deflstm_cell_forward(xt, a_prev, c_prev, parameters):"""
Implement a single forward step of the LSTM-cell as described in Figure (4)
Arguments:
xt -- your input data at timestep "t", numpy array of shape (n_x, m).
a_prev -- Hidden state at timestep "t-1", numpy array of shape (n_a, m)
c_prev -- Memory state at timestep "t-1", numpy array of shape (n_a, m)
parameters -- python dictionary containing:
Wf -- Weight matrix of the forget gate, numpy array of shape (n_a, n_a + n_x)
bf -- Bias of the forget gate, numpy array of shape (n_a, 1)
Wi -- Weight matrix of the save gate, numpy array of shape (n_a, n_a + n_x)
bi -- Bias of the save gate, numpy array of shape (n_a, 1)
Wc -- Weight matrix of the first "tanh", numpy array of shape (n_a, n_a + n_x)
bc -- Bias of the first "tanh", numpy array of shape (n_a, 1)
Wo -- Weight matrix of the focus gate, numpy array of shape (n_a, n_a + n_x)
bo -- Bias of the focus gate, numpy array of shape (n_a, 1)
Wy -- Weight matrix relating the hidden-state to the output, numpy array of shape (n_y, n_a)
by -- Bias relating the hidden-state to the output, numpy array of shape (n_y, 1)
Returns:
a_next -- next hidden state, of shape (n_a, m)
c_next -- next memory state, of shape (n_a, m)
yt_pred -- prediction at timestep "t", numpy array of shape (n_y, m)
cache -- tuple of values needed for the backward pass, contains (a_next, c_next, a_prev, c_prev, xt, parameters)
Note: ft/it/ot stand for the forget/update/output gates, cct stands for the candidate value (c tilda),
c stands for the memory value
"""# Retrieve parameters from "parameters"
Wf = parameters["Wf"]
bf = parameters["bf"]
Wi = parameters["Wi"]
bi = parameters["bi"]
Wc = parameters["Wc"]
bc = parameters["bc"]
Wo = parameters["Wo"]
bo = parameters["bo"]
Wy = parameters["Wy"]
by = parameters["by"]
# Retrieve dimensions from shapes of xt and Wy
n_x, m = xt.shape
n_y, n_a = Wy.shape
# Concatenate a_prev and xt (≈3 lines)
concat = np.zeros((n_x+n_a,m))
concat[: n_a, :] = a_prev
concat[n_a :, :] = xt
# Compute values for ft, it, cct, c_next, ot, a_next using the formulas given figure (4) (≈6 lines)
ft = sigmoid(np.dot(Wf,concat)+bf)
it = sigmoid(np.dot(Wi,concat)+bi)
cct = np.tanh(np.dot(Wc,concat)+bc)
c_next = ft*c_prev + it*cct
ot = sigmoid(np.dot(Wo,concat)+bo)
a_next = ot*np.tanh(c_next)
# Compute prediction of the LSTM cell (≈1 line)
yt_pred = softmax(np.dot(Wy, a_next) + by)
# store values needed for backward propagation in cache
cache = (a_next, c_next, a_prev, c_prev, ft, it, cct, ot, xt, parameters)
return a_next, c_next, yt_pred, cache
back propagation
deflstm_cell_backward(da_next, dc_next, cache):"""
Implement the backward pass for the LSTM-cell (single time-step).
Arguments:
da_next -- Gradients of next hidden state, of shape (n_a, m)
dc_next -- Gradients of next cell state, of shape (n_a, m)
cache -- cache storing information from the forward pass
Returns:
gradients -- python dictionary containing:
dxt -- Gradient of input data at time-step t, of shape (n_x, m)
da_prev -- Gradient w.r.t. the previous hidden state, numpy array of shape (n_a, m)
dc_prev -- Gradient w.r.t. the previous memory state, of shape (n_a, m, T_x)
dWf -- Gradient w.r.t. the weight matrix of the forget gate, numpy array of shape (n_a, n_a + n_x)
dWi -- Gradient w.r.t. the weight matrix of the input gate, numpy array of shape (n_a, n_a + n_x)
dWc -- Gradient w.r.t. the weight matrix of the memory gate, numpy array of shape (n_a, n_a + n_x)
dWo -- Gradient w.r.t. the weight matrix of the save gate, numpy array of shape (n_a, n_a + n_x)
dbf -- Gradient w.r.t. biases of the forget gate, of shape (n_a, 1)
dbi -- Gradient w.r.t. biases of the update gate, of shape (n_a, 1)
dbc -- Gradient w.r.t. biases of the memory gate, of shape (n_a, 1)
dbo -- Gradient w.r.t. biases of the save gate, of shape (n_a, 1)
"""# Retrieve information from "cache"
(a_next, c_next, a_prev, c_prev, ft, it, cct, ot, xt, parameters) = cache
# Retrieve dimensions from xt's and a_next's shape (≈2 lines)
n_x, m = xt.shape
n_a, m = a_next.shape
# Compute gates related derivatives, you can find their values can be found by looking carefully at equations (7) to (10) (≈4 lines)
dot = da_next * np.tanh(c_next) * ot * (1 - ot)
dcct = (dc_next * it + ot * (1 - np.square(np.tanh(c_next))) * it * da_next) * (1 - np.square(cct))
dit = (dc_next * cct + ot * (1 - np.square(np.tanh(c_next))) * cct * da_next) * it * (1 - it)
dft = (dc_next * c_prev + ot *(1 - np.square(np.tanh(c_next))) * c_prev * da_next) * ft * (1 - ft)
# Compute parameters related derivatives. Use equations (11)-(14) (≈8 lines)
dWf = np.dot(dft,np.concatenate((a_prev, xt), axis=0).T)
dWi = np.dot(dit,np.concatenate((a_prev, xt), axis=0).T)
dWc = np.dot(dcct,np.concatenate((a_prev, xt), axis=0).T)
dWo = np.dot(dot,np.concatenate((a_prev, xt), axis=0).T)
dbf = np.sum(dft, axis=1 ,keepdims = True)
dbi = np.sum(dit, axis=1, keepdims = True)
dbc = np.sum(dcct, axis=1, keepdims = True)
dbo = np.sum(dot, axis=1, keepdims = True)
# Compute derivatives w.r.t previous hidden state, previous memory state and input. Use equations (15)-(17). (≈3 lines)
da_prev = np.dot(parameters['Wf'][:,:n_a].T,dft)+np.dot(parameters['Wi'][:,:n_a].T,dit)+np.dot(parameters['Wc'][:,:n_a].T,dcct)+np.dot(parameters['Wo'][:,:n_a].T,dot)
dc_prev = dc_next*ft+ot*(1-np.square(np.tanh(c_next)))*ft*da_next
dxt = np.dot(parameters['Wf'][:,n_a:].T,dft)+np.dot(parameters['Wi'][:,n_a:].T,dit)+np.dot(parameters['Wc'][:,n_a:].T,dcct)+np.dot(parameters['Wo'][:,n_a:].T,dot)
# parameters['Wf'][:, :n_a].T 每一行的 第 0 到 n_a-1 列的数据取出来# parameters['Wf'][:, n_a:].T 每一行的 第 n_a 到最后列的数据取出来# Save gradients in dictionary
gradients = {"dxt": dxt, "da_prev": da_prev, "dc_prev": dc_prev, "dWf": dWf,"dbf": dbf, "dWi": dWi,"dbi": dbi,
"dWc": dWc,"dbc": dbc, "dWo": dWo,"dbo": dbo}
return gradients
