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.
- 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 \(\mathcal{D}\).
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.
- 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
- 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 \(|M| \ll |\mathcal{D}|\).
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?
why \(\sum_{i=1}^{\infty} \eta_i = \infty\) and \(\sum_{i=1}^{\infty} \eta_i^2 < \infty\) is the condition for convergence? Based on what assumption of \(J(\theta)\)?
