RNN神经元理解
单个RNN神经元行为
括号中表示的是维度
向前传播
def rnn_step_forward(x, prev_h, Wx, Wh, b):
"""
Run the forward pass for a single timestep of a vanilla RNN that uses a tanh
activation function. The input data has dimension D, the hidden state has dimension H, and we use
a minibatch size of N. Inputs:
- x: Input data for this timestep, of shape (N, D).
- prev_h: Hidden state from previous timestep, of shape (N, H)
- Wx: Weight matrix for input-to-hidden connections, of shape (D, H)
- Wh: Weight matrix for hidden-to-hidden connections, of shape (H, H)
- b: Biases of shape (H,) Returns a tuple of:
- next_h: Next hidden state, of shape (N, H)
- cache: Tuple of values needed for the backward pass.
"""
next_h, cache = None, None
##############################################################################
# TODO: Implement a single forward step for the vanilla RNN. Store the next #
# hidden state and any values you need for the backward pass in the next_h #
# and cache variables respectively. #
##############################################################################
next_h = np.tanh(x.dot(Wx) + prev_h.dot(Wh) + b)
cache = (x, Wx, Wh, prev_h, next_h)
##############################################################################
# END OF YOUR CODE #
##############################################################################
return next_h, cache
反向传播
def rnn_step_backward(dnext_h, cache):
"""
Backward pass for a single timestep of a vanilla RNN. Inputs:
- dnext_h: Gradient of loss with respect to next hidden state
- cache: Cache object from the forward pass Returns a tuple of:
- dx: Gradients of input data, of shape (N, D)
- dprev_h: Gradients of previous hidden state, of shape (N, H)
- dWx: Gradients of input-to-hidden weights, of shape (N, H)
- dWh: Gradients of hidden-to-hidden weights, of shape (H, H)
- db: Gradients of bias vector, of shape (H,)
"""
dx, dprev_h, dWx, dWh, db = None, None, None, None, None
##############################################################################
# TODO: Implement the backward pass for a single step of a vanilla RNN. #
# #
# HINT: For the tanh function, you can compute the local derivative in terms #
# of the output value from tanh. #
##############################################################################
x, Wx, Wh, prev_h, next_h = cache
dtanh = 1 - next_h**2
dx = (dnext_h*dtanh).dot(Wx.T)
dWx = x.T.dot(dnext_h*dtanh)
dprev_h = (dnext_h*dtanh).dot(Wh.T)
dWh = prev_h.T.dot(dnext_h*dtanh)
db = np.sum(dnext_h*dtanh,axis=0)
##############################################################################
# END OF YOUR CODE #
##############################################################################
return dx, dprev_h, dWx, dWh, db
单层RNN神经元行为
x（N，T，D）表示N样本的batch中有T个字符向量，每个响亮H维度。
RNN输出有两个方向，一个向上一层（输出层），一个向同层下一个时序，所以反向传播时两个梯度需要相加，输出层梯度可以直接求出（或是上一层中递归求出），所以使用dh(N,T,H)保存好，而同层时序梯度必须在同层中递归计算。
正向传播
def rnn_forward(x, h0, Wx, Wh, b):
"""
Run a vanilla RNN forward on an entire sequence of data. We assume an input
sequence composed of T vectors, each of dimension D. The RNN uses a hidden
size of H, and we work over a minibatch containing N sequences. After running
the RNN forward, we return the hidden states for all timesteps. Inputs:
- x: Input data for the entire timeseries, of shape (N, T, D).
- h0: Initial hidden state, of shape (N, H)
- Wx: Weight matrix for input-to-hidden connections, of shape (D, H)
- Wh: Weight matrix for hidden-to-hidden connections, of shape (H, H)
- b: Biases of shape (H,) Returns a tuple of:
- h: Hidden states for the entire timeseries, of shape (N, T, H).
- cache: Values needed in the backward pass
"""
h, cache = None, None
##############################################################################
# TODO: Implement forward pass for a vanilla RNN running on a sequence of #
# input data. You should use the rnn_step_forward function that you defined #
# above. #
##############################################################################
N, T, D = x.shape
_, H = h0.shape
h = np.zeros((N,T,H))
h_next = h0
cache = []
for i in range(T):
h[:,i,:], cache_next = rnn_step_forward(x[:,i,:], h_next, Wx, Wh, b)
h_next = h[:,i,:]
cache.append(cache_next)
##############################################################################
# END OF YOUR CODE #
##############################################################################
return h, cache
单层RNN反向传播
def rnn_backward(dh, cache):
"""
Compute the backward pass for a vanilla RNN over an entire sequence of data. Inputs:
- dh: Upstream gradients of all hidden states, of shape (N, T, H) Returns a tuple of:
- dx: Gradient of inputs, of shape (N, T, D)
- dh0: Gradient of initial hidden state, of shape (N, H)
- dWx: Gradient of input-to-hidden weights, of shape (D, H)
- dWh: Gradient of hidden-to-hidden weights, of shape (H, H)
- db: Gradient of biases, of shape (H,)
"""
dx, dh0, dWx, dWh, db = None, None, None, None, None
##############################################################################
# TODO: Implement the backward pass for a vanilla RNN running an entire #
# sequence of data. You should use the rnn_step_backward function that you #
# defined above. #
##############################################################################
x, Wx, Wh, prev_h, next_h = cache[-1]
_, D = x.shape
N, T, H = dh.shape
dx = np.zeros((N,T,D))
dh0 = np.zeros((N,H))
dWx = np.zeros((D,H))
dWh = np.zeros((H,H))
db = np.zeros(H)
dprev_h_ = np.zeros((N,H))
for i in range(T-1,-1,-1):
dx_, dprev_h_, dWx_, dWh_, db_ = rnn_step_backward(dh[:,i,:] + dprev_h_, cache.pop())
dx[:,i,:] = dx_
dh0 = dprev_h_
dWx += dWx_
dWh += dWh_
db += db_
##############################################################################
# END OF YOUR CODE #
##############################################################################
return dx, dh0, dWx, dWh, db
图像标注过程理解
正向传播流程如下，
几个有意思的点
字符和向量的映射
涉及两个映射，
- 一个是caption_in到输出节点维度向量的映射，映射矩阵是需要学习的参数
- 一个是输出节点向量到字符的映射，这里面有专门的映射函数，输出节点本身是变化的（被学习的）
第一个映射：
caption_in和caption_out是输入和标准（caption_in=caption[:-1],caption_out=caption[1:]），不考虑batch的话是一维数组，通过We矩阵可以映射到字符向量空间，转换以及反向传播过程如下，
def word_embedding_forward(x, W):
"""
Forward pass for word embeddings. We operate on minibatches of size N where
each sequence has length T. We assume a vocabulary of V words, assigning each
to a vector of dimension D. Inputs:
- x: Integer array of shape (N, T) giving indices of words. Each element idx
of x muxt be in the range 0 <= idx < V.
- W: Weight matrix of shape (V, D) giving word vectors for all words. Returns a tuple of:
- out: Array of shape (N, T, D) giving word vectors for all input words.
- cache: Values needed for the backward pass
""" out = W[x, :]
cache = (W, x) return out, cache
反向传播注意，这不是个标准意义上的链式传播的门，按照逻辑分析这个映射过程的梯度是叠加的，注意函数np.func.at()的用法
def word_embedding_backward(dout, cache):
"""
Backward pass for word embeddings. We cannot back-propagate into the words
since they are integers, so we only return gradient for the word embedding
matrix. HINT: Look up the function np.add.at Inputs:
- dout: Upstream gradients of shape (N, T, D)
- cache: Values from the forward pass Returns:
- dW: Gradient of word embedding matrix, of shape (V, D).
"""
W, x = cache
dW = np.zeros_like(W)
#dW[x] += dout # this will not work, see the doc of np.add.at
np.add.at(dW, x, dout) return dW
第二个映射：
正常的多维y=xW计算，
y = x.reshape(x.shape[0], -1).dot(w) + b # 保留N，后面的数据化为一维
这里的y=xW计算，
y = x.reshape(N * T, D).dot(w).reshape(N, T, M) + b # 其实使用y = x.dot(w) + b 效果是一样的，自动广播到最低维度
上面问题不大，问题在求梯度的时候，两者处理有一定差别，注意到这一点的话只要在演算的时候较对好各个维度的值就好了（保证相乘的两项维度可以相乘，而且理论结果维度和算式相符）
情况2的代码，
def affine_forward(x, w, b):
"""
Computes the forward pass for an affine (fully-connected) layer. The input x has shape (N, d_1, ..., d_k) where x[i] is the ith input.
We multiply this against a weight matrix of shape (D, M) where
D = \prod_i d_i Inputs:
x - Input data, of shape (N, d_1, ..., d_k)
w - Weights, of shape (D, M)
b - Biases, of shape (M,) Returns a tuple of:
- out: output, of shape (N, M)
- cache: (x, w, b)
"""
out = x.reshape(x.shape[0], -1).dot(w) + b
cache = (x, w, b)
return out, cache def affine_backward(dout, cache):
"""
Computes the backward pass for an affine layer. Inputs:
- dout: Upstream derivative, of shape (N, M)
- cache: Tuple of:
- x: Input data, of shape (N, d_1, ... d_k)
- w: Weights, of shape (D, M) Returns a tuple of:
- dx: Gradient with respect to x, of shape (N, d1, ..., d_k)
- dw: Gradient with respect to w, of shape (D, M)
- db: Gradient with respect to b, of shape (M,)
"""
x, w, b = cache
dx = dout.dot(w.T).reshape(x.shape)
dw = x.reshape(x.shape[0], -1).T.dot(dout)
db = np.sum(dout, axis=0)
return dx, dw, db
情况2的代码，
def temporal_affine_forward(x, w, b):
"""
Forward pass for a temporal affine layer. The input is a set of D-dimensional
vectors arranged into a minibatch of N timeseries, each of length T. We use
an affine function to transform each of those vectors into a new vector of
dimension M. Inputs:
- x: Input data of shape (N, T, D)
- w: Weights of shape (D, M)
- b: Biases of shape (M,) Returns a tuple of:
- out: Output data of shape (N, T, M)
- cache: Values needed for the backward pass
"""
N, T, D = x.shape
M = b.shape[0]
# out = x.reshape(N * T, D).dot(w).reshape(N, T, M) + b
out = x.dot(w) + b
cache = x, w, b, out
return out, cache def temporal_affine_backward(dout, cache):
"""
Backward pass for temporal affine layer. Input:
- dout: Upstream gradients of shape (N, T, M)
- cache: Values from forward pass Returns a tuple of:
- dx: Gradient of input, of shape (N, T, D)
- dw: Gradient of weights, of shape (D, M)
- db: Gradient of biases, of shape (M,)
"""
x, w, b, out = cache
N, T, D = x.shape
M = b.shape[0] #dx = dout.reshape(N * T, M).dot(w.T).reshape(N, T, D)
#dw = dout.reshape(N * T, M).T.dot(x.reshape(N * T, D)).T
dx = dout.dot(w.T)
dw = x.reshape(N * T, D).T.dot(dout.reshape(N * T, M))
db = dout.sum(axis=(0, 1)) return dx, dw, db
进行一次向前传播&向后传播的测试函数，用于训练
def loss(self, features, captions):
"""
Compute training-time loss for the RNN. We input image features and
ground-truth captions for those images, and use an RNN (or LSTM) to compute
loss and gradients on all parameters. Inputs:
- features: Input image features, of shape (N, D)
- captions: Ground-truth captions; an integer array of shape (N, T) where
each element is in the range 0 <= y[i, t] < V Returns a tuple of:
- loss: Scalar loss
- grads: Dictionary of gradients parallel to self.params
"""
# Cut captions into two pieces: captions_in has everything but the last word
# and will be input to the RNN; captions_out has everything but the first
# word and this is what we will expect the RNN to generate. These are offset
# by one relative to each other because the RNN should produce word (t+1)
# after receiving word t. The first element of captions_in will be the START
# token, and the first element of captions_out will be the first word.
captions_in = captions[:, :-1]
captions_out = captions[:, 1:] # You'll need this
mask = (captions_out != self._null) # Weight and bias for the affine transform from image features to initial
# hidden state
W_proj, b_proj = self.params['W_proj'], self.params['b_proj'] # Word embedding matrix
W_embed = self.params['W_embed'] # Input-to-hidden, hidden-to-hidden, and biases for the RNN
Wx, Wh, b = self.params['Wx'], self.params['Wh'], self.params['b'] # Weight and bias for the hidden-to-vocab transformation.
W_vocab, b_vocab = self.params['W_vocab'], self.params['b_vocab'] loss, grads = 0.0, {}
############################################################################
# TODO: Implement the forward and backward passes for the CaptioningRNN. #
# In the forward pass you will need to do the following: #
# (1) Use an affine transformation to compute the initial hidden state #
# from the image features. This should produce an array of shape (N, H)#
# (2) Use a word embedding layer to transform the words in captions_in #
# from indices to vectors, giving an array of shape (N, T, W). #
# (3) Use either a vanilla RNN or LSTM (depending on self.cell_type) to #
# process the sequence of input word vectors and produce hidden state #
# vectors for all timesteps, producing an array of shape (N, T, H). #
# (4) Use a (temporal) affine transformation to compute scores over the #
# vocabulary at every timestep using the hidden states, giving an #
# array of shape (N, T, V). #
# (5) Use (temporal) softmax to compute loss using captions_out, ignoring #
# the points where the output word is <NULL> using the mask above. #
# #
# In the backward pass you will need to compute the gradient of the loss #
# with respect to all model parameters. Use the loss and grads variables #
# defined above to store loss and gradients; grads[k] should give the #
# gradients for self.params[k]. #
############################################################################
captions_in_emb, emb_cache = word_embedding_forward(captions_in, W_embed) h_0, feature_cache = affine_forward(features, W_proj, b_proj)
h, rnn_cache = rnn_forward(captions_in_emb, h_0, Wx, Wh, b)
temporal_out, temporal_cache = temporal_affine_forward(h, W_vocab, b_vocab) loss, dout = temporal_softmax_loss(temporal_out, captions_out, mask) dtemp, grads['W_vocab'], grads['b_vocab'] = temporal_affine_backward(dout, temporal_cache)
drnn, dh0, grads['Wx'], grads['Wh'], grads['b'] = rnn_backward(dtemp, rnn_cache)
dfeatures, grads['W_proj'], grads['b_proj'] = affine_backward(dh0, feature_cache) grads['W_embed'] = word_embedding_backward(drnn, emb_cache) return loss, grads
用于预测的函数
补充一下 RNN 是怎么生成一句话的，一般 RNN 的训练（train）和预测（Inference or test）是不一样的，因为训练时有 label，每时刻的输入都是 ground-truth 的单词；而预测只能把自己上时刻的输出（就是预测概率最大的那个单词）当做输入，顶多只是取前几个 sampling 一下。对比如下：
- Train
- 把 ground-truth word 当做 RNN 的输入，也有把这种做法叫做 teacher forcing 的
- 比较常见的做法
- Inference
- 把 RNN 上时刻的输出，当做下时刻的输入。
- 比如训练的时候也这样做，就会比较难训练
def sample(self, features, max_length=30):
"""
Run a test-time forward pass for the model, sampling captions for input
feature vectors. At each timestep, we embed the current word, pass it and the previous hidden
state to the RNN to get the next hidden state, use the hidden state to get
scores for all vocab words, and choose the word with the highest score as
the next word. The initial hidden state is computed by applying an affine
transform to the input image features, and the initial word is the <START>
token. For LSTMs you will also have to keep track of the cell state; in that case
the initial cell state should be zero. Inputs:
- features: Array of input image features of shape (N, D).
- max_length: Maximum length T of generated captions. Returns:
- captions: Array of shape (N, max_length) giving sampled captions,
where each element is an integer in the range [0, V). The first element
of captions should be the first sampled word, not the <START> token.
"""
N = features.shape[0]
captions = self._null * np.ones((N, max_length), dtype=np.int32) # Unpack parameters
W_proj, b_proj = self.params['W_proj'], self.params['b_proj']
W_embed = self.params['W_embed']
Wx, Wh, b = self.params['Wx'], self.params['Wh'], self.params['b']
W_vocab, b_vocab = self.params['W_vocab'], self.params['b_vocab'] ###########################################################################
# TODO: Implement test-time sampling for the model. You will need to #
# initialize the hidden state of the RNN by applying the learned affine #
# transform to the input image features. The first word that you feed to #
# the RNN should be the <START> token; its value is stored in the #
# variable self._start. At each timestep you will need to do to: #
# (1) Embed the previous word using the learned word embeddings #
# (2) Make an RNN step using the previous hidden state and the embedded #
# current word to get the next hidden state. #
# (3) Apply the learned affine transformation to the next hidden state to #
# get scores for all words in the vocabulary #
# (4) Select the word with the highest score as the next word, writing it #
# to the appropriate slot in the captions variable #
# #
# For simplicity, you do not need to stop generating after an <END> token #
# is sampled, but you can if you want to. #
# #
# HINT: You will not be able to use the rnn_forward or lstm_forward #
# functions; you'll need to call rnn_step_forward or lstm_step_forward in #
# a loop. #
########################################################################### # initialize the hidden state of the RNN by applying the learned affine transform to the input image features.
prev_h, _ = affine_forward(features, W_proj, b_proj)
# The first word that you feed to the RNN should be the <START> token
x = np.array([self._start for i in range(N)]) # 输出N个self._start
captions[:, 0] = self._start for t in range(1, max_length):
x_emb, _ = word_embedding_forward(x, W_embed)
next_h, cache = rnn_step_forward(x_emb, prev_h, Wx, Wh, b)
prev_h = next_h
vocab_out, vocab_cache = affine_forward(next_h, W_vocab, b_vocab)
x = vocab_out.argmax(1)
captions[:, t] = x ############################################################################
# END OF YOUR CODE #
############################################################################
return captions