LSTM的推导与实现

最近在看CS224d，这里主要介绍LSTM(Long Short-Term Memory)的推导过程以及用Python进行简单的实现。LSTM是一种时间递归神经网络，是RNN的一个变种，非常适合处理和预测时间序列中间隔和延迟非常长的事件。假设我们去试着预测‘I grew up in France...（很长间隔）...I speak fluent French’最后的单词，当前的信息建议下一个此可能是一种语言的名字（因为speak嘛），但是要准确预测出‘French’我们就需要前面的离当前位置较远的‘France’作为上下文，当这个间隔比较大的时候RNN就会难以处理，而LSTM则没有这个问题。

LSTM的原理

为了弄明白LSTM的实现，我下载了alex的原文，但是被论文上图片和公式弄的晕头转向，无奈最后在网上收集了一些资料才总算弄明白。我这里不介绍就LSTM的前置RNN了，不懂的童鞋自己了解一下吧。

LSTM的前向过程

首先看一张LSTM节点的内部示意图：图片来自一篇讲解LSTM的blog(http://colah.github.io/posts/2015-08-Understanding-LSTMs/)这是我认为网上画的最好的LSTM网络节点图（比论文里面画的容易理解多了），LSTM前向过程就是看图说话，关键的函数节点已经在图中标出,这里我们忽略了其中一个tanh计算过程。

g(t)i(t)f(t)o(t)s(t)h(t)======ϕ(Wgxx(t)+Wghh(t−1)+bgσ(Wixx(t)+Wihh(t−1)+biσ(Wfxx(t)+Wfhh(t−1)+bfσ(Woxx(t)+Wohh(t−1)+bog(t)∗i(t)+s(t−1)∗f(t)s(t)∗o(t)(1)(2)(3)(4)(5)(6)

这里ϕ(x)=tanh(x),σ(x)=11+e−x

， x(t),h(t) 分别是我们的输入序列和输出序列。如果我们把 x(t) 与 h(t−1)

这两个向量进行合并:

xc(t)=[x(t),h(t−1)]

那么可以上面的方程组可以重写为:

g(t)i(t)f(t)o(t)s(t)h(t)======ϕ(Wgxc(t))+bgσ(Wixc(t))+biσ(Wfxc(t))+bfσ(Woxc(t))+bog(t)∗i(t)+s(t−1)∗f(t)s(t)∗o(t)(7)(8)(9)(10)(11)(12)

其中f(t)

被称为 忘记门，所表达的含义是决定我们会从以前状态中丢弃什么信息。 i(t),g(t) 构成了 输入门，决定什么样的新信息被存放在细胞状态中。 o(t)

所在位置被称作输出门，决定我们要输出什么值。这里表述的不是很准确，感兴趣的读者可以去http://colah.github.io/posts/2015-08-Understanding-LSTMs/ NLP这块我也不太懂。

前向过程的代码如下：

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

LSTM的反向过程

LSTM的正向过程比较容易，反向过程则比较复杂，我们先定义一个loss function l(t)=f(h(t),y(t)))=||h(t)−y(t)||2

， h(t),y(t) 分别为输出序列与样本标签，我们要做的就是最小化整个时间序列上的 l(t)

，即最小化

L=∑t=1Tl(t)

其中T

代表整个时间序列，下面我们通过 L 来计算梯度，假设我们要计算 dLdw ，其中 w 是一个标量(例如是矩阵 Wgx 的一个元素)，由链式法则可以导出 dLdw=∑t=1T∑i=1MdLdhi(t)dhi(t)dw 其中 hi(t) 是第i个单元的输出， M 是LSTM单元的个数，网络随着时间t前向传播， hi(t) 的改变不影响t时刻之前的loss，我们可以写出： dLdhi(t)=∑s=1Tdl(s)dhi(t)=∑s=tTdl(s)dhi(t) 为了书写方便我们令 L(t)=∑Ts=tl(s) 来简化我们的书写，这样 L(1) 就是整个序列的loss，重写上式有： dLdhi(t)=∑s=1Tdl(s)dhi(t)=dL(t)dhi(t)

这样我们就可以将梯度重写为：

dLdw=∑t=1T∑i=1MdL(t)dhi(t)dhi(t)dw

我们知道L(t)=l(t)+L(t+1)

，那么 dL(t)dhi(t)=dl(t)dhi(t)+dL(t+1)dhi(t) ，这说明得到下一时序的导数后可以直接得出当前时序的导数，所以我们可以计算 T 时刻的导数然后往前推，在 T 时刻有 dL(T)dhi(T)=dl(T)dhi(T)

。

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

从上面公式可以很容易理解diff_h的计算过程。这里的loss_layer.bottom_diff定义如下：

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

该函数结合上文的loss function很明显。下面来推导dL(t)ds(t)

，结合前面的前向公式我们可以很容易得出 s(t) 的变化会直接影响 h(t) 和 h(t+1) ，进而影响 L(t) ，即有： dL(t)dhi(t)=dL(t)dhi(t)∗dhi(t)dsi(t)+dL(t)dhi(t+1)∗dhi(t+1)dsi(t) 因为 h(t+1) 不影响 l(t) 所以有 dL(t)dhi(t+1)=dL(t+1)dhi(t+1)

，因此有：

dL(t)dhi(t)=dL(t)dhi(t)∗dhi(t)dsi(t)+dL(t+1)dhi(t+1)∗dhi(t+1)dsi(t)=dL(t)dhi(t)∗dhi(t)dsi(t)+dL(t+1)dsi(t)

同样的我们可以通过后面的导数逐级反推得到前面的导数，代码即diff_s的计算过程。

下面我们计算dL(t)dhi(t)∗dhi(t)dsi(t)

，

因为h(t)=s(t)∗o(t)，那么dL(t)dhi(t)∗dhi(t)dsi(t)=dL(t)dhi(t)∗oi(t)=oi(t)[diff_h]，即dL(t)dsi(t)=o(t)[diff_h]i+[diff_s]i，其中[diff_h]i,[diff_s]i分别表述当前t时序的dL(t)dhi(t)和t+1时序的dL(t)dsi(t)

。同样的，结合上面的代码应该比较容易理解。

下面我们根据前向过程挨个计算导数：

dL(t)do(t)dL(t)di(t)dL(t)dg(t)dL(t)df(t)====dL(t)dh(t)∗s(t)dL(t)ds(t)∗ds(t)di(t)=dL(t)ds(t)∗g(t)dL(t)ds(t)∗ds(t)dg(t)=dL(t)ds(t)∗i(t)dL(t)ds(t)∗ds(t)df(t)=dL(t)ds(t)∗s(t−1)(13)(14)(15)(16)

因此有以下代码：

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 #sigmoid diff 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 #tanh diff # 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:]

这里top_diff_h，top_diff_s分别是上文的diff_h，diff_s。这里我们讲解下wi_diff的求解过程，其他变量类似。

dL(t)dWi=dL(t)di(t)∗di(t)d(Wixc(t))∗d(Wixc(t))dxc(t)

上式化简之后即得到以下代码

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

其它的导数可以同样得到，这里就不赘述了。

LSTM完整例子

#lstm在输入一串连续质数时预估下一个质数 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 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) 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) 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) 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 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 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:] 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 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)

测试代码

import numpy as np 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 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() if __name__ == "__main__": example_0()