Gradient Question: Why Does Direction Maximize Function?

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In summary, the gradient vector of a function is always perpendicular to the level curve and the level surface. This means that the direction of the gradient is also the direction of maximum change for the function. This can be seen by considering the directional derivative of the function and setting it equal to 0, which shows that the direction of maximum change is in the direction of the gradient. Thus, the gradient vector is a useful tool for determining the direction of maximum change and optimizing a function.
  • #1
kidsasd987
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This is a bit counterintuitive to me that the gradient vector is always normal to the level curve
and the level surface.

lets say we have a function f(x,y)=z

then the gradient is,

f(x,y) partial derivative with respect to x*i +f(x,y) partial derivative with respect to y*j


what we actually get is,

dz/dx*i+dz/dy*j=grad_f(x,y)

then,

[(dz/dx*i)+(dz/dy)*j]/sqrt((dz/dx*i)^2+(dz/dy*j)^2)

this is always perpendicular to the level curve. But why does that direction always maximize the function?
 
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  • #2
Think of going up a hill. You always have to go perpendicular to the level curve. You're not going to be winding around the side if you want the most direct route.
 
  • #3
Another way of looking at it: if z= f(x, y) then the "directional derivative", [itex]f_{\theta}[/itex], the rate of change of f as you move in the direction that makes angle [itex]\theta[/itex] with the positive x axis, is [itex]\nabla f\cdot \vec{e}_{\theta}[/itex] where [itex]\vec{e}_{\theta}[/itex] is the unit vector in that direction, which, in turn, is equal to [tex]\left(\frac{\partial f}{\partial x}\vec{i}+ \frac{\partial f}{\partial y}\vec{j}\right)\cdot \left(\vec{i}cos(\theta)+ \vec{j}sin(\theta)\right)= cos(\theta)\frac{\partial f}{\partial x}+ sin(\theta)\frac{\partial f}{\partial x}[/tex].

To maximize (or minimize) that with respect to [itex]\theta[/itex], take the derivative with respect to [itex]\theta[/itex], set it equal to 0 and solve for [itex]\theta[/itex]. you will get
[tex]tan(\theta)= \frac{\frac{\partial f}{\partial y}}{\frac{\partial f}{\partial x}}[/tex]
showing that [itex]\theta[/itex] for the maximum is, indeed, the direction in which [itex]\nabla f[/itex] points while the minimum is in the opposite direction.
 

1. What is gradient descent and why is it important?

Gradient descent is an optimization algorithm used in machine learning and other mathematical models to find the minimum value of a function. It is important because it allows us to efficiently optimize complex models and find the best set of parameters to minimize the overall error or loss.

2. How does gradient descent work?

Gradient descent works by calculating the gradient (slope) of the function at a given point and taking small steps in the direction that decreases the value of the function. This process is repeated until the minimum value is reached or a certain threshold is met.

3. Why does direction matter in gradient descent?

Direction matters in gradient descent because it determines the path that the algorithm takes to reach the minimum value of the function. The direction is determined by the gradient, which points towards the steepest descent at that point.

4. Can gradient descent get stuck in local minima?

Yes, gradient descent can get stuck in local minima, especially in complex functions with many local minima. This means that the algorithm may not be able to find the global minimum and instead settles for a local minimum.

5. How do we know when to stop gradient descent?

Stopping criteria for gradient descent can vary depending on the problem and the desired accuracy. Some common methods include setting a maximum number of iterations, monitoring the change in the value of the function, or setting a threshold for the gradient magnitude. When the criteria are met, the algorithm stops and the current values for the parameters are considered the solution.

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