How does the derivation of gradient descent work?

How does the derivation of gradient descent work?

In Andrew Ng’s Machine Learning course, there is one small modification to this derivation. We multiply our MSE cost function by 1/2 so that when we take the derivative, the 2s cancel out. Multiplying the cost function by a scalar does not affect the location of its minimum, so we can get away with this.

How to do gradient descent on MSE cost function?

Gradient Descent of MSE. Now that we know how to perform gradient descent on an equation with multiple variables, we can return to looking at gradient descent on our MSE cost function. The MSE cost function is labeled as equation [1.0] below. Taking the derivative of this equation is a little more tricky.

When to use a separate update rule for gradient descent?

When there are multiple variables in the minimization objective, gradient descent defines a separate update rule for each variable. The update rule for θ 1 uses the partial derivative of J with respect to θ 1 .

How does error backpropagation work in gradient descent?

As we’ll see shortly, the process of “backpropagating” the error signal can repeated all the way back to the input layer by successively projecting δk δ k back through wjk w j k, then through the activation function g′ j(zj) g j ′ ( z j) for the hidden layer to give the error signal δj δ j, and so on.

How is gradient descent used in iterative optimization?

Gradient descent is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. To find a local minimum of a function using gradient descent, we take steps proportional to the negative of the gradient (or approximate gradient) of the function at the current point.

Is the learning rate for gradient descent too small?

The only concern with using too small of a learning rate is that you will need to run more iterations of gradient descent, increasing your training time. Note in the above example that gradient descent will never actually converge on the minimum, θ = 0 .

How to find a local minimum of a function using gradient descent?

To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient (or approximate gradient) of the function at the current point.

Is the gradient descent algorithm guaranteed to converge?

We can use fixed learning rate during training without worrying about learning rate decay. It has straight trajectory towards the minimum and it is guaranteed to converge in theory to the global minimum if the loss function is convex and to a local minimum if the loss function is not convex. It has unbiased estimate of gradients.

How to find the derivative of the cost function?

We found that our cost function is: In Part 2, we learned how to find the partial derivative. This is important because there are more than one parameter (variable) in this function that we can tweak. We need to find the derivative of the cost function with respect to both the weights and biases, and partial derivatives come into play.