EmpiricalRisks

This Julia package provides a collection of predictors and loss functions, mainly to support the implementation of (regularized) empirical risk minimization methods.

Test Status:

Currently, the following higher-level packages are depending on EmpiricalRisks:

• Regression: solving moderate-size problem using conventional optimization techniques.
• SGDOptim: solving large-scale problem using stochastic gradient descent or its variants.

Overview

This package provides basic components for implementing regularized empirical risk minimization:

• Prediction models u = f(x; θ)

  • linear prediction
  • affine prediction
  • multivariate linear prediction
  • multivariate affine prediction

• Loss functions loss(u, y)

  • squared loss
  • absolute loss
  • quantile loss
  • huber loss
  • hinge loss
  • squared hinge loss
  • smoothed hinge loss
  • logistic loss
  • sum squared loss (for multivariate prediction)
  • multinomial logistic loss

  Notes:

  • For each loss function, we provide methods to compute the loss value, the derivative/gradient, or both (at the same time).
  • For each (consistent) combination of loss function and prediction model (which together are referred to as a risk model), we provide methods to compute the total risk and the gradient w.r.t. the parameter.

• Regularizers

  • squared L2
  • L1
  • elastic net (L1 + squared L2)

  Notes:

  • For each regularizer, we provide methods to evaluate the regularization value, the gradient, and the proximal operator.

Remarks: All functions in this package are carefully optimized and tested.

Documentation

Here is the Detailed Documentation.