Machine learning
Gradient Descent :
Finding a local minimum or maximum of a given function is done using the iterative first-order optimization process known as gradient descent (GD).
To minimize a cost/loss function, this technique is frequently used in machine learning (ML) and deep learning (DL) (e.g. in a linear regression).
The gradient descent algorithm may not always work. There are two distinct prerequisites. A function needs to:
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differentiable
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convex
Not all functions meet these requirements, but if a function is differentiable, it has a derivative for every point in its domain.
Some examples of Differentiable equations:
f(x) = x2 = d(x)/dx = 2x
f(x) = 3sin(x) = d(x)/dx = 3cos(x)
f(x) = x3-5x = d(x)/dx = 3x2-5
Figure E.1. f(x) = x2 = d(x)/dx = 2x ; graphed equation
Figure E.2. f(x) = 3sin(x) = d(x)/dx = 3cos(x) ; graphed equation
Figure E.3. f(x) = x3-5x = d(x)/dx = 3x2-5 ; graphed equation
Some non-differentiable equations are :
f(x) = x/|x|
f(x) = √|x|
f(x) = 1/x
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Figure E.4. Some un-differentiable graphed equations.
A convex function is required.
This implies that the line segment joining the points of a univariate function lies on or above the curve rather than across it.
If it does, it has a local minimum rather than a global minimum.
Two functions with model section lines are shown below.
Figure E.5. Depiction of some Convex and non convex functions.
Lets get into some maths :
Calculating the second derivative and determining whether or not its value is always greater than zero is another approach to quantitatively determine whether a univariate function is convex.
Figure E.6. Second order derivatives.
for x<0: function is convex
for 0<x<1: function is concave (the 2nd derivative < 0)
for x>1: function is convex again
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Codeblock E.1. Standard Gradient Descent demonstration.
Download. Exam results csv file.
Codeblock E.2. Standard Gradient Descent demonstration with Exam result csv.
---- Summary ----
As of now you know all basics of Protein Structures.
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Linear regression
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Y=mx +b.
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intercept.
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Slope.
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etc..
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