Julia implementation of predictors and loss functions for empirical risk minimization
19 Stars
Updated Last
2 Years Ago
Started In
April 2015


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

Test Status: Build Status EmpiricalRisks EmpiricalRisks

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.


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


    • 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)


    • 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.


Here is the Detailed Documentation.