state.g * ds
        dg = self.state.i * ds
        df = self.s_prev * ds

前缀d表达误差L对某一项导数(directive)。
ds是在根据公式dL(t)/ds(t)计算当前t时序dL(t)/ds(t)。
do是计算dL(t)/do(t)，h(t) = s(t) o(t)，dh(t)/do(t) = s(t)，dL(t)/do(t) = (dL(t)/dh(t)) (dh(t)/do(t)) = top_diff_h * s(t)。
di是计算dL(t)/di(t)。s(t) = f(t) s(t-1) + i(t) g(t)。dL(t)/di(t) = (dL(t)/ds(t)) (ds(t)/di(t)) = ds g(t)。
dg是计算dL(t)/dg(t)，dL(t)/dg(t) = (dL(t)/ds(t)) (ds(t)/dg(t)) = ds i(t)。
df是计算dL(t)/df(t)，dL(t)/df(t) = (dL(t)/ds(t)) (ds(t)/df(t)) = ds s(t-1)。

        di_input = (1. - self.state.i) * self.state.i * di
        df_input = (1. - self.state.f) * self.state.f * df
        do_input = (1. - self.state.o) * self.state.o * do
        dg_input = (1. - self.state.g ** 2) * dg

sigmoid函数导数，tanh函数导数。di_input，(1. - self.state.i) * self.state.i，sigmoid导数，当i神经元输入发生单位变化时输出值有多大变化，再乘di表示当i神经元输入发生单位变化时误差L(t)发生多大变化，dL(t)/d i_input(t)。

        self.param.wi_diff += np.outer(di_input, self.xc)
        self.param.wf_diff += np.outer(df_input, self.xc)
        self.param.wo_diff += np.outer(do_input, self.xc)
        self.param.wg_diff += np.outer(dg_input, self.xc)
        self.param.bi_diff += di_input
        self.param.bf_diff += df_input
        self.param.bo_diff += do_input
        self.param.bg_diff += dg_input

w_diff是权重矩阵误差，b_diff是偏置误差，用于更新。

        dxc = np.zeros_like(self.xc)
        dxc += np.dot(self.param.wi.T, di_input)
        dxc += np.dot(self.param.wf.T, df_input)
        dxc += np.dot(self.param.wo.T, do_input)
        dxc += np.dot(self.param.wg.T, dg_input)

累加输入xdiff，x在四处起作用，四处diff加和后作xdiff。

        self.state.bottom_diff_s = ds * self.state.f
        self.state.bottom_diff_x = dxc[:self.param.x_dim]
        self.state.bottom_diff_h = dxc[self.param.x_dim:]

bottom_diff_s是在t-1时序上s变化和t时序上s变化时f倍关系。dxc是x和h横向合并矩阵，分别取两部分diff信息bottom_diff_x和bottom_diff_h。

def x_list_clear(self):
        self.x_list = []

    def x_list_add(self, x):
        self.x_list.append(x)
        if len(self.x_list) > len(self.lstm_node_list):
            # need to add new lstm node, create new state mem
            lstm_state = LstmState(self.lstm_param.mem_cell_ct, self.lstm_param.x_dim)
            self.lstm_node_list.append(LstmNode(self.lstm_param, lstm_state))

        # get index of most recent x input
        idx = len(self.x_list) - 1
        if idx == 0:
            # no recurrent inputs yet
            self.lstm_node_list[idx].bottom_data_is(x)
        else:
            s_prev = self.lstm_node_list[idx - 1].state.s
            h_prev = self.lstm_node_list[idx - 1].state.h
            self.lstm_node_list[idx].bottom_data_is(x, s_prev, h_prev)

添加训练样本，输入x数据。

def example_0():
    # learns to repeat simple sequence from random inputs
    np.random.seed(0)

    # parameters for input data dimension and lstm cell count
    mem_cell_ct = 100
    x_dim = 50
    concat_len = x_dim + mem_cell_ct
    lstm_param = LstmParam(mem_cell_ct, x_dim)
    lstm_net = LstmNetwork(lstm_param)
    y_list = [-0.5,0.2,0.1, -0.5]
    input_val_arr = [np.random.random(x_dim) for _ in y_list]

    for cur_iter in range(100):
        print "cur iter: ", cur_iter
        for ind in range(len(y_list)):
            lstm_net.x_list_add(input_val_arr[ind])
            print "y_pred[%d] : %f" % (ind, lstm_net.lstm_node_list[ind].state.h[0])

        loss = lstm_net.y_list_is(y_list, ToyLossLayer)
        print "loss: ", loss
        lstm_param.apply_diff(lr=0.1)
        lstm_net.x_list_clear()

初始化LstmParam，指定记忆存储单元数为100，指定输入样本x维度是50。初始化LstmNetwork训练模型，生成4组各50个随机数，分别以[-0.5,0.2,0.1, -0.5]作为y值训练，每次喂50个随机数和一个y值，迭代100次。
lstm输入一串连续质数预估下一个质数。小测试，生成100以内质数，循环拿出50个质数序列作x，第51个质数作y，拿出10个样本参与训练1w次，均方误差由0.17973最终达到了1.05172e-06，几乎完全正确：

import numpy as np
import sys

from lstm import LstmParam, LstmNetwork

class ToyLossLayer:
    """
    Computes square loss with first element of hidden layer array.
    """
    @classmethod
    def loss(self, pred, label):
        return (pred[0] - label) ** 2

    @classmethod
    def bottom_diff(self, pred, label):
        diff = np.zeros_like(pred)
        diff[0] = 2 * (pred[0] - label)
        return diff

class Primes:
    def __init__(self):
        self.primes = list()
        for i in range(2, 100):
            is_prime = True
            for j in range(2, i-1):
                if i % j == 0:
                    is_prime = False
            if is_prime:
                self.primes.append(i)
        self.primes_count = len(self.primes)
    def get_sample(self, x_dim, y_dim, index):
        result = np.zeros((x_dim+y_dim))
        for i in range(index, index + x_dim + y_dim):
            result[i-index] = self.primes[i%self.primes_count]/100.0
        return result

def example_0():
    mem_cell_ct = 100
    x_dim = 50
    concat_len = x_dim + mem_cell_ct
    lstm_param = LstmParam(mem_cell_ct, x_dim)
    lstm_net = LstmNetwork(lstm_param)

    primes = Primes()
    x_list = []
    y_list = []
    for i in range(0, 10):
        sample = primes.get_sample(x_dim, 1, i)
        x = sample[0:x_dim]
        y = sample[x_dim:x_dim+1].tolist()[0]
        x_list.append(x)
        y_list.append(y)

    for cur_iter in range(10000):
        if cur_iter % 1000 == 0:
            print "y_list=", y_list
        for ind in range(len(y_list)):
            lstm_net.x_list_add(x_list[ind])
            if cur_iter % 1000 == 0:
                print "y_pred[%d] : %f" % (ind, lstm_net.lstm_node_list[ind].state.h[0])

        loss = lstm_net.y_list_is(y_list, ToyLossLayer)
        if cur_iter % 1000 == 0:
            print "loss: ", loss
        lstm_param.apply_diff(lr=0.01)
        lstm_net.x_list_clear()

if __name__ == "__main__":
    example_0()

质数列表全都除以100，这个代码训练数据必须是小于1数值。

torch是深度学习框架。1）tensorflow，谷歌主推，时下最火，小型试验和大型计算都可以，基于python，缺点是上手相对较难，速度一般；2）torch，facebook主推，用于小型试验，开源应用较多，基于lua，上手较快，网上文档较全，缺点是lua语言相对冷门；3）mxnet，Amazon主推，主要用于大型计算，基于python和R，缺点是网上开源项目较少；4）caffe，facebook主推，用于大型计算，基于c++、python，缺点是开发不是很方便；5）theano，速度一般，基于python，评价很好。

torch github上lstm实现项目比较多。

在mac上安装torch。https://github.com/torch/torch7/wiki/Cheatsheet#installing-and-running-torch 。

git clone https://github.com/torch/distro.git ~/torch --recursive
cd ~/torch; bash install-deps;
./install.sh

qt安装不成功问题，自己单独安装。

brew install cartr/qt4/qt

安装后需要手工加到~/.bash_profile中。

. ~/torch/install/bin/torch-activate

source ~/.bash_profile后执行th使用torch。
安装itorch，安装依赖

brew install zeromq
brew install openssl
luarocks install luacrypto OPENSSL_DIR=/usr/local/opt/openssl/

git clone https://github.com/facebook/iTorch.git
cd iTorch
luarocks make 

用卷积神经网络实现图像识别。
创建pattern_recognition.lua：

require 'nn'
require 'paths'
if (not paths.filep("cifar10torchsmall.zip")) then
    os.execute('wget -c https://s3.amazonaws.com/torch7/data/cifar10torchsmall.zip')
    os.execute('unzip cifar10torchsmall.zip')
end
trainset = torch.load('cifar10-train.t7')
testset = torch.load('cifar10-test.t7')
classes = {'airplane', 'automobile', 'bird', 'cat',
'deer', 'dog', 'frog', 'horse', 'ship', 'truck'}
setmetatable(trainset,
{__index = function(t, i)
    return {t.data[i], t.label[i]}
end}
);
trainset.data = trainset.data:double() -- convert the data from a ByteTensor to a DoubleTensor.

function trainset:size()
    return self.data:size(1)
end
mean = {} -- store the mean, to normalize the test set in the future
stdv  = {} -- store the standard-deviation for the future
for i=1,3 do -- over each image channel
    mean[i] = trainset.data[{ {}, {i}, {}, {}  }]:mean() -- mean estimation
    print('Channel ' .. i .. ', Mean: ' .. mean[i])
    trainset.data[{ {}, {i}, {}, {}  }]:add(-mean[i]) -- mean subtraction

    stdv[i] = trainset.data[{ {}, {i}, {}, {}  }]:std() -- std estimation
    print('Channel ' .. i .. ', Standard Deviation: ' .. stdv[i])
    trainset.data[{ {}, {i}, {}, {}  }]:div(stdv[i]) -- std scaling
end
net = nn.Sequential()
net:add(nn.SpatialConvolution(3, 6, 5, 5)) -- 3 input image channels, 6 output channels, 5x5 convolution kernel
net:add(nn.ReLU())                       -- non-linearity
net:add(nn.SpatialMaxPooling(2,2,2,2))     -- A max-pooling operation that looks at 2x2 windows and finds the max.
net:add(nn.SpatialConvolution(6, 16, 5, 5))
net:add(nn.ReLU())                       -- non-linearity
net:add(nn.SpatialMaxPooling(2,2,2,2))
net:add(nn.View(16*5*5))                    -- reshapes from a 3D tensor of 16x5x5 into 1D tensor of 16*5*5
net:add(nn.Linear(16*5*5, 120))             -- fully connected layer (matrix multiplication between input and weights)
net:add(nn.ReLU())                       -- non-linearity
net:add(nn.Linear(120, 84))
net:add(nn.ReLU())                       -- non-linearity
net:add(nn.Linear(84, 10))                   -- 10 is the number of outputs of the network (in this case, 10 digits)
net:add(nn.LogSoftMax())                     -- converts the output to a log-probability. Useful for classification problems
criterion = nn.ClassNLLCriterion()
trainer = nn.StochasticGradient(net, criterion)
trainer.learningRate = 0.001
trainer.maxIteration = 5
trainer:train(trainset)
testset.data = testset.data:double()   -- convert from Byte tensor to Double tensor
for i=1,3 do -- over each image channel
    testset.data[{ {}, {i}, {}, {}  }]:add(-mean[i]) -- mean subtraction
    testset.data[{ {}, {i}, {}, {}  }]:div(stdv[i]) -- std scaling
end
predicted = net:forward(testset.data[100])
print(classes[testset.label[100]])
print(predicted:exp())
for i=1,predicted:size(1) do
    print(classes[i], predicted[i])
end
correct = 0
for i=1,10000 do
    local groundtruth = testset.label[i]
    local prediction = net:forward(testset.data[i])
    local confidences, indices = torch.sort(prediction, true)  -- true means sort in descending order
    if groundtruth == indices[1] then
        correct = correct + 1
    end
end

print(correct, 100*correct/10000 .. ' % ')
class_performance = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
for i=1,10000 do
    local groundtruth = testset.label[i]
    local prediction = net:forward(testset.data[i])
    local confidences, indices = torch.sort(prediction, true)  -- true means sort in descending order
    if groundtruth == indices[1] then
        class_performance[groundtruth] = class_performance[groundtruth] + 1
    end
end

for i=1,#classes do
    print(classes[i], 100*class_performance[i]/1000 .. ' %')
end

执行th pattern_recognition.lua。

首先下载cifar10torchsmall.zip样本，有50000张训练用图片，10000张测试用图片，分别都标注，包括airplane、automobile等10种分类，对trainset绑定__index和size方法，兼容nn.Sequential使用，绑定函数看lua教程：http://tylerneylon.com/a/learn-lua/ ,trainset数据正规化，数据转成均值为1方差为1的double类型张量。初始化卷积神经网络模型，包括两层卷积、两层池化、一个全连接以及一个softmax层，进行训练，学习率为0.001，迭代5次，模型训练好后对测试机第100号图片做预测，打印出整体正确率以及每种分类准确率。https://github.com/soumith/cvpr2015/blob/master/Deep%20Learning%20with%20Torch.ipynb 。

torch可以方便支持gpu计算，需要对代码做修改。

比较流行的seq2seq基本都用lstm组成编码器解码器模型实现，开源实现大都基于one-hot embedding(没有词向量表达信息量大)。word2vec词向量 seq2seq模型，只有一个lstm单元机器人。

下载《甄环传》小说原文。上网随便百度“甄环传 txt”，下载下来，把文件转码成utf-8编码，把windows回车符都替换成n，以便后续处理。

对甄环传切词。切词工具word_segment.py到github下载，地址在https://github.com/warmheartli/ChatBotCourse/blob/master/word_segment.py 。

python ./word_segment.py zhenhuanzhuan.txt zhenhuanzhuan.segment

生成词向量。用word2vec，word2vec源码 https://github.com/warmheartli/ChatBotCourse/tree/master/word2vec 。make编译即可执行。

./word2vec -train ./zhenhuanzhuan.segment -output vectors.bin -cbow 1 -size 200 -window 8 -negative 25 -hs 0 -sample 1e-4 -threads 20 -binary 1 -iter 15

生成一个vectors.bin文件，基于甄环传原文生成的词向量文件。

训练代码。

# -*- coding: utf-8 -*-

import sys
import math
import tflearn
import chardet
import numpy as np
import struct

seq = []

max_w = 50
float_size = 4
word_vector_dict = {}

def load_vectors(input):
    """从vectors.bin加载词向量，返回一个word_vector_dict的词典，key是词，value是200维的向量
    """
    print "begin load vectors"

    input_file = open(input, "rb")

    # 获取词表数目及向量维度
    words_and_size = input_file.readline()
    words_and_size = words_and_size.strip()
    words = long(words_and_size.split(' ')[0])
    size = long(words_and_size.split(' ')[1])
    print "words =", words
    print "size =", size

    for b in range(0, words):
        a = 0
        word = ''
        # 读取一个词
        while True:
            c = input_file.read(1)
            word = word + c
            if False == c or c == ' ':
                break
            if a < max_w and c != 'n':
                a = a + 1
        word = word.strip()

        vector = []
        for index in range(0, size):
            m = input_file.read(float_size)
            (weight,) = struct.unpack('f', m)
            vector.append(weight)

        # 将词及其对应的向量存到dict中
        word_vector_dict[word.decode('utf-8')] = vector

    input_file.close()
    print "load vectors finish"

def init_seq():
    """读取切好词的文本文件，加载全部词序列
    """
    file_object = open('zhenhuanzhuan.segment', 'r')
    vocab_dict = {}
    while True:
        line = file_object.readline()
        if line:
            for word in line.decode('utf-8').split(' '):
                if word_vector_dict.has_key(word):
                    seq.append(word_vector_dict[word])
        else:
            break
    file_object.close()

def vector_sqrtlen(vector):
    len = 0
    for item in vector:
        len += item * item
    len = math.sqrt(len)
    return len

def vector_cosine(v1, v2):
    if len(v1) != len(v2):
        sys.exit(1)
    sqrtlen1 = vector_sqrtlen(v1)
    sqrtlen2 = vector_sqrtlen(v2)
    value = 0
    for item1, item2 in zip(v1, v2):
        value += item1 * item2
    return value / (sqrtlen1*sqrtlen2)

def vector2word(vector):
    max_cos = -10000
    match_word = ''
    for word in word_vector_dict:
        v = word_vector_dict[word]
        cosine = vector_cosine(vector, v)
        if cosine > max_cos:
            max_cos = cosine
            match_word = word
    return (match_word, max_cos)

def main():
    load_vectors("./vectors.bin")
    init_seq()
    xlist = []
    ylist = []
    test_X = None
    #for i in range(len(seq)-100):
    for i in range(10):
        sequence = seq[i:i+20]
        xlist.append(sequence)
        ylist.append(seq[i+20])
        if test_X is None:
            test_X = np.array(sequence)
            (match_word, max_cos) = vector2word(seq[i+20])
            print "right answer=", match_word, max_cos

    X = np.array(xlist)
    Y = np.array(ylist)
    net = tflearn.input_data([None, 20, 200])
    net = tflearn.lstm(net, 200)
    net = tflearn.fully_connected(net, 200, activation='linear')
    net = tflearn.regression(net, optimizer='sgd', learning_rate=0.1,
                                     loss='mean_square')
    model = tflearn.DNN(net)
    model.fit(X, Y, n_epoch=500, batch_size=10,snapshot_epoch=False,show_metric=True)
    model.save("model")
    predict = model.predict([test_X])
    #print predict
    #for v in test_X:
    #    print vector2word(v)
    (match_word, max_cos) = vector2word(predict[0])
    print "predict=", match_word, max_cos

main()
 

load_vectors从vectors.bin加载词向量，init_seq加载甄环传切词文本并存到一个序列里，vector2word求距离某向量最近词，模型只有一个lstm单元。
经过500个epoch训练，均方损失降到0.33673，以0.941794432002余弦相似度预测出下一个字。
强大gpu，调整参数，整篇文章都训练，修改代码predict部分，不断输出下一个字，自动吐出甄环体。基于tflearn实现，tflearn官方文档examples实现seq2seq直接调用tensorflow中的tensorflow/python/ops/seq2seq.py，基于one-hot embedding方法，一定没有词向量效果好。

参考资料：

《Python 自然语言处理》
http://www.shareditor.com/blogshow?blogId=116
http://www.shareditor.com/blogshow?blogId=117
http://www.shareditor.com/blogshow?blogId=118

欢迎推荐上海机器学习工作机会，我的微信：qingxingfengzi