深度学习优化算法实现(Momentum, Adam)
目录
- Momentum
- 初始化
- 更新参数
- Adam
- 初始化
- 更新参数
除了常见的梯度下降法外，还有几种比较通用的优化算法；表现都优于梯度下降法。本文只记录完成吴恩达深度学习作业时遇到的Momentum和Adam算法，而且只有简要的代码。具体原理请看深度学习优化算法解析(Momentum, RMSProp, Adam),比较具体的说明了吴恩达版本的三种优化算法的讲述！
Momentum
初始化
def initialize_velocity(parameters):
"""
Initializes the velocity as a python dictionary with:
- keys: "dW1", "db1", ..., "dWL", "dbL"
- values: numpy arrays of zeros
of the same shape as the corresponding
gradients/parameters.
Arguments:
parameters -- python dictionary containing your parameters.
parameters['W' + str(l)] = Wl
parameters['b' + str(l)] = bl
Returns:
v -- python dictionary containing the current velocity.
v['dW' + str(l)] = velocity of dWl
v['db' + str(l)] = velocity of dbl
"""
L = len(parameters) // 2 # number of layers in the neural networks
v = {}
# Initialize velocity
for l in range(L):
### START CODE HERE ### (approx. 2 lines)
v['dW' + str(l + 1)] =
np.zeros(np.shape(parameters['W' + str(l + 1)]))
v['db' + str(l + 1)] =
np.zeros(np.shape(parameters['b' + str(l + 1)]))
### END CODE HERE ###
return v
更新参数
def update_parameters_with_momentum(parameters, grads,
v, beta, learning_rate):
"""
Update parameters using Momentum
Arguments:
parameters -- python dictionary containing your parameters:
parameters['W' + str(l)] = Wl
parameters['b' + str(l)] = bl
grads -- python dictionary containing your gradients for
each parameters:
grads['dW' + str(l)] = dWl
grads['db' + str(l)] = dbl
v -- python dictionary containing the current velocity:
v['dW' + str(l)] = ...
v['db' + str(l)] = ...
beta -- the momentum hyperparameter, scalar
learning_rate -- the learning rate, scalar
Returns:
parameters -- python dictionary containing your updated parameters
v -- python dictionary containing your updated velocities
"""
L = len(parameters) // 2 # number of layers in the neural networks
# Momentum update for each parameter
for l in range(L):
### START CODE HERE ### (approx. 4 lines)
# compute velocities
v['dW' + str(l + 1)] =
beta * v['dW' + str(l + 1)] +
(1 - beta) * grads['dW' + str(l + 1)]
v['db' + str(l + 1)] =
beta * v['db' + str(l + 1)] +
(1 - beta) * grads['db' + str(l + 1)]
# update parameters
parameters['W' + str(l + 1)] +=
-learning_rate * v['dW' + str(l + 1)]
parameters['b' + str(l + 1)] +=
-learning_rate * v['db' + str(l + 1)]
### END CODE HERE ###
return parameters, v
Adam
初始化
def initialize_adam(parameters) :
"""
Initializes v and s as two python dictionaries with:
- keys: "dW1", "db1", ..., "dWL", "dbL"
- values: numpy arrays of zeros of the same shape as the
corresponding gradients/parameters.
Arguments:
parameters -- python dictionary containing your parameters.
parameters["W" + str(l)] = Wl
parameters["b" + str(l)] = bl
Returns:
v -- python dictionary that will contain the exponentially weighted
average of the gradient.
v["dW" + str(l)] = ...
v["db" + str(l)] = ...
s -- python dictionary that will contain the exponentially weighted
average of the squared gradient.
s["dW" + str(l)] = ...
s["db" + str(l)] = ...
"""
L = len(parameters) // 2 # number of layers in the neural networks
v = {}
s = {}
# Initialize v, s. Input: "parameters". Outputs: "v, s".
for l in range(L):
### START CODE HERE ### (approx. 4 lines)
v['dW' + str(l + 1)] =
np.zeros(np.shape(parameters['W' + str(l + 1)]))
v['db' + str(l + 1)] =
np.zeros(np.shape(parameters['b' + str(l + 1)]))
s['dW' + str(l + 1)] =
np.zeros(np.shape(parameters['W' + str(l + 1)]))
s['db' + str(l + 1)] =
np.zeros(np.shape(parameters['b' + str(l + 1)]))
### END CODE HERE ###
return v, s
更新参数
def update_parameters_with_adam(parameters, grads, v, s, t,
learning_rate = 0.01,beta1 = 0.9, beta2 = 0.999, epsilon = 1e-8):
"""
Update parameters using Adam
Arguments:
parameters -- python dictionary containing your parameters:
parameters['W' + str(l)] = Wl
parameters['b' + str(l)] = bl
grads -- python dictionary containing your gradients
for each parameters:
grads['dW' + str(l)] = dWl
grads['db' + str(l)] = dbl
v -- Adam variable, moving average of the first gradient,
python dictionary
s -- Adam variable, moving average of the squared gradient,
python dictionary
learning_rate -- the learning rate, scalar.
beta1 -- Exponential decay hyperparameter for
the first moment estimates
beta2 -- Exponential decay hyperparameter for
the second moment estimates
epsilon -- hyperparameter preventing division
by zero in Adam updates
Returns:
parameters -- python dictionary containing your updated parameters
v -- Adam variable, moving average of the first gradient
, python dictionary
s -- Adam variable, moving average of the squared gradient
, python dictionary
"""
L = len(parameters) // 2 # number of layers in the neural networks
# Initializing first moment estimate,python dictionary
v_corrected = {}
# Initializing second moment estimate,python dictionary
s_corrected = {}
# Perform Adam update on all parameters
for l in range(L):
### START CODE HERE ### (approx. 2 lines)
v['dW' + str(l + 1)] =
beta1 * v['dW' + str(l + 1)]
+ (1 - beta1) * grads['dW' + str(l + 1)]
v['db' + str(l + 1)] =
beta1 * v['db' + str(l + 1)]
+ (1 - beta1) * grads['db' + str(l + 1)]
### END CODE HERE ###
### START CODE HERE ### (approx. 2 lines)
v_corrected['dW' + str(l + 1)] =
v['dW' + str(l + 1)] / (1 - beta1 ** t)
v_corrected['db' + str(l + 1)] =
v['db' + str(l + 1)] / (1 - beta1 ** t)
### END CODE HERE ###
### START CODE HERE ### (approx. 2 lines)
s['dW' + str(l + 1)] =
beta2 * s['dW' + str(l + 1)]
+ (1 - beta2) * np.square(grads['dW' + str(l + 1)])
s['db' + str(l + 1)] =
beta2 * s['db' + str(l + 1)]
+ (1 - beta2) * np.square(grads['db' + str(l + 1)])
### END CODE HERE ###
### START CODE HERE ### (approx. 2 lines)
s_corrected['dW' + str(l + 1)] =
s['dW' + str(l + 1)] / (1 - beta2 ** t)
s_corrected['db' + str(l + 1)] =
s['db' + str(l + 1)] / (1 - beta2 ** t)
### END CODE HERE ###
### START CODE HERE ### (approx. 2 lines)
parameters['W' + str(l + 1)] +=
-learning_rate * v_corrected['dW' + str(l + 1)]
/ (np.sqrt(s['dW' + str(l + 1)]) + epsilon)
parameters['b' + str(l + 1)] +=
-learning_rate * v_corrected['db' + str(l + 1)]
/ (np.sqrt(s['db' + str(l + 1)]) + epsilon)
### END CODE HERE ###
return parameters, v, s