Description Usage Arguments Details Value References

Subgradient-based quasi-Newton method for non-differentiable optimization.

1 2 | ```
optimization_L1(w,X,y,nHidden, verbose= FALSE,lambda,lambda2, optimtol, prgmtol,
maxIter, decsuff)
``` |

`w` |
(numeric, |

`X` |
(numeric, |

`y` |
(numeric, |

`nHidden` |
(positive integer, |

`verbose` |
logical, if TRUE prints detail history. |

`lambda` |
numeric, lagrange multiplier for L1 norm penalty on parameters. |

`lambda2` |
numeric, lagrange multiplier for L2 norm penalty on parameters. |

`optimtol` |
numeric, a tiny number useful for checking convergenge of subgradients. |

`prgmtol` |
numeric, a tiny number useful for checking convergenge of parameters of NN. |

`maxIter` |
positive integer, maximum number of epochs(iterations) to train, default 100. |

`decsuff` |
numeric, a tiny number useful for checking change of loss function. |

It is based on choosing a sub-gradient with minimum norm as a steepest descent direction and taking a step resembling Newton iteration in this direction with a Hessian approximation (Nocedal, 1980). An active-set method is adopted to set some parameters to exactly zero (Krishnan et al., 2007). At each iteration, the non-zero parameters are divided into two sets: the working set containing the non-zero variables, and the active set containing the sufficiently zero-values variables. Then a Newton step is taken along the working set. A subgradient-based quasi-Newton method ensures that the step size taken in the active variables is such that they do not violate the sufficiently zero-value variables constraint. A projected steepest descent is taken to set some parameters to exactly zero.

A vector of weights and biases.

Nocedal, J. 1980. Updating quasi-newton matrices with limited storage. *Mathematics of Computation*, 35(35), 773-782.

Krishnan, D., Lin, P., and Yip, A., M. 2007. A primal-dual active-set method for non-negativity constrained total variation deblurring problems. *IEEE Transactions on Image Processing*, 16(11), 2766-2777.

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