Loss functions and gradients

This module contains the loss functions and the gradients

Loss function

MSE

 MSE (x:numpy.ndarray, y:numpy.ndarray, fun:<built-infunctioncallable>,
      params=None)

Given the data \(x\) and \(y\), this function computes the mean square error between \(y\) and \(y'=f(x)\)

	Type	Default	Details
x	ndarray		x data of N elements
y	ndarray		y data of N elements
fun	callable		function \(y=f(x)\)
params	NoneType	None	Parameters of the function in the form of a dictionary
Returns	float

The mean square error is defined as \[ MSE(y,y')=\frac{1}{N}\sum_{i=1}^{N}(y'_i-y_i)^2.\]

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grad_MSE_lr

 grad_MSE_lr (x:numpy.ndarray, y:numpy.ndarray, params:dict)

Computes the gradient of the mean square error loss function with respect to \(a\) and \(b\) and returns np.array([\(\partial_a\) MSE,\(\partial_b\) MSE])

	Type	Details
x	ndarray	x data of N elements
y	ndarray	y data of N elements
params	dict	Parameters of the function
Returns	ndarray	gradients

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grad_MSE_pr

 grad_MSE_pr (x:numpy.ndarray, y:numpy.ndarray, params:dict)

Computes the gradient of the mean square error loss function with respect to \(a\) and \(b\) and returns np.array([\(\partial_a\) MSE,\(\partial_b\) MSE])

	Type	Details
x	ndarray	x data of N elements
y	ndarray	y data of N elements
params	dict	parameters of the function
Returns	ndarray	gradients

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BCE

 BCE (x:numpy.ndarray, y:numpy.ndarray, fun:<built-infunctioncallable>,
      params:dict)

Given the data \(x\) and \(y\), this function computes the mean binary cross entropy \(y\) and \(y'=f(x)\)

	Type	Details
x	ndarray	x data of N elements
y	ndarray	y data of N elements
fun	callable	function \(y=f(x)\)
params	dict	Parameters of the function
Returns	float

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grad_BCE

 grad_BCE (x:numpy.ndarray, y:numpy.ndarray, params:dict)

Computes the gradient of the binary cross entropy loss function with respect to \(a\) and \(b\) and returns np.array([\(\partial_a\) BCE,\(\partial_b\) BCE])

	Type	Details
x	ndarray	x data of N elements
y	ndarray	y data of N elements
params	dict	Parameters of the function
Returns	ndarray	gradients

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L_per

 L_per (x:numpy.ndarray, y:numpy.ndarray, fun:<built-infunctioncallable>,
        params:dict)

Given the data \(x\) and \(y\), this function computes the Loss of the perceptron algorithm

	Type	Details
x	ndarray	x data of N elements
y	ndarray	y data of N elements
fun	callable	function \(y=f(x)\)
params	dict	Parameters of the function
Returns	float

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grad_per

 grad_per (x:numpy.ndarray, y:numpy.ndarray, params:dict)

Computes the gradient of the perceptron loss function and returns np.array(grad_w)

	Type	Details
x	ndarray	x data of N elements
y	ndarray	y data of N elements
params	dict	Parameters of the function
Returns	ndarray	gradients