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.

## 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 $$\boldsymbol{g}_t$$:

$\boldsymbol{v}_t \leftarrow \beta_1 \boldsymbol{v}_{t-1} + (1 - \beta_1) \boldsymbol{g}_t.$

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 $$\boldsymbol{s}_t$$:

$\boldsymbol{s}_t \leftarrow \beta_2 \boldsymbol{s}_{t-1} + (1 - \beta_2) \boldsymbol{g}_t \odot \boldsymbol{g}_t.$

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 $$\boldsymbol{s}_t$$:

$\hat{\boldsymbol{v}}_t \leftarrow \frac{\boldsymbol{v}_t}{1 - \beta_1^t},$
$\hat{\boldsymbol{s}}_t \leftarrow \frac{\boldsymbol{s}_t}{1 - \beta_2^t}.$

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.

$\boldsymbol{g}_t' \leftarrow \frac{\eta \hat{\boldsymbol{v}}_t}{\sqrt{\hat{\boldsymbol{s}}_t + \epsilon}},$

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:

$\boldsymbol{x}_t \leftarrow \boldsymbol{x}_{t-1} - \boldsymbol{g}_t'.$

## 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 gluonbook as gb
from mxnet import nd

features, labels = gb.get_data_ch7()

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))

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]:

gb.train_ch7(adam, init_adam_states(), {'lr': 0.01, 't': 1}, features, labels)

loss: 0.243957, 0.449113 sec per epoch


## Implementation with Gluon¶

From the Trainer instance of the algorithm named “adam”, we can implement Adam with Gluon.

In [3]:

gb.train_gluon_ch7('adam', {'learning_rate': 0.01}, features, labels)

loss: 0.242933, 0.182535 sec per epoch


## Summary¶

• Created on the basis of RMSProp, Adam also uses EWMA on the mini-batch stochastic gradient