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What does the second derivative of a function measure?
The second derivative measures the instantaneous rate of change of the first derivative. The sign of the second derivative tells us whether the slope of the tangent line to f is increasing or decreasing. In other words, the second derivative tells us the rate of change of the rate of change of the original function.
How do you know if the second derivative is positive or negative from a graph?
The second derivative tells whether the curve is concave up or concave down at that point. If the second derivative is positive at a point, the graph is bending upwards at that point. Similarly if the second derivative is negative, the graph is concave down.
Why is the derivative of x2 2x?
Since f(x) = x², the ‘x’ on the x-axis results in an x² on the y-axis. The ((x+δ)² — x²) / δ) represents that we are measuring the change on the y-axis and dividing it by the change on the x-axis. Then we simplify the question, which results in 2x. We have now proved that the differential of x² is equal to 2x.
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 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.
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 .