7.8. Adam¶
Created on the basis of RMSProp, Adam also uses EWMA on the mini-batch stochastic gradient[1]. Here, we are going to introduce this algorithm.
7.8.1. The Algorithm¶
Adam uses the momentum variable \(\boldsymbol{v}_t\) and variable \(\boldsymbol{s}_t\), which is an EWMA on the squares of elements in the mini-batch stochastic gradient from RMSProp, and initializes each element of the variables to 0 at time step 0. Given the hyperparameter \(0 \leq \beta_1 < 1\) (the author of the algorithm suggests a value of 0.9), the momentum variable \(\boldsymbol{v}_t\) at time step \(t\) is the EWMA of the mini-batch stochastic gradient \(\boldsymbol{g}_t\):
Just as in RMSProp, given the hyperparameter \(0 \leq \beta_2 < 1\) (the author of the algorithm suggests a value of 0.999), After taken the squares of elements in the mini-batch stochastic gradient, find \(\boldsymbol{g}_t \odot \boldsymbol{g}_t\) and perform EWMA on it to obtain \(\boldsymbol{s}_t\):
Since we initialized elements in \(\boldsymbol{v}_0\) and \(\boldsymbol{s}_0\) to 0, we get \(\boldsymbol{v}_t = (1-\beta_1) \sum_{i=1}^t \beta_1^{t-i} \boldsymbol{g}_i\) at time step \(t\). Sum the mini-batch stochastic gradient weights from each previous time step to get \((1-\beta_1) \sum_{i=1}^t \beta_1^{t-i} = 1 - \beta_1^t\). Notice that when \(t\) is small, the sum of the mini-batch stochastic gradient weights from each previous time step will be small. For example, when \(\beta_1 = 0.9\), \(\boldsymbol{v}_1 = 0.1\boldsymbol{g}_1\). To eliminate this effect, for any time step \(t\), we can divide \(\boldsymbol{v}_t\) by \(1 - \beta_1^t\), so that the sum of the mini-batch stochastic gradient weights from each previous time step is 1. This is also called bias correction. In the Adam algorithm, we perform bias corrections for variables \(\boldsymbol{v}_t\) and \(\boldsymbol{s}_t\):
Next, the Adam algorithm will use the bias-corrected variables \(\hat{\boldsymbol{v}}_t\) and \(\hat{\boldsymbol{s}}_t\) from above to re-adjust the learning rate of each element in the model parameters using element operations.
Here, \(eta\) is the learning rate while \(\epsilon\) is a constant added to maintain numerical stability, such as \(10^{-8}\). Just as for Adagrad, RMSProp, and Adadelta, each element in the independent variable of the objective function has its own learning rate. Finally, use \(\boldsymbol{g}_t'\) to iterate the independent variable:
7.8.2. Implementation from Scratch¶
We use the formula from the algorithm to implement Adam. Here, time step
\(t\) uses hyperparams to input parameters to the adam
function.
In [1]:
%matplotlib inline
import d2l
from mxnet import nd
features, labels = d2l.get_data_ch7()
def init_adam_states():
v_w, v_b = nd.zeros((features.shape[1], 1)), nd.zeros(1)
s_w, s_b = nd.zeros((features.shape[1], 1)), nd.zeros(1)
return ((v_w, s_w), (v_b, s_b))
def adam(params, states, hyperparams):
beta1, beta2, eps = 0.9, 0.999, 1e-6
for p, (v, s) in zip(params, states):
v[:] = beta1 * v + (1 - beta1) * p.grad
s[:] = beta2 * s + (1 - beta2) * p.grad.square()
v_bias_corr = v / (1 - beta1 ** hyperparams['t'])
s_bias_corr = s / (1 - beta2 ** hyperparams['t'])
p[:] -= hyperparams['lr'] * v_bias_corr / (s_bias_corr.sqrt() + eps)
hyperparams['t'] += 1
Use Adam to train the model with a learning rate of \(0.01\).
In [2]:
d2l.train_ch7(adam, init_adam_states(), {'lr': 0.01, 't': 1}, features, labels)
loss: 0.246920, 0.375252 sec per epoch
7.8.3. Implementation with Gluon¶
From the Trainer instance of the algorithm named “adam”, we can
implement Adam with Gluon.
In [3]:
d2l.train_gluon_ch7('adam', {'learning_rate': 0.01}, features, labels)
loss: 0.245241, 0.266137 sec per epoch
7.8.4. Summary¶
- Created on the basis of RMSProp, Adam also uses EWMA on the mini-batch stochastic gradient
- Adam uses bias correction.
7.8.5. Problems¶
- Adjust the learning rate and observe and analyze the experimental results.
- Some people say that Adam is a combination of RMSProp and momentum. Why do you think they say this?
7.8.6. Reference¶
[1] Kingma, D. P., & Ba, J. (2014). Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980.