Ml notes matrix fundamental, gradient descent

matrix fundamental

basic concepts:



Wrong questions ：



matrix calculation:






Gradient Descent:






We can speed up gradient descent by having each of our input values in roughly the same range. This is because θ will descend quickly on small ranges and slowly on large ranges, and so will oscillate inefficiently down to the optimum when the variables are very uneven.

The way to prevent this is to modify the ranges of our input variables so that they are all roughly the same. Ideally:
−1 ≤ x(i)​ ≤ 1
or
−0.5 ≤ x(i)​ ≤ 0.5

Two techniques to help with this are feature scaling and mean normalization. Feature scaling involves dividing the input values by the range (i.e. the maximum value minus the minimum value) of the input variable, resulting in a new range of just 1. Mean normalization involves subtracting the average value for an input variable from the values for that input variable resulting in a new average value for the input variable of just zero. To implement both of these techniques, adjust your input values as shown in this formula:

Where μ_iμi​ is the average of all the values for feature (i) and s_isi​ is the range of values (max - min), or s_isi​ is the standard deviation.