Gradient Descent Optimization Algorithms with Momentum, RMSProp, and Adam

In practice, simple gradient descent (SGD) can face difficulties. Local minimum is rarely a problem because with many features, there hardly is a local minimum where a gradient descent might be trapped in. However, plateau, where gradients are close to 0 is a problem. SGD takes a long time to descend from a plateau.

saddle point

As you can see, there are another optimization algorithms that are faster than SGD.

Here, we will look at Momentum, RMSProp amd Adam.

Momentum

In SGD, only current $dW$ was considered in one gradient descent step($W:=W-\alpha dW$).

However in Momentum, average of previous $dW$s also have impact in current gradient descent step. We calculate ‘exponentially weighted averages’ of $dW$s and $db$s then use them to update $W$ and $b$.

This is Momentum algorithm;

  1. Initialize $v_{dW}$ and $v_{db}$ to 0s.
  2. $v_{dW} = \beta_1 v_{dW} + (1-\beta_1)dW$
  3. $v_{db} = \beta_1 v_{db} + (1-\beta_1)db$
  4. $W = W-\alpha v_{dW}$
  5. $b = b-\alpha v_{db}$

Momentum

Look at the contour line of cost with respect to $W$(x-axis) and $b$(y-axis). If we optimize with SGD, our gradient descent will look like blue line, and when we overtune $\alpha$, it might overshoot like purple line on the left. It is reasonable to gradient descent like red line. We are doing this by Momentum. Average of $db$s is close to 0 but average of $dW$s is large. Thus we take big steps in updating $W$, and small steps in updating $b$.

RMSProp

RMSProp is similar to Momentum but it considers squared average of previous parameters.

This is RMSProp algorithm;

  1. Initialize $s_{dW}$ and $s_{db}$ to 0s.
  2. $s_{dW} = \beta_2 s_{dW} + (1-\beta_2)dW^2$
  3. $s_{db} = \beta_2 s_{db} + (1-\beta_2)db^2$
  4. $W = W-\alpha \frac{dW}{\sqrt{s_{dW}+\epsilon}}$
  5. $b = b-\alpha \frac{db}{\sqrt{s_{db}+\epsilon}}$

Explanations for step 4 and 5.

  • We add $\epsilon$ to the denominator to prevent calculation error from dividing with a value too close to 0.
  • In the above image, average of $db^2$ is large, so $s_{db}$ is large and $\frac{db}{\sqrt{s_{db}+\epsilon}}$ is small. Thus $b$ moves a little. It’s the opposite for $W$.

Adam

In practice in training neural networks, we use Adam optimizer for default.

Adam optimization algorithm is a combination of Momentum and RMSProp.

This is Adam algorithm;

  1. Initialize $s_{dW}, s_{db}, v_{dW}, v_{db}$ to 0s.
  2. $v_{dW}=\beta_1v_W+(1-\beta_1)dW$
  3. $v_{dW}^{corrected}=\frac{v_{dW}}{1-{\beta_1}^t}$
  4. $s_{dW} = \beta_2 s_{dW} + (1-\beta_2)dW^2$
  5. $s_{dW}^{corrected}=\frac{s_{dW}}{1-{\beta_2}^t}$
  6. $W:=W-\alpha \frac{v_{dW}^{corrected}}{\sqrt{s_{dW}+\epsilon}}$
  7. Do the same with $b$.

We do step 3 and 5 to correct bias of exponentially weighted average.

Choice of Hyperparameters

This is the typical value of hyperparameters;

$\beta_1=0.9$

$\beta_2=0.999$

$\epsilon =10^{-8}$

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