Mean Squared Error Algorithm

Note

Mean Squared Error Algorithm is not supported on GPU.

Details

Given \(x = (x_{i1}, \ldots, x_{ip}) \in R^p\), a set of feature vectors \(i \in \{1, \ldots, n\}\), and a set of respective responses \(y_i\), the mean squared error (MSE) objective function \(F(\theta; x, y)\) is a function that has the format:

\[F(\theta; x, y) = \sum _{i=1}^{n} F_i(\theta; x, y) = \frac {1}{2n} \sum _{i=1}^{n} (y_i - h(\theta, x_i))^2\]

\[M(\theta) = 0\]

\[\mathrm{prox}_\gamma^M (\theta_j) = \theta_j, j = 1, \ldots, p\]

In oneDAL implementation of the MSE, the \(h(\theta, y_i)\) is represented as:

\[h(\theta, y_i) = \theta_0 + \sum _{j=1}^{p} \theta_j x_{ij}\]

For a given set of the indices \(I = \{i_1, i_2, \ldots, i_m\}\), \(1 \leq i_r < n\), \(l \in \{1, \ldots, m\}\), \(|I| = m\), the value and the gradient of the sum of functions in the argument \(x\) respectively have the format:

\[F_I(\theta; x, y) = \frac {1}{2m} \sum_{i_k \in I} (y_{i_k} - h(\theta, x_{i_k}))^2\]

\[\nabla F_I(\theta; x, y) = \left\{ \frac{\partial F_I}{\partial \theta_0}, \ldots, \frac{\partial F_I}{\partial \theta_p} \right\}\]

where

\[\frac{\partial F_I}{\partial \theta_0} = \frac{1}{m} \sum_{i_k \in I} (y_{i_k} - h(\theta, x_{i_k}))\]

\[\frac{\partial F_I}{\partial \theta_j} = \frac{1}{m} \sum_{i_k \in I} (y_{i_k} - h(\theta, x_{i_k})) x_{i_k j}, j = 1, \ldots, p\]

\(lipschitzConstant = \underset{i = 1, \ldots, n} \max \| x_i \|_2\)

Computation

Algorithm Input

The mean squared error algorithm accepts the input described below. Pass the Input ID as a parameter to the methods that provide input for your algorithm. For more details, see Algorithms.

Algorithm Input for MSE Computation

Input ID	Input
argument	A numeric table of size \((p + 1) \times 1\) with the input argument \(\theta\) of the objective function.
data	A numeric table of size \(n \times p\) with the data \(x_{ij}\).
dependentVariables	A numeric table of size \(n \times 1\) with dependent variables \(y_i\).

Optional Algorithm Input

The mean squared error algorithm accepts the optional input described below. Pass the Optional Input ID as a parameter to the methods that provide input for your algorithm. For more details, see Algorithms.

Optional Algorithm Input for MSE Computation

Input ID	Input
weights	Optional input. Pointer to the \(1 \times n\) numeric table with weights of samples. The input can be an object of any class derived from NumericTable except for PackedTriangularMatrix and PackedSymmetricMatrix. By default, all weights are equal to \(1\).
gramMatrix	Optional input. Pointer to the \(p \times p\) numeric table with pre-computed Gram matrix. The input can be an object of any class derived from NumericTable except for PackedTriangularMatrix and PackedSymmetricMatrix. By default, the table is set to empty numeric table.

Algorithm Parameters

The mean squared error algorithm has the following parameters. Some of them are required only for specific values of the computation method parameter method:

Algorithm Parameters for MSE Computation

Parameter	Default value	Description
penaltyL1	\(0\)	The numeric table of size \(1 \times \mathrm{nDependentVariables}\) with L1 regularized coefficients.
penaltyL2	\(0\)	The numeric table of size \(1 \times \mathrm{nDependentVariables}\) with L2 regularized coefficients.
interceptFlag	true	Flag to indicate whether or not to compute the intercept.
algorithmFPType	float	The floating-point type that the algorithm uses for intermediate computations. Can be float or double.
method	defaultDense	Performance-oriented computation method.
numberOfTerms	Not applicable	The number of terms in the objective function.
batchIndices	Not applicable	The numeric table of size \(1 \times m\), where \(m\) is the batch size, with a batch of indices to be used to compute the function results. If no indices are provided, the implementation uses all the terms in the computation. Note This parameter can be an object of any class derived from NumericTable except for PackedTriangularMatrix and PackedSymmetricMatrix.
resultsToCompute	gradient	The 64-bit integer flag that specifies which characteristics of the objective function to compute. Provide one of the following values to request a single characteristic or use bitwise OR to request a combination of the characteristics: value Value of the objective function nonSmoothTermValue Value of non-smooth term of the objective function gradient Gradient of the smooth term of the objective function hessian Hessian of smooth term of the objective function proximalProjection Projection of proximal operator for non-smooth term of the objective function lipschitzConstant Lipschitz constant of the smooth term of the objective function

Algorithm Output

For the output of the mean squared error algorithm, see Output for objective functions.

Examples

C++ (CPU)

• mse_dense_batch.cpp