LSTM学习

2021-08-06 本文已影响0人斐硕人

RNN&LSTM

当前输出的产生与之前的信息（输出、状态）相关
RNN：过去发生的事件可以预测现在
LSTM（进阶RNN）：过去发生的事情太多了，全部记住的话，处理起来过于复杂，因此选择遗忘不重要事件，记住重要的（解决长期依赖问题）

LSTM 核心思想

Sigmoid 层输出 0 到 1 之间的数值，可以用来描述每个部分有多少需要记住。0 指代“全忘记”，1 指“记住所有”
原本RNN只有 tanh层 来处理当前输出（过去和现在输入），LSTM将 Sigmoid 层运用到 tanh层，增添了状态信息，并进一步将其设计为门结构，来决定当前输出。

LSTM 有通过精心设计的称作为“门”的结构来去除或者增加信息到细胞状态的能力。
门是一种让信息选择式通过的方法。他们包含一个 sigmoid 神经网络层和一个pointwise 乘法操作。

门
LSTM 拥有三个门，来保护和控制细胞状态。f，i，o 分别表示遗忘门、输入门、输出门
分别表示遗忘门、输入门、输出门

前向传播

当前状态

LSTM 设计了状态信息Ct来选择是否遗忘，因而三个门中，有两个是用来产生状态信息的，如下：

1. (遗忘门 f )sigmoid (Ht-1,Xt)（pointwise）C t-1 ；——乘：有 0为 0 ,并且以后所有乘0都为0
1. (输入门 i )sigmoid (Ht-1,Xt)（pointwise） tanh(Ht-1,Xt) ；
C t = 1输出 + 2输出；*——与：有0为0，同1为 1 // 加号：同0为0，异为1，同1为2
状态信息的产生

当前输出

(输出门 o )sigmoid (Ht-1,Xt)（pointwise） tanh(Ct)

其中 sigmoid (x,y) 和 tanh(x,y) 等涉及的两个向量x,y处理为 W(x,y)+b，即 sigmoid (W(x,y)+b)、tanh(W(x,y)+b)。再将两个参数向量合并 Xc(t) = [X(t),H(t-1)]，则 sigmoid (W(Xc)+b)、tanh(W(Xc)+b)。

则当前cell相关方程如下（s 为状态 C）：

参数方程

LSTM当前节点参数图

sigmoid和tanh的区别

sigmoid 在特征相差比较复杂或是相差不大时效果比较好
tanh 在特征相差明显时的效果会很好，在循环过程中会不断扩大特征效果，0均值

sigmoid在输入处于[-1,1]之间时，函数值变化敏感，一旦接近或者超出区间就失去敏感性，处于饱和状态，影响神经网络预测的精度值。tanh的输出和输入能够保持非线性单调上升和下降关系，符合BP网络的梯度求解，容错性好，有界，渐进于0、1，符合人脑神经饱和的规律，但比 sigmoid 函数延迟了饱和期。

sigmoid

sigmoid函数及其导数

tanh

tanh函数及其导数

反向传播

已知部分参数基于损失函数 t+1 时刻的偏导数，计算所有参数基于损失函数 t 时刻的偏导数
反向传播计算误差的传播沿两个方向，分别为从输出层传递至输入层，以及沿时间 t 的反向传播
先计算h输出误差，用h表述s状态的误差，再用h和s计算其他参数误差

diff_h

损失函数
h(t),y(t) 分别为输出序列与样本标签
diff_h = 2·sum(h(t)-y(t))
输出误差计算

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

while idx >= 0:
  diff_h = 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

diff_s

        ds = self.state.o * top_diff_h + top_diff_s

其他 diff_o i g f

反向传播导数

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

wi_diff

权重 diff

        wi_diff += np.outer( (1. - i) * i *di, xc)

损失 J 对 h 的导数—— sigmoid 型
1. - i 相比较 1 - i ，若 i 为整数，结果为 float

以下代码主要来自参考文章7

LSTM 代码实现

import random

import numpy as np
import math

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

# createst uniform random array w/ values in [a,b) and shape args
def rand_arr(a, b, *args): 
    np.random.seed(0)
    return np.random.rand(*args) * (b - a) + a
# 参数
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
        # 权重矩阵 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)
        # 偏置 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) 
        # 梯度 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) 

    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) 
# 状态
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)
# LSTM各节点    
class LstmNode:
    def __init__(self, lstm_param, lstm_state):
        # 状态和参数 store reference to parameters and to activations
        self.state = lstm_state
        self.param = lstm_param
        # 输入x(t) 节点的非经常性输入 non-recurrent input to node
        self.x = None
        # 输入x(t)和 h(t-1) 非经常性输入与经常性输入同时进行 non-recurrent input concatenated with recurrent input
        self.xc = None
    # 前向传播
    def bottom_data_is(self, x, s_prev = None, h_prev = None):
        # 首节点
        if s_prev is None:
            s_prev = np.zeros_like(self.state.s)
        if h_prev is None:
            h_prev = np.zeros_like(self.state.h)
        # 为反向传播存储数据
        self.s_prev = s_prev
        self.h_prev = h_prev

        # 拼接 x(t) and h(t-1)为 xc
        xc = np.hstack((x,  h_prev))
        # 定义参数方程
        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
# 导数计算    
    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
        # sigmoid导数
        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 
        # tanh 导数
        dg_input = (1. - self.state.g ** 2) * dg

        # 权重和偏置导数 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:]
# LSTM网络
class LstmNetwork():
    def __init__(self, lstm_param):
        self.lstm_param = lstm_param
        self.lstm_node_list = []
        # input sequence
        self.x_list = []
    # 输出节点遍历
    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

    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)
# 损失函数
class ToyLossLayer:
    """
    Computes square loss with first element of hidden layer array.
    """
    # 损失函数定义
    @classmethod
    def loss(self, pred, label):
        return (pred[0] - label) ** 2
    # 节点输出求导
    @classmethod
    def bottom_diff(self, pred, label):
        diff = np.zeros_like(pred)
        diff[0] = 2 * (pred[0] - label)
        return diff

输入一串连续质数，预估下一个质数

#lstm在输入一串连续质数时预估下一个质数
# 测试
import numpy as np

# from lstm import LstmParam, LstmNetwork, ToyLossLayer

def example_0():
    # learns to repeat simple sequence from random inputs
    np.random.seed(0)
    # 隐藏层节点数和词向量维度
    # parameters for input data dimension and lstm cell count 
    mem_cell_ct = 100
    x_dim = 50

    # 实例化 LSTM 参数及网络
    lstm_param = LstmParam(mem_cell_ct, x_dim) 
    lstm_net = LstmNetwork(lstm_param)
    
    # 前项输出
    y_list = [-0.5,0.2,0.1, -0.5]
    # y_list个x_dim维0-1数组
    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()

if __name__ == "__main__":
    example_0()

参考文章：