摘要：和分別是樣本輸入和輸出二進制值第位，對于每個樣本有兩個值，分別是和對應第位。最簡單實現，沒有考慮偏置變量，只有兩個神經元。存儲神經元狀態，包括，是內部狀態矩陣記憶，是隱藏層神經元輸出矩陣。表示當前時序表示時序記憶單元。下載甄環傳小說原文。

真正掌握一種算法，最實際的方法，完全手寫出來。

LSTM（Long Short Tem Memory）特殊遞歸神經網絡，神經元保存歷史記憶，解決自然語言處理統計方法只能考慮最近n個詞語而忽略更久前詞語的問題。用途：word representation（embedding）(詞語向量)、sequence to sequence learning（輸入句子預測句子）、機器翻譯、語音識別等。

100多行原始python代碼實現基于LSTM二進制加法器。https://iamtrask.github.io/20... ，翻譯http://blog.csdn.net/zzukun/a... ：

import copy, numpy as np
np.random.seed(0)
最開始引入numpy庫，矩陣操作。

def sigmoid(x):

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

聲明sigmoid激活函數，神經網絡基礎內容，常用激活函數sigmoid、tan、relu等，sigmoid取值范圍[0, 1]，tan取值范圍[-1,1]，x是向量，返回output是向量。

def sigmoid_output_to_derivative(output):

return output*(1-output)

聲明sigmoid求導函數。
加法器思路：二進制加法是二進制位相加，記錄滿二進一進位，訓練時隨機c=a+b樣本，輸入a、b輸出c是整個lstm預測過程，訓練由a、b二進制向c各種轉換矩陣和權重，神經網絡。

int2binary = {}
聲明詞典，由整型數字轉成二進制，存起來不用隨時計算，提前存好讀取更快。

binary_dim = 8
largest_number = pow(2,binary_dim)
聲明二進制數字維度，8，二進制能表達最大整數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]

預先把整數到二進制轉換詞典存起來。

alpha = 0.1
input_dim = 2
hidden_dim = 16
output_dim = 1
設置參數，alpha是學習速度，input_dim是輸入層向量維度，輸入a、b兩個數，是2，hidden_dim是隱藏層向量維度，隱藏層神經元個數，output_dim是輸出層向量維度，輸出一個c，是1維。從輸入層到隱藏層權重矩陣是216維，從隱藏層到輸出層權重矩陣是161維，隱藏層到隱藏層權重矩陣是16*16維：

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生成從0到1之間隨機浮點數，2x-1使其取值范圍在[-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)
聲明三個矩陣更新，Delta。

for j in range(10000):
進行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]
隨機生成樣本，包含二進制a、b、c，c=a+b，a_int、b_int、c_int分別是a、b、c對應整數格式。

d = np.zeros_like(c)
d存模型對c預測值。

overallError = 0
全局誤差，觀察模型效果。
layer_2_deltas = list()
存儲第二層(輸出層)殘差，輸出層殘差計算公式推導公式http://deeplearning.stanford.... 。

layer_1_values = list()
layer_1_values.append(np.zeros(hidden_dim))
存儲第一層(隱藏層)輸出值，賦0值作為上一個時間值。

for position in range(binary_dim):
遍歷二進制每一位。

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分別是樣本輸入和輸出二進制值第position位，X對于每個樣本有兩個值，分別是a和b對應第position位。把樣本拆成每個二進制位用于訓練，二進制加法存在進位標記正好適合利用LSTM長短期記憶訓練，每個樣本8個二進制位是一個時間序列。

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))
這里使用的公式是C2 = sigma(W1·C1)，

layer_2_error = y - layer_2
計算預測值和真實值誤差。

layer_2_deltas.append((layer_2_error)*sigmoid_output_to_derivative(layer_2))
反向傳導，計算delta，添加到數組layer_2_deltas

overallError += np.abs(layer_2_error[0])
計算累加總誤差，用于展示和觀察。

d[binary_dim - position - 1] = np.round(layer_20)
存儲預測position位輸出值。

layer_1_values.append(copy.deepcopy(layer_1))
存儲中間過程生成隱藏層值。

future_layer_1_delta = np.zeros(hidden_dim)
存儲下一個時間周期隱藏層歷史記憶值，先賦一個空值。

for position in range(binary_dim):
遍歷二進制每一位。

X = np.array([[a[position],b[position]]])
取出X值，從大位開始更新，反向傳導按時序逆著一級一級更新。

layer_1 = layer_1_values[-position-1]
取出位對應隱藏層輸出。

prev_layer_1 = layer_1_values[-position-2]
取出位對應隱藏層上一時序輸出。

layer_2_delta = layer_2_deltas[-position-1]
取出位對應輸出層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)
神經網絡反向傳導公式，加上隱藏層?值。

synapse_1_update += np.atleast_2d(layer_1).T.dot(layer_2_delta)
累加權重矩陣更新，對權重(權重矩陣)偏導等于本層輸出與下一層delta點乘。

synapse_h_update += np.atleast_2d(prev_layer_1).T.dot(layer_1_delta)
前一時序隱藏層權重矩陣更新，前一時序隱藏層輸出與本時序delta點乘。

synapse_0_update += X.T.dot(layer_1_delta)
輸入層權重矩陣更新。

future_layer_1_delta = layer_1_delta
記錄本時序隱藏層delta。

synapse_0 += synapse_0_update * alpha
synapse_1 += synapse_1_update * alpha
synapse_h += synapse_h_update * alpha
權重矩陣更新。

synapse_0_update *= 0
synapse_1_update *= 0
synapse_h_update *= 0
更新變量歸零。

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 "------------"

每訓練1000個樣本輸出總誤差信息，運行時看收斂過程。
LSTM最簡單實現，沒有考慮偏置變量，只有兩個神經元。

完整LSTM python實現。完全參照論文great intro paper實現,代碼來源https://github.com/nicodjimen... ，作者解釋http://nicodjimenez.github.io... ，具體過程參考http://colah.github.io/posts/... 圖。

import random
import numpy as np
import math

def sigmoid(x):

return 1. / (1 + np.exp(-x))

聲明sigmoid函數。

def rand_arr(a, b, *args):

np.random.seed(0)
return np.random.rand(*args) * (b - a) + a

生成隨機矩陣，取值范圍[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類傳遞參數，mem_cell_ct是lstm神經元數目，x_dim是輸入數據維度，concat_len是mem_cell_ct與x_dim長度和，wg是輸入節點權重矩陣，wi是輸入門權重矩陣，wf是忘記門權重矩陣，wo是輸出門權重矩陣，bg、bi、bf、bo分別是輸入節點、輸入門、忘記門、輸出門偏置，wg_diff、wi_diff、wf_diff、wo_diff分別是輸入節點、輸入門、忘記門、輸出門權重損失，bg_diff、bi_diff、bf_diff、bo_diff分別是輸入節點、輸入門、忘記門、輸出門偏置損失，初始化按照矩陣維度初始化，損失矩陣歸零。

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)

定義權重更新過程，先減損失，再把損失矩陣歸零。

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存儲LSTM神經元狀態，包括g、i、f、o、s、h，s是內部狀態矩陣(記憶)，h是隱藏層神經元輸出矩陣。

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對應樣本輸入，x是輸入樣本x，xc是用hstack把x和遞歸輸入節點拼接矩陣（hstack是橫拼矩陣，vstack是縱拼矩陣）。

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是兩個方向，輸入樣本從底部輸入，反向傳導從頂部向底部傳導，bottom_data_is是輸入樣本過程，把x和先前輸入拼接成矩陣，用公式wx+b分別計算g、i、f、o值，激活函數tanh和sigmoid。
每個時序神經網絡有四個神經網絡層(激活函數)，最左邊忘記門，直接生效到記憶C，第二個輸入門，依賴輸入樣本數據，按照一定“比例”影響記憶C，“比例”通過第三個層(tanh)實現，取值范圍是[-1,1]可以正向影響也可以負向影響，最后一個輸出門，每一時序產生輸出既依賴輸入樣本x和上一時序輸出，還依賴記憶C，設計模仿生物神經元記憶功能。

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:]

反向傳導，整個訓練過程核心。假設在t時刻lstm輸出預測值h(t)，實際輸出值是y(t)，之間差別是損失，假設損失函數為l(t) = f(h(t), y(t)) = ||h(t) - y(t)||^2，歐式距離，整體損失函數是L(t) = ∑l(t)，t從1到T，T表示整個事件序列最大長度。最終目標是用梯度下降法讓L(t)最小化，找到一個最優權重w使得L(t)最小，當w發生微小變化L(t)不再變化，達到局部最優，即L對w偏導梯度為0。
dL/dw表示當w發生單位變化L變化多少，dh(t)/dw表示當w發生單位變化h(t)變化多少，dL/dh(t)表示當h(t)發生單位變化時L變化多少，(dL/dh(t)) * (dh(t)/dw)表示第t時序第i個記憶單元w發生單位變化L變化多少，把所有由1到M的i和所有由1到T的t累加是整體dL/dw。

第i個記憶單元，h(t)發生單位變化，整個從1到T時序所有局部損失l的累加和，是dL/dh(t)，h(t)只影響從t到T時序局部損失l。

假設L(t)表示從t到T損失和，L(t) = ∑l(s)。

h(t)對w導數。

L(t) = l(t) + L(t+1)，dL(t)/dh(t) = dl(t)/dh(t) + dL(t+1)/dh(t)，用下一時序導數得出當前時序導數，規律推導，計算T時刻導數往前推，在T時刻，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(預測結果誤差發生單位變化損失L多少，dL(t)/dh(t)數值計算)，由idx從T往前遍歷到1，計算loss_layer.bottom_diff和下一個時序bottom_diff_h和作為diff_h(第一次遍歷即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導數l"(t) = 2 * (h(t) - y(t))
。當s(t)發生變化，L(t)變化來源s(t)影響h(t)和h(t+1)，影響L(t)。
h(t+1)不會影響l(t)。
左邊式子(dL(t)/dh(t)) * (dh(t)/ds(t))，由t+1到t來逐級反推dL(t)/ds(t)。
神經元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表示神經網絡層輸入，top表示神經網絡層輸出，和caffe概念一致。
def top_diff_is(self, top_diff_h, top_diff_s):
top_diff_h表示當前t時序dL(t)/dh(t), top_diff_s表示t+1時序記憶單元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

前綴d表達誤差L對某一項導數(directive)。
ds是在根據公式dL(t)/ds(t)計算當前t時序dL(t)/ds(t)。
do是計算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是計算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是計算dL(t)/dg(t)，dL(t)/dg(t) = (dL(t)/ds(t)) (ds(t)/dg(t)) = ds i(t)。
df是計算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函數導數，tanh函數導數。di_input，(1. - self.state.i) * self.state.i，sigmoid導數，當i神經元輸入發生單位變化時輸出值有多大變化，再乘di表示當i神經元輸入發生單位變化時誤差L(t)發生多大變化，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是權重矩陣誤差，b_diff是偏置誤差，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)

累加輸入xdiff，x在四處起作用，四處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時序上s變化和t時序上s變化時f倍關系。dxc是x和h橫向合并矩陣，分別取兩部分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)

添加訓練樣本，輸入x數據。

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，指定記憶存儲單元數為100，指定輸入樣本x維度是50。初始化LstmNetwork訓練模型，生成4組各50個隨機數，分別以[-0.5,0.2,0.1, -0.5]作為y值訓練，每次喂50個隨機數和一個y值，迭代100次。
lstm輸入一串連續質數預估下一個質數。小測試，生成100以內質數，循環拿出50個質數序列作x，第51個質數作y，拿出10個樣本參與訓練1w次，均方誤差由0.17973最終達到了1.05172e-06，幾乎完全正確：

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()

質數列表全都除以100，這個代碼訓練數據必須是小于1數值。

torch是深度學習框架。1）tensorflow，谷歌主推，時下最火，小型試驗和大型計算都可以，基于python，缺點是上手相對較難，速度一般；2）torch，facebook主推，用于小型試驗，開源應用較多，基于lua，上手較快，網上文檔較全，缺點是lua語言相對冷門；3）mxnet，Amazon主推，主要用于大型計算，基于python和R，缺點是網上開源項目較少；4）caffe，facebook主推，用于大型計算，基于c++、python，缺點是開發不是很方便；5）theano，速度一般，基于python，評價很好。

torch github上lstm實現項目比較多。

在mac上安裝torch。https://github.com/torch/torc... 。

git clone https://github.com/torch/dist... ~/torch --recursive
cd ~/torch; bash install-deps;
./install.sh
qt安裝不成功問題，自己多帶帶安裝。

brew install cartr/qt4/qt
安裝后需要手工加到~/.bash_profile中。

. ~/torch/install/bin/torch-activate
source ~/.bash_profile后執行th使用torch。
安裝itorch，安裝依賴

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

git clone https://github.com/facebook/i...
cd iTorch
luarocks make
用卷積神經網絡實現圖像識別。
創建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(1655)) -- reshapes from a 3D tensor of 16x5x5 into 1D tensor of 1655
net:add(nn.Linear(1655, 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
執行th pattern_recognition.lua。

首先下載cifar10torchsmall.zip樣本，有50000張訓練用圖片，10000張測試用圖片，分別都標注，包括airplane、automobile等10種分類，對trainset綁定__index和size方法，兼容nn.Sequential使用，綁定函數看lua教程：http://tylerneylon.com/a/lear... ,trainset數據正規化，數據轉成均值為1方差為1的double類型張量。初始化卷積神經網絡模型，包括兩層卷積、兩層池化、一個全連接以及一個softmax層，進行訓練，學習率為0.001，迭代5次，模型訓練好后對測試機第100號圖片做預測，打印出整體正確率以及每種分類準確率。https://github.com/soumith/cv... 。

torch可以方便支持gpu計算，需要對代碼做修改。

比較流行的seq2seq基本都用lstm組成編碼器解碼器模型實現，開源實現大都基于one-hot embedding(沒有詞向量表達信息量大)。word2vec詞向量 seq2seq模型，只有一個lstm單元機器人。

下載《甄環傳》小說原文。上網隨便百度“甄環傳 txt”，下載下來，把文件轉碼成utf-8編碼，把windows回車符都替換成n，以便后續處理。

對甄環傳切詞。切詞工具word_segment.py到github下載，地址在https://github.com/warmheartl... 。

python ./word_segment.py zhenhuanzhuan.txt zhenhuanzhuan.segment
生成詞向量。用word2vec，word2vec源碼 https://github.com/warmheartl... 。make編譯即可執行。

./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
生成一個vectors.bin文件，基于甄環傳原文生成的詞向量文件。

訓練代碼。

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):

"""從vectors.bin加載詞向量，返回一個word_vector_dict的詞典，key是詞，value是200維的向量
"""
print "begin load vectors"

input_file = open(input, "rb")

# 獲取詞表數目及向量維度
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 = ""
    # 讀取一個詞
    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)

    # 將詞及其對應的向量存到dict中
    word_vector_dict[word.decode("utf-8")] = vector

input_file.close()
print "load vectors finish"

def init_seq():

"""讀取切好詞的文本文件，加載全部詞序列
"""
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從vectors.bin加載詞向量，init_seq加載甄環傳切詞文本并存到一個序列里，vector2word求距離某向量最近詞，模型只有一個lstm單元。
經過500個epoch訓練，均方損失降到0.33673，以0.941794432002余弦相似度預測出下一個字。
強大gpu，調整參數，整篇文章都訓練，修改代碼predict部分，不斷輸出下一個字，自動吐出甄環體?；趖flearn實現，tflearn官方文檔examples實現seq2seq直接調用tensorflow中的tensorflow/python/ops/seq2seq.py，基于one-hot embedding方法，一定沒有詞向量效果好。
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