In many machine learning algorithm, the goal is to find a function or parameters that allows us to approximate or modelize unknown observable data. Those data could come from device measurement, web crawling, empirical observations etc. Generally speaking we have  samples  of the observation vector . For example such a vector  could be the coordinates  of an object in space.  We want to approximate these data with some parameters  through a known function  so that



One common way to do that is trying to minimize an error function, for example the root mean square (RMS)



# Linear Regression Example

Without loss of generality, we'll focus on the simple case of linear regression. For example, imagine you have a dataset  of  points and you want to fit a line

The regression function have then the form of



but more generally for a linear regression this function will have the form of



where  is the input vector, the vector on which we want to regress, the observable variables, and  are the regression parameters. To be more generic, we can also define vector  to be  with  so that the regression function become now



We are looking for the vector  that minimizes the error function . The error function is continuous and differentiable so that at the minimal error, we have



As there is no analytic solution of this optimization problem in the general case, one technic is to iteratively update the weights in the direction of the gradient



where  is the learning rate. The learning rate has to be small enough not to pass trough the minimum or oscillate around it, but large enough not to converge rapidely. In the special case of linear regression with a polynom of order 2, this takes the form



Practically, we don't look for  that satisfies , but we iterate over  as long as the vector  significantly changes, or we can also fix the number of iterations. I propose here an implementation of such an algorithm in Python

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