A review of gradient descent optimization methods

Posted 2020-11-13 gaoqichao

Suppose we are going to optimize a parameterized function (J( heta)), where ( heta in mathbb{R}^d), for example, ( heta) could be a neural net.

More specifically, we want to (mbox{ minimize } J( heta; mathcal{D})) on dataset (mathcal{D}), where each point in (mathcal{D}) is a pair ((x_i, y_i)).

There are different ways to apply gradient descent.

Let (eta) be the learning rate.

1. Vanilla batch update
   ( heta gets heta - eta abla J( heta; mathcal{D}))
   Note that ( abla J( heta; mathcal{D})) computes the gradient on of the whole dataset (mathcal{D}).
   ```python
   for i in range(n_epochs):
   gradient = compute_gradient(J, theta, D)
   theta = theta - eta * gradient
   eta = eta * 0.95

```
It is obvious that when (mathcal{D}) is too large, this approach is unfeasible.

1. Stochastic Gradient Descent
   Stochastic Gradient, on the other hand, update the parameters example by example.
   ( heta gets heta - eta *J( heta, x_i, y_i)), where ((x_i, y_i) in mathcal{D}).

   for n in range(n_epochs):
   for x_i, y_i in D: 
       gradient=compute_gradient(J, theta, x_i, y_i)
       theta = theta - eta * gradient 
   eta = eta * 0.95 

2. Mini-batch Stochastic Gradient Descent
   Update ( heta) example by example could lead to high variance, the alternative approach is to update ( heta) by mini-batches (M) where (|M| ll |mathcal{D}|).

   for n in range(n_epochs):
   for M in D: 
       gradient = compute_gradient(J, M)
       theta = theta - eta * gradient 
   eta = eta * 0.95

Question? Why decaying the learning rate leads to convergence?