A review of gradient descent optimization methods

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

More specifically, we want to \(\mbox{ minimize } J(\theta; \mathcal{D})\) 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
   \(\theta \gets \theta - \eta \nabla J(\theta; \mathcal{D})\)
   Note that \(\nabla J(\theta; \mathcal{D})\) 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.
   \(\theta \gets \theta - \eta *J(\theta, x_i, y_i)\), 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 \(\theta\) example by example could lead to high variance, the alternative approach is to update \(\theta\) by mini-batches \(M\) where \(|M| << |\mathcal{D}|\).

   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?