Negative Gradient and Gradient Descent Method

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What is "Negative Gradient" ? and what is "Gradient Descent Method" ? What is the difference and relationship between them ?
What is the benefit each of them ?
 
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jtbell
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Do you know how to calculate the gradient of a function, in vector calculus, and what it means geometrically?
 
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Do you know how to calculate the gradient of a function, in vector calculus, and what it means geometrically?
thanks a lot

No, I am not so good in math , I know this is a stupid question, but if you can answer it, I will appreciated that
 
  • #4
jtbell
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You can find out how to calculate the gradient in any calculus textbook that includes multivariable calculus (vector calculus), and probably on hundreds of web sites including Wikipedia (http://en.wikipedia.org/wiki/Gradient), so I won't do that here. I'll just talk about the meaning of the gradient.

Suppose you have a function h(x,y) that tells you the elevation (height) of the land at horizontal coordinates (x,y). The gradient of this function, ##\vec \nabla h(x,y)##, is a vector function that gives you a vector for each point (x,y). This gradient vector points in the direction of steepest uphill slope, and its magnitude is the value of that slope (like the slope of a straight-line graph).

The opposite direction, the negative gradient ##-\vec \nabla h(x,y)## tells you the direction of steepest downhill slope.

If you want to find the location (x,y) at which h(x,y) is minimum (e.g. the bottom of a valley), one way is to follow the negative gradient vector downhill. Calculate ##-\vec \nabla h## at your starting point (x0, y0), take a step downhill in that direction to the point (x1, y1), calculate ##-\vec \nabla h## at that point, take a step in the new downhill direction, etc. Keep going until you find yourself at a higher elevation at the end of a step, indicating that you have gone past the bottom.

http://en.wikipedia.org/wiki/Gradient_descent
 

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