optimization
:: DeveloperApi :: Class used to compute the gradient for a loss function, given a single data point.
Class used to solve an optimization problem using Gradient Descent.
:: DeveloperApi :: Compute gradient and loss for a Hinge loss function, as used in SVM binary classification.
:: DeveloperApi :: Compute gradient and loss for a Hinge loss function, as used in SVM binary classification. See also the documentation for the precise formulation.
This assumes that the labels are {0,1}
:: DeveloperApi :: Updater for L1 regularized problems.
:: DeveloperApi :: Updater for L1 regularized problems. R(w) = ||w||_1 Uses a step-size decreasing with the square root of the number of iterations.
Instead of subgradient of the regularizer, the proximal operator for the L1 regularization is applied after the gradient step. This is known to result in better sparsity of the intermediate solution.
The corresponding proximal operator for the L1 norm is the soft-thresholding function. That is, each weight component is shrunk towards 0 by shrinkageVal.
If w is greater than shrinkageVal, set weight component to w-shrinkageVal. If w is less than -shrinkageVal, set weight component to w+shrinkageVal. If w is (-shrinkageVal, shrinkageVal), set weight component to 0.
Equivalently, set weight component to signum(w) * max(0.0, abs(w) - shrinkageVal)
:: DeveloperApi :: Class used to solve an optimization problem using Limited-memory BFGS.
:: DeveloperApi :: Class used to solve an optimization problem using Limited-memory BFGS. Reference: Wikipedia on Limited-memory BFGS
:: DeveloperApi :: Compute gradient and loss for a Least-squared loss function, as used in linear regression.
:: DeveloperApi :: Compute gradient and loss for a Least-squared loss function, as used in linear regression. This is correct for the averaged least squares loss function (mean squared error) L = 1/2n ||A weights-y||^2 See also the documentation for the precise formulation.
:: DeveloperApi :: Compute gradient and loss for a multinomial logistic loss function, as used in multi-class classification (it is also used in binary logistic regression).
:: DeveloperApi :: Compute gradient and loss for a multinomial logistic loss function, as used in multi-class classification (it is also used in binary logistic regression).
In The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd Edition
by Trevor Hastie, Robert Tibshirani, and Jerome Friedman, which can be downloaded from
http://statweb.stanford.edu/~tibs/ElemStatLearn/ , Eq. (4.17) on page 119 gives the formula of
multinomial logistic regression model. A simple calculation shows that
$$ P(y=0|x, w) = 1 / (1 + \sum_i^{K-1} \exp(x w_i))\\ P(y=1|x, w) = exp(x w_1) / (1 + \sum_i^{K-1} \exp(x w_i))\\ ...\\ P(y=K-1|x, w) = exp(x w_{K-1}) / (1 + \sum_i^{K-1} \exp(x w_i))\\ $$
for K classes multiclass classification problem.
The model weights \(w = (w_1, w_2, ..., w_{K-1})^T\) becomes a matrix which has dimension of (K-1) * (N+1) if the intercepts are added. If the intercepts are not added, the dimension will be (K-1) * N.
As a result, the loss of objective function for a single instance of data can be written as
$$ \begin{align} l(w, x) &= -log P(y|x, w) = -\alpha(y) log P(y=0|x, w) - (1-\alpha(y)) log P(y|x, w) \\ &= log(1 + \sum_i^{K-1}\exp(x w_i)) - (1-\alpha(y)) x w_{y-1} \\ &= log(1 + \sum_i^{K-1}\exp(margins_i)) - (1-\alpha(y)) margins_{y-1} \end{align} $$
where $\alpha(i) = 1$ if \(i \ne 0\), and $\alpha(i) = 0$ if \(i == 0\), \(margins_i = x w_i\).
For optimization, we have to calculate the first derivative of the loss function, and a simple calculation shows that
$$ \begin{align} \frac{\partial l(w, x)}{\partial w_{ij}} &= (\exp(x w_i) / (1 + \sum_k^{K-1} \exp(x w_k)) - (1-\alpha(y)\delta_{y, i+1})) * x_j \\ &= multiplier_i * x_j \end{align} $$
where $\delta_{i, j} = 1$ if \(i == j\), $\delta_{i, j} = 0$ if \(i != j\), and multiplier = $\exp(margins_i) / (1 + \sum_k^{K-1} \exp(margins_i)) - (1-\alpha(y)\delta_{y, i+1})$
If any of margins is larger than 709.78, the numerical computation of multiplier and loss
function will be suffered from arithmetic overflow. This issue occurs when there are outliers
in data which are far away from hyperplane, and this will cause the failing of training once
infinity / infinity is introduced. Note that this is only a concern when max(margins)
> 0.
Fortunately, when max(margins) = maxMargin > 0, the loss function and the multiplier
can be easily rewritten into the following equivalent numerically stable formula.
$$ \begin{align} l(w, x) &= log(1 + \sum_i^{K-1}\exp(margins_i)) - (1-\alpha(y)) margins_{y-1} \\ &= log(\exp(-maxMargin) + \sum_i^{K-1}\exp(margins_i - maxMargin)) + maxMargin - (1-\alpha(y)) margins_{y-1} \\ &= log(1 + sum) + maxMargin - (1-\alpha(y)) margins_{y-1} \end{align} $$
where sum = $\exp(-maxMargin) + \sum_i^{K-1}\exp(margins_i - maxMargin) - 1$.
Note that each term, $(margins_i - maxMargin)$ in $\exp$ is smaller than zero; as a result, overflow will not happen with this formula.
For multiplier, similar trick can be applied as the following,
$$ \begin{align} multiplier &= \exp(margins_i) / (1 + \sum_k^{K-1} \exp(margins_i)) - (1-\alpha(y)\delta_{y, i+1}) \\ &= \exp(margins_i - maxMargin) / (1 + sum) - (1-\alpha(y)\delta_{y, i+1}) \end{align} $$
where each term in $\exp$ is also smaller than zero, so overflow is not a concern.
For the detailed mathematical derivation, see the reference at http://www.slideshare.net/dbtsai/2014-0620-mlor-36132297
:: DeveloperApi :: Trait for optimization problem solvers.
:: DeveloperApi :: Trait for optimization problem solvers.
:: DeveloperApi :: A simple updater for gradient descent *without* any regularization.
:: DeveloperApi :: A simple updater for gradient descent *without* any regularization. Uses a step-size decreasing with the square root of the number of iterations.
:: DeveloperApi :: Updater for L2 regularized problems.
:: DeveloperApi :: Updater for L2 regularized problems. R(w) = 1/2 ||w||^2 Uses a step-size decreasing with the square root of the number of iterations.
:: DeveloperApi :: Class used to perform steps (weight update) using Gradient Descent methods.
:: DeveloperApi :: Class used to perform steps (weight update) using Gradient Descent methods.
For general minimization problems, or for regularized problems of the form min L(w) + regParam * R(w), the compute function performs the actual update step, when given some (e.g. stochastic) gradient direction for the loss L(w), and a desired step-size (learning rate).
The updater is responsible to also perform the update coming from the regularization term R(w) (if any regularization is used).
:: DeveloperApi :: Top-level method to run gradient descent.
:: DeveloperApi :: Top-level method to run gradient descent.
:: DeveloperApi :: Top-level method to run L-BFGS.
:: DeveloperApi :: Top-level method to run L-BFGS.
:: DeveloperApi :: Class used to compute the gradient for a loss function, given a single data point.