What is the precise intuition behind the gradient being the steepest direction of ascent? Thanks.
motoole2
Try to picture how the gradient would be evaluated for the function shown in this slide.
The first component of the gradient represents how much the value of f(x) changes when increasing the first component of x (aligned with the x_1 axis); in this case, we would observe a large, positive change to f(x). Repeating the same procedure to the second component (aligned with the x_2 axis) would result in a small (or no) change to f(x). The corresponding gradient would therefore be a 2D vector pointing primarily in the direction of x_1.
OtB_BlueBerry
In mathematics, the (total) derivative of $f$ can be defined to be a linear map, which is essentially $\langle\nabla f(x_0), \cdot\rangle$.
A-star
Can we model our interpretation of a gradient for a minimization function?
For example, in supervised learning (ML), we often use a convex cost function and then use gradient descent to find the point of minimum cost.
What is the precise intuition behind the gradient being the steepest direction of ascent? Thanks.
Try to picture how the gradient would be evaluated for the function shown in this slide.
The first component of the gradient represents how much the value of
f(x)changes when increasing the first component ofx(aligned with thex_1axis); in this case, we would observe a large, positive change tof(x). Repeating the same procedure to the second component (aligned with thex_2axis) would result in a small (or no) change tof(x). The corresponding gradient would therefore be a 2D vector pointing primarily in the direction ofx_1.In mathematics, the (total) derivative of $f$ can be defined to be a linear map, which is essentially $\langle\nabla f(x_0), \cdot\rangle$.
Can we model our interpretation of a gradient for a minimization function?
For example, in supervised learning (ML), we often use a convex cost function and then use gradient descent to find the point of minimum cost.